/sci/ - Science & Math

Unanswered questions from the previous thread:
Math questions:
>>11424623
>>11426913 [lad in /mg/ solved it]
>>11428072
>>11430348
>>11432228
>>11432463 [Very convoluted.]
>>11435174 [lad in /mg/ solved it]
>>11437535

Physics questions:
>>11427896

Stupid questions:
>>11423676
>>11424021
>>11426013
>>11426028 [5 questions]
>>11428698
>>11428989
>>11429265 [Dude made his own thread later and solved it.]
>>11429376 specified by >>11429408
>>11430113
>>11432537
>>11433635
>>11434338
>>11434532
>>11435410 [I have literally no idea what he's talking about.]
>>11435982
>>11437303
>>11437543

>>11437575
Where do I even start with this bros? Just give me a hint

The image is in Portuguese, sorry for that.

I'm at a loss.
How did "X v ~X ^ Y" become "(X v X ^ Y) v ~X ^ Y"? How is X = (X v X ^ Y)?

X v ~X ^ Y =
(X v X ^ Y) v ~X ^ Y =
X v X ^ Y v ~X ^ Y =
X v Y ^ (X v ~X) =
X v Y ^ 1 =
X v Y.

>>11437861
Consider this:
If x is true, then x is true, obviously. Same vice versa.

If x OR x AND y is true, then x must be true, or x AND y must be true, thus y must be true, which is superfluous.
Other way around, if x OR x AND y is false, then x must be false. Same way of reasoning as above.

It literally more than trivial...

Is there way to genetically engineer bacterium or something to react to the coronavirus so that I can make a test for it I can sell? I love money so much and am desperate to make it.

>>11437861
can u tell all portugeuse people to stop taking spanish and then randomly swapping l's and r's around please thank you

>Vasodilatation: The blood vessels under your skin get wider. This increases blood flow to your skin where it is cooler — away from your warm inner body. This lets your body release heat through heat radiation.

If the outside temperature was hotter than your body temperature, would vasodilation make you warmer? Heat travels from hot to cold, all vasodilation does is increase the exchange rate. In 50C weather, would the air dump more heat into your blood than your blood releases to the air?

how do i replicate the clearheadedness i get when im sleep deprived? when im stoned, havent slept in 30 hours, or are otherwise "disabled", its like my inhibitions go way down and i can think easily and freely about math and science,solving problems. but other issues of the situation get in the way and hinder me so its not sustainable. how do i think freely like i do when sleep deprived or drugged normally? i feel like im stressed and caged during normal life

>>11438032
>Anonymous 03/03/20(Tue)18:20:49 No.1143
healthy amount of stress+eutsress. i.e. always stay on your feet, but don't back yourself into a corner or get complacent
Also, if "well rested" means 9hrs sleep, aim for 7-8. On the short side of a full night's sleep, without being deprived.

>>11437535
Suppose [math]u: M\rightarrow N[/math] is a smooth map and [math]p:x\mapsto x(t)[/math] a path on [math]M[/math], then we push [math]p[/math] into [math]N[/math] by [math]u^*p(t) = u(x(t))[/math]. Now just apply the chain rule [math]\dot{x} = \dot{u} \cdot dx(u)[/math], which taking away [math]x[/math] and in coordinates reads [math]\frac{d}{dt} = \partial_t + \dot{u}_i \cdot \nabla[/math].
Now in general [math]\nabla u[/math] is tangent to the [math]level~sets[/math] of [math]u[/math], on which [math]u[/math] is constant, so as [math]x[/math] varies [math](\nabla u)(x) \cdot u(x) = 0[/math]. This gives us [math]\nabla(\rho u^2) = u\cdot \nabla(\rho u)[/math] by Leibniz.

>>11438217
> [math]p: x \rightarrow x(t) [/math]
Are you drunk?

>>11438234
Look closer hun

>>11438234
Meant [math]t[/math] ofc

>>11438248
You mean [math]p : t \rightarrow p(t)[/math]

>>11438251
No, I meant [math]p: t\mapsto x(t)[/math]

I'm 4 years into my Engineering graduation, only 1 year left, and i'm thinking of picking up Software development, to have something else to do if i can't find a job as an engineer
Do you think it would be worth it for me to start learning it now? where would i even start? need some opinions

if race doesn't exist, then how come I'm racist? surely racism isn't real if race doesn't exist

The real redpill is to pretend you're kind of retarded around your peers. They will not feel intellectually threatened by you and you will not feel the stress in need to maintain your smart status.
Plus when you solve a hard problem, people around you who couldn't do it will be humiliated that someone as stupid as you did it but they couldn't.

>>11439156
unironically this

Someone ask something, I'm bored.
>>11438032
You stop with the not sleeping for 30 hours and getting stoned thing until your mind returns to regular working conditions.
>>11439156
The realest redpill is having healthy human interactions with friends and family.

What's the formula to calculate how many seconds it takes for a number of electrons to exit a wire?

Is this correct? Couldn't the constants from the integration just be inverse values, like -5 and 5?

>>11439580
No, it's not. Not unless you specify bounds of integration. [math] \int \fac{\text{d}x}{x}=\int \text{d}t\implies \ln|x|=t+C [/math]

>>11439592
why doesn't the x integral get a constant

>>11439592
No. Not unless you specify limits of integration. [math] \int \frac{\text{d}x}{x}=\int \text{d}t\implies \ln|x|=t+C [/math
>>11439630
I combined the constants into one. An arbitrary constant minus an arbitrary constant is just another arbitrary constant.
>>11439560
[math] t=\frac{Nq}{i} [/math]. t is time, N is number of charges, i is current, q is the elementary charge. This probably makes no sense.
>>11438003
>through heat radiation
That's not radiation. It's just conduction thru the skin and then convection with the air. But your logic makes sense. Heat conducts in a material in the direction opposed to the temperature gradient, and a hot fluid will transfer heat to a relatively cool surface it encounters.
>>11435410
But that isn't true. "Duty cycle" depends entirely on the device you are powering. The entire signal can potentially be used to do work, including voltage negative relative to earth.
>>11423676
Yes.

>>11439580
No. Not unless you specify limits of integration. [math] \int \frac{\text{d}x}{x}=\int \text{d}t\implies \ln|x|=t+C [/math]
>>11439630
I combined the constants into one. An arbitrary constant minus an arbitrary constant is just another arbitrary constant.
>>11439560
[math] t=\frac{Nq}{i} [/math]. t is time, N is number of charges, i is current, q is the elementary charge. This probably makes no sense.
>>11438003
>through heat radiation
That's not radiation. It's just conduction thru the skin and then convection with the air. But your logic makes sense. Heat conducts in a material in the direction opposed to the temperature gradient, and a hot fluid will transfer heat to a relatively cool surface it encounters.
>>11423676
Yes.

fucking damn, third time is the charm

>>11439580
The first step is kinda sorta correct.
I can't make sense of either the third one or the second one.
>-5 and 5
What do you mean?

>>11439526
>Someone ask something
When someone refers to a scalar particle, does this simply mean the field has no spinor part? Also what about the terminology "scalar guage theory" (hopefully remembering the wording correctly)? I see "theory" used to refer to the lagrangian, but I don't see the connection here.

>>11439650
third step just looks like him plugging in given init. conditions.

So, I started an apprenticeship for an electrician, and in the theoretical part of the course we are learning about boolean circuits, and we got this shitty introductory book on it, and in it the boolean term "¬P" (notP), is presented as pic related.

I couldn't find any resource online on why it is presented that way and how to make sense of it whatsoever. Any ideas, or is the book just that shitty?

>>11439758
After googling to check, the answer is
>kinda, it mostly means the field transforms like a scalar

>Also what about the terminology "scalar guage theory" (hopefully remembering the wording correctly)? I see "theory" used to refer to the lagrangian, but I don't see the connection here.
It's like abelian gauge theory or anabelian gauge theory, I think, except abelian refers to the gauge group and scalar refers to the field.

>>11439779
Actually I might be wrong, scalar particle might or not be specific to the fields being literal scalar rather than just transforming like scalars.

>>11437575
How do I transition from mathematics to physics through self-study alone?

Hi can someone explain exercise 4 in Fuchs-Fomenko p. 128?
The original Whitehead paper seems to contradict the exercise (contradiction highlighted in red).
Anyway, how would you prove this result?
Also what did Whitehead mean by the green highlight and how does this relate to the result?

>>11439895
I think the E^m x E^n = (-1)^mn E^n x E^m refers to the fact that the natural homeomorphism (x,y) -> (y,x) considered as an endomorphism of E^(n+m) has determinant (-1)^(mn)

>>11439895
>Hi can someone explain exercise 4 in Fuchs-Fomenko p. 128?
Dunno.
>Anyway, how would you prove this result?
Other anon's proof strat and degrees.
>Also what did Whitehead mean by the green highlight and how does this relate to the result?
That looks like singular homology.
>>11439907
>the virgin rigorously making sure your definitions make sense
>the chad taking the determinant of a map from the wedge of two manifolds of potentially different dimension
Proof strat seems essentially correct, tho. Muh degrees and stuff.

>>11439644
>t=Nqi. t is time, N is number of charges, i is current, q is the elementary charge. This probably makes no sense.
thanks

Suppose I want to minimize the squared sum of several linear fractional functions. That is, a least squares problem with a linear fractional function modeling data.
If I use the transformation of pic related, will the resulting problem be convex?

Is there more to this question or am I missing something? Wouldn't the answer just be because it is surjective you have for every element in B an element in A so the equivalence classes of A is just the elements of B?

So I'm trying to solve this Statics problem on forces applied to a cable (right side of the picture) but then the book applies the equation on the left and I have no idea what it means. I'm guessing it's fc+something else=fd, but that makes no sense to me.

>>11440244
There is a natural bijection given by f between the equivalence classes of A and the elements of B, yes.

>>11440363
what exactly are you solving for? all statics problems are the same

Do all real vector spaces except R2 have an eigenvector? im thinking this should be true if all linear operators refer to a rotation, mirror, and/or magnification, but im not sure if that notion is true

how do i get an intuition for vector spaces over the complexes? i can understand higher dimensional reals just fine, but i dont know how to parse 2 separate planes where each plane is a single number, whole within itself

Suppose I want to minimize a squared sum of fractional linear functions, that is, a least squares problem in which the data is modeled by a linear fractional function.
If I use the transformation of pic related, will the transformed problem have the same minimum of the original problem?

[math]
\text{Let the }\mathbb{R}^3\text{ vector }\vec{C} = \begin{bmatrix}C_r\\C_g\\C_b\end{bmatrix}\text{ be a color of }light\text{ in Cartesian form,}\\
\text{and the vector }\vec{P} = \begin{bmatrix}P_r\\P_y\\P_b\end{bmatrix}\text{ be a color of }pigment\text{ in Cartesian form.}\\
\text{Then, the magnitude of }\vec{C}\text{, }|\vec{C}| = \sqrt{C_r^2 + C_g^2 + C_b^2}\text{,}\\
\text{might be considered }\vec{C}\text{'s luminous intensity.}\\
\text{Likewise, }|\vec{P}| = \sqrt{P_r^2 + P_y^2 + P_b^2}\text{ represents pigment intensity,}\\
\text{though I don't even know what exactly }``\text{pigment intensity"}\text{ means.}\\
\text{Assume the constraint }\begin{bmatrix}0&&0\\0&&0\\0&&0\end{bmatrix} \le \begin{bmatrix}\vec{C}&&\vec{P}\end{bmatrix} \le \begin{bmatrix}1&&1\\1&&1\\1&&1\end{bmatrix}\\
\text{(where matrix comparison is used euphemistically as shorthand}\\
\text{for comparing each pair of corresponding elements).}\\
\text{Then, }\vec{C} = \begin{bmatrix}0\\0\\0\end{bmatrix}\text{ is black, }\vec{C} = \begin{bmatrix}1\\1\\1\end{bmatrix}\text{ is white, and the upper bound of luminous intensity is }\sqrt{3}\text{.}\\
\text{Consider the vector rejection }\vec{C}_{\bot \vec{W}}\text{ of }\vec{C}\text{ against }\vec{W}\text{, where }\vec{W} = \begin{bmatrix}1\\1\\1\end{bmatrix}\text{.}\\
\text{Then, the angle }\theta_{\vec{C}\bot\vec{W}}\text{ of }\vec{C}\text{ around the axis }\vec{W}\text{ should equal the hue of }\vec{C}\text{, }h_{\vec{C}}\text{.}\\
\text{The same cannot be said for }h_{\vec{P}}\text{:}\\
\text{let }\vec{B} = \vec{W}\text{, but interpreted as a pigment color (B for brown, or something);}\\
\text{then, }h_{\vec{P}} \ne \theta_{\vec{P}\bot\vec{B}}\text{.}\\
\underline{\textbf{My question, then, is this:}}\\
\text{How do I find a function }f(\sigma)\text{, and its inverse }f^{-1}(\phi)\text{,}\\
\text{such that if }h_{\vec{C}} = h_{\vec{P}}\text{,}\\
\text{then }f(\theta_{\vec{C}\bot\vec{W}}) = \theta_{\vec{P}\bot\vec{B}}\text{?}
[/math]

>>11437575
Hey /sci/, I'm looking for help from someone who is smarter than myself since I'm not good with statistics.
On World of Warcraft Classic there is a simple gambling game called deathrolling. Two players declare some bet amount, like 5 gold, and one player rolls first. Their roll is a randomly generated number between 1 and 100. The second player then rolls a number between 1 and whatever the 1st player rolled. This continues back and forth until one player rolls a 1. Whoever rolls a 1 first loses.

Quick example:
1st rolls 1 - 100 and rolls 53
2nd rolls 1 - 53 and rolls 32
1st rolls 1 -32 and rolls 6
2nd rolls 1 - 6 and rolls 2
1st rolls 1 - 2 and rolls 2
2nd rolls 1 - 2 and rolls 1. 2nd player lost and owes 1st player 5 gold.

My question is which player has the greater advantage in winning? The 1st player starts with the highest chances of not rolling a 1, however they start with a chance of immediately losing before the 2nd player even rolls.
I wrote something in python really quickly to see what happened and it looks like playing first has a slight advantage, but that could be due to my poor coding or any myriad of issues with the online compiler or random number generator available through python.

Any thoughts or advice on actually calculating what advantage player one has vs player two?

How do I convert between colors of light and colors of pigment in both cartesian and polar notation?

[math]
\text{That is,}\\
\text{let }\vec{L} = \begin{bmatrix}L_r \in [0,1]\\L_g \in [0,1]\\L_b \in [0,1]\end{bmatrix}
\text{ be a color in a luminous interpretation of }\mathbb{R}^3\text{,}\\
\text{and }\vec{P} = \begin{bmatrix}P_r \in [0, 1]\\P_y \in [0, 1]\\P_b \in [0, 1]\end{bmatrix}
\text{ be a color in a pigment interpretation of }\mathbb{R}^3\text{.}\\
\text{Consider the function }C_L(\vec{v})\text{ defined, abstractly, as how the }luminous\text{ color }\vec{v}\text{ appears to the eye,}\\
\text{and the function }C_P(\vec{u})\text{ defined as how the }pigment\text{ color }\vec{u}\text{ appears to the eye.}\\
\text{We don't know the mathematical categories or representations of the }C\text{ functions,}\\
\text{only their meanings in this abstract sense.}\\
\text{We can at least draw, from analytical consequence, the inference that }range(C_P) \in range(C_L)\text{.}\\
\text{Suppose, then, that }C_L(\vec{L}) = C_P(\vec{P})\text{.}\\
\underline{\textbf{Then my question is this:}}\\
\text{What function }F(\vec{v}) = \vec{u}\text{, and its inverse }F^{-1}(\vec{u}) = \vec{v}\text{, relate }\vec{L}\text{ and }\vec{P}\text{?}
[/math]

>>11440244
You need to, at the absolute least, construct the obvious bijection.
>>11440684
For [math]\mathbb{R} ^n[/math] with [math]n[/math] odd, the characteristic polynomial has odd degree, and hence at least one real root (which is an eigenvalue).
For even degree, consider [math]\mathbb{C}^n[/math] as a real vector space and the [math]J[/math] operator of multiplication by [math]i[/math].

>>11440788
>treat C^N as R^n/2
Does that actually work out though? Sure C^N has twice the dimensions but the dimensions come in pairs and "move" together so to speak. I just cant see how it should obviously work out

>>11440883
You've left the coordinates, but the coordinates haven't left you.
Literally just restrict the multiplication by a scalar map to the reals and check the vector space axioms. It takes all of three seconds.

>>11440927
oh if you restrict the multiplication to reals.. its just taking a vector and vecting it. and they add vectily. okay. that.. makes sense, but what about ith complex multiples? or complex operators?

how did they get from the top expression to the bottom one?
applying associativity i get
[math] \displaystyle (( (( \cdots (a_1 \circ a_2) \circ a_3) \circ \cdots) \circ a_m) \circ (((\cdots (a_{m + 1} \circ a_{m + 2} ) \circ a_{m + 3} ) \circ \cdots) \circ a_n) ) \\= (((((a_1\circ a_2)\circ a_3)\circ\cdots)\circ a_m)\circ((\cdots(a_{m+1}\circ a_{m+2})\circ a_{m+3})\circ \cdots))\circ a_n [/math]

im about to go for grad school for maths and i fucking forgot how acceleration works

this is a graph of velocity. is the object speeding up for all nonzero values of v since it is not linear?

>>11439526
>Someone ask something, I'm bored.
Alright. Define [math]\mathcal{S}[/math] as the smooth subalgebra of trace-class operators in [math]\mathcal{B}(\mathscr{H}) \cap \mathcal{A}[/math] where [math]\mathcal{A}[/math] is the algebra of many-body observables, I'm trying to prove that the inclusion [math]\mathcal{S}\hookrightarrow \mathcal{A}[/math] is a quasi-isomorphism in operator [math]K[/math]-theory, meaning that [math]K_0(\mathcal{S}) \cong K_0(\mathcal{A})[/math]. Reason being I need to be able to take the von Neumann trace for an index formula but I also don't want to lose any topological info. I know in the one-body case I can do this if [math]\mathcal{S}=\mathcal{R}[/math] is a smooth subalgebra and [math]\mathcal{A}=C^*_r(G)[/math] is the reduced [math]C^*[/math]-algebra because of the Haagerupp inequality, but [math]C^*_r(G)[/math] is finitely generated for finite [math]G[/math] while the many-body observables [math]\mathcal{A}[/math] is literally the image of the Sobolev [math]H^1[/math] under the second quantization functor. Haagerupp inequality doesn't help here. Anyone knows of any ideas?
>>11440690
>Suppose I want to minimize a squared sum of fractional linear functions
Against what? What are the minimization parameters?
>linear fractional function
What? The Mobius transforms are projective, not linear.
Anyways suppose you want to minimize the sum with respect to the coefficients, i.e. [math]f(z) = \int_A g^2 d\mu(g) |g(z)|^2[/math] for some collection [math]A\subset PSL_2(\mathbb{C})[/math], then by the left-invariance of the Haar measure we have [math]d\mu(g) = -g^2d\mu(g^{-1})[/math] hence [math]f(z) = -|z|^2\int_{A^{-1}}d\mu(g) g^2[/math], then it's just an issue of maximizing the average [math]\mu(\cdot^2)[/math] of the squaring function [math]g\mapsto g^2[/math] as [math]g[/math] moves through [math]A^{-1}[/math].

>>11440402
I'm solving for the tension of each segment, but I'm getting filtered by trigonometry.

>>11441180
>is the object speeding up
Acceleration is the derivative of speed.

>>11438651
Being able to program is literally the most important basic skill you can have.
If you can't program and are bored, learn to program.

I am actually surprised that there exists a single university were Software development isn't an essential part of any engineering curriculum.
As, even in engineering, it is really important.

What causes alpha cells in the pancreas to produce glucagon? My teacher told me that it's primarily because of norepinephrine, but I can't find anything describing that in detail.

Being [math] a,b \in \mathbb{R}_{>0} [/math] and [math] D^* = \{ (x,y) : \frac{x^2}{a^2}+\frac{y^2}{b^2}\leq1\} [/math], consider the following change of variables: [math] g(r,\theta)=(a\cdot r\cdot cos(\theta), b \cdot r\cdot sin(\theta)) [/math]
a) Describe [math] D\subset \mathbb{R}^2 [/math] such that [math] g(D)=D^* [/math] and [math] g:D \rightarrow D^* [/math] is bijective (except in the area 0).
b) Calculate g's Jacobian, and using that change of variables calculate: [math] \phi(a,b):=\iint_{D^*}exp \left( \frac{x^2}{a^2}+\frac{y^2}{b^2}\right) [/math]
I'm a massive brainlet, I'm not even sure where to start. I guess D will be bijective in [-a,0]x[0,b], [-a,0]x[-b,0], [0,a]x[0,b] and [0,a]x[-b,0] or something like that, but that's all. About g jacobian I calculated it and I got a.b.r.

If you got a function that goes from R^2 ---> R, can you say that since R^2 is convex the domain of that function is convex?

>>11441823
If A is a convex set, is then A a convex set?

>>11441118
The first step into the red square is plugging in the definition, that is, introducing parenthesis because multiplication is left to right.
For the second step, we set [math]C = ((( \cdots (a_{m+1} \circ a_{m+2} ) \circ a_{m+3}) \circ \cdots ) \circ a_{n-1} )[/math] to obtain [math]B = C \circ a_n[/math], then [math]A \circ (C \circ a_n ) = (A \circ C) \circ a_n )[/math] and I think you just throw in the induction hypothesis.
Then again, I can't see the entire proof.
>>11441241
>>linear fractional function
>What? The Mobius transforms are projective, not linear.
https://en.wikipedia.org/wiki/Linear_fractional_transformation

Thank you for the question, I'll throw away some time trying to figure it out.
>>11441713
Compose the changes of variables [math](x, y) \rightarrow (r, \theta) \rightarrow (ar \cos \theta , br \sin \theta )[/math]
You should get a linear transformation. From then, it's pretty simple.

I am peeling potatoes after spreading "scotts weed & feed" on my lawn. I rinsed off but my hands still smell. I won't be the only one eating this. Will plant killer kill humans?

If you have a group with identity element e, is abc=e the same as bca=e? I thought groups were associative but then I saw this thing where you have to insert the identity element as a set + its inverse and then kind of cycle around the sets.

>>11442462
abc=e the same as bca=e
yes.
if ab=e, then aba=(ab)a=a, su multiplying by a^-1 on both sides you get ba = e.

>>11442462
Any element [math] a [/math] in a group has an identity element [math] a' [/math]. So [math] abc=e \implies a'abc=a'e \implies ebc=a'e \implies bc=a' \implies bca=a'a=aa'=e [/math]
>>11441512
>>11440363
It's pretty simple, just the definition of tangent. [math] \tan\alpha_1=f_c/a\ ;\ \tan\alpha_2=(f_d-f_c)/b\ ;\ \tan\alpha_3=f_d/c [/math]
[eqn] a\tan\alpha_1+b\tan\alpha_2=a\cdot\frac{f_c}{a}+b\cdot\frac{f_d-f_c}{b}=c\cdot\frac{f_d}{c}=c\tan\alpha_3 [/eqn] Genuinely don't know what the "g" is tho. Probably a typo.
>>11440771
I can't help you, but you deserve a thoughtful and informative reply for the gorgeous TeX along.

>>11442462
Set [math]d=bc[/math]. Then [math]ad=e[/math] if and only if [math]d=a^{-1}[/math] if and only if [math]da=e[/math], by basic properties of the inverse.

Is [math]0[/math] a discontinuity point for the function [math]1/x[/math]? If so, is [math]-5[/math] a discontinuity point for the function [math]\sqrt{x}[/math]

Let G be a group of order (p^n)*m, where p is prime and p does not divide m.
Suppose G has a normal subgroup P of order p^n.
Show that Theta(P) = P for every automorphism Theta of G.

>>11442880
-5 is not in the domain of f(x)=sqrt(x), so no

>>11442916
But 0 is not in the domain of f(x) = 1/x, however people call it a discontinuity point of it anyways

>>11442938
1/x is a continuous function. 0 is not a discontinuity point.

>>11442903
[math]P[/math] is a Sylow p-subgroup. Any isomorphism [math]\theta [/math] would necessarily induce an isomorphism onto its image. Thus, the image would also be a Sylow p-subgroup of [math]G[/math]
But any two Sylow p-subgroups are conjugate, and since [math]P[/math] is normal, it isn't conjugate to anything else.

This is probably wrong, my group theory is rusty.

>>11443077
That's an elegant proof but we haven't touched any Sylow stuff yet so I'm not allowed to use it unfortunately :(

Originally asked 2 threads ago but didn't get any explanation.
How is the parallax angle found? I understand the earth rotating around the sun causing the star to appear to move but how is the angle found? My guess is it's the difference in angle of the telescope observing the star which is measured, but all the sources I read just hand wave it.

>>11442943
>>11442938
I could be wrong but as far as I know 1/x is not a continuous function because its domain is not a interval but a union of intervals (-infinity, 0) U (0,infinity). It is continous on its domain but it is not a continuous function. Like I said, I could be very wrong.

>>11442164
>From then, it's pretty simple.
Please explain I'm a brainlet.

I have a piecewise function that goes from R^2 to R. I have to prove the symmetry in all points except in the (0,0). I know how to prove the symmetry, if the function is C^2, should I first prove that the function is c^2 in it's whole domain except for the 0,0?

>>11443505
the symmetry of the second derivatives*

how did physicists use diff eqs to predict the stock market? What are the sources of input i can find for reliable stock modeling? i want to become a billionaire

second of all waht chemicals could i make in my kitchen other than meth to make lots of money from?

Can a tiger breed with a house cat?

any good books on simulation theory?

So I'm finishing up my last semester of undergrad with a dual major in CS and math. Currently trying to get a job doing embedded software design, but I want to keep self-studying math after I graduate.

I havent really taken any "advanced" (by autistic /sci/ standards) courses besides intro to Real Analysis, but over the next year or so I plan on reviewing Calculus, linear algebra, ODEs etc, and then becoming acquainted with Number Theory, Algebra and Topology.

What are my chances of passing GREs and getting into any grad programs? Is it even worth the time and monetary investment of going back to school?

>>11442164
>links the wiki page
I'm saying it's a stupid name to call Mobius transforms since it's [math]PSL[/math] not [math]SL[/math].
I'll wait for your deliberations on my problem, hun.

Am I retarded or is this question retarded? The answer is just the height of the man, right?

And no, this isn't my homework, I know how to set up the kinematic y = H - (g/2)t^2

How should I go about solving congruences like [math]x = 3^{8411} [/math] mod 1000?

And how do you use congruence here?
\equiv doesn't work.

>>11443796
Depends on what you want to do. What do you want to do?

>>11443741
No because that's scifi shit and not science.

>>11442943
>1/x is a continuous function.
It isn't even defined. But if you assign any value to it at zero it is discontinuous as a function from R to R.

>>11442880
These questions only make sense if you explicitly include the domains and images.
1/x is continuous from R without zero to R.

>>11444183
It's 147. Use a calculator, but there is also some stupid algebra trick to go it.

>>11444192
fuck bitches and get money

>>11444239
Ya, I'm asking for the stupid trick mostly.

>>11444241
Then it's probably not worth it, no.

>>11443505
If the piecewise function is C^2 it's piecewise second derivative is it's second derivative.

What precisely you need to do depends on the concrete definition.

>>11444192
Idk I enjoy studying math but I dont know if a masters or doing grad research would really be worth it desu. I guess I just wanna have the option.

>>11444183
[math] \phi (1000) = 400 = 3^{8411} = 3^{400(21)+11}
= 3^{11}
= 3^8 3^2 3
=147 \text{ mod 1000}
[/math]

>>11444314
Ignore the equal sign directly after the euler function = 400.

>>11443653
>how did physicists use diff eqs to predict the stock market?
Did they?
>>11444179
Find how much time T it takes for the pot to fall 18.25 meters. Then determine how far away from the sidewalk the pot is at T-0.3 seconds.

>>11443301
You are incorrect. Continuity as a definition applies only to the domain of the function. The function which is zero on (-1, 1) and also zero on (2,3) is continuous even though its domain is not an interval.

>>11443104
How much do you know about quotient groups? By the way, if Theta is an automorphism and P is normal, so is Theta(P).

Why do the eigenvectors and eigenvectors of the shape operator determine the direction that the surface bends? Can you explain this to me geometrically?

Is it possible to turn metal complexes to chelates?

Whats better if my math background is very shaky.
1. Learn everything from the ground up again and then Start with linear Algebra and Analysis (proof based)
2. Just dive into linear Algebra and Analysis and fill my knowledge holes along the way

Help me /sci/ i'm stuck. What's the area of the trapezoid? I have already calculated the radius and it is 3.5

>>11444776
whats (a+b)/2 or x+y?

>>11442588
>>11444390
Refrain from posting further, avatarfagging furshit.

>>11444725
Yes? Easy, even.

>>11444776
You now have two trapezoids

>>11445063
Here this is better
Also, I assume you meant b-a = 49

>>>/wsr/784888
Start by letting [math] \mathbf{F}(x)=\mathbf{U}'(x) [/math]. Then with the fundamental theorem of calculus and a very small amount of rearranging, you get [eqn] \mathbf{U}'(x)-\frac{1}{x}\mathbf{U}(x)=xe^x\mathbf{A}=\mathbf{U}'(x)+f(x)\mathbf{U}(x)=\mathbf{g}(x) [/eqn] which is just a typical linear ODE. Use an integrating factor to solve this. Specifically, if [math] \mu(x)=\exp\int f(x)\text{ d}x [/math] then [math] \mathbf{U}=[\int \mu \mathbf{g} \text{ d}x+\mathbf{C}]/\mu[/math] (you can find a derivation of this in any ODE textbook). So we have [math] \mu=\exp\int\frac{\text{d} x}{-x}=-x [/math] then [eqn] \mathbf{U}(x)=\frac{\int-x\cdot xe^x\mathbf{A}\text{ d}x+\mathbf{C}}{-x}=\frac{\mathbf{A}e^x(x^2-2x+2)+\mathbf{K}}{x} [/eqn] where K is the arbitrary constant vector we picked up thru integration. Now differentiate U to get F.

>>11445037
You must be new here.

>>11441241
>>11444095
The identity (actually, the matrix with the identity in the first entry) should have non-trivial class in [math]B( \mathbb{H} )/J[/math] when [math] \mathbb{H} [/math] is separable and has infinite dimension (this is actually my basic intuition for why your result should be false, the projections in [math]S[/math] all have finite dimension, so the structures of their projection lattices are too different for a quasi-isomorphism, and "the identity shouldn't be the image of any similarity class"). You might be able to prove it by including [math]S[/math] into the ideal of compact operators and borrowing results from Calkin algebras.
If so, you just use strong excision and the (short) exact sequence.

The definitions are fucking confusing, tho, so I might have mixed something up.

>>11444776
There is no single solution.
There are infinitely many solutions for the area of a trapezoid where a*b = 49, of which the smallest one is 49 itself. To sum it up you cannot tell for sure what the area of the trapezoid is without knowing more about it. generally we can state that the area of the trapezoid you are searching for has to be greater than or equal to 49.

theoretically, how much student debt is too much student debt?

>>11445354
>Rainfurrest patron cries newfag
Like pottery. Sorry Tumblr got shut down.

>>11445354
Thanks.
I'll suck your furry cock for this.

>>11444095
Deliberations as follows:
[math]K_0 (S) = K_0 (S^+ , S)[/math]. Strong excision gives [math]K_0 ( S^+ , S) \cong K_0 (B(H), S)[/math].
So, way I look at it, you need to show that every equivalence class in [math]K_0 (B(H))[/math] has a matrix representative in [math]K_0 (S(H))[/math]. Honestly, if the class of the matrix with the identity in the first entry does, you're golden, but it's also pretty likely to be a counterexample for [math]H[/math] infinite dimensional.

Also, compact projections are always finite. Just reminding you.

Also try to mess around with actions of [math]M_n (S)[/math] on [math]H^n[/math]. I think [math]M_n(S)[/math] ends up inside the algebra of compact operators in [math]H^n[/math], which should facilitate computations.

>>11445470
New idea: Because [math]S[/math] and [math]K[/math] (compact operators) have the same projections (finite ones), you might be able to conjure an isomorphism between their [math]K_0 [/math].
Specifically, you put the Hilbert structure on [math]H^n[/math] and then the projections for the compacts and trace class in [math]H^n[/math] might coincide for every step.

Then, you factor [math]S \rightarrow K \rightarrow B(H)[/math], and google results about K-theory of the Calkin algebra.

>>11441241
>>11442164
Linear fractional optimization anon here, I'm confused. So you did understand my question and is thinking about it?

Anyway, to make it clear, I'm trying to solve the problem stated on pic related, but I'm wondering whether it is possible to use the Charnes-Cooper transformation (see pic related here >>11440690) to reformulate the problem as linear least squares.

Nothing really gets answered here but anyway:

Is there a way to improve my memory? Is it possible for me to read a book and memorize most of it?

Does each human have a base cap on their intelligence? I've realized that back in school I was good in math (compared to you college and uni freaks i'm dogshit), but the stuff I understood easily some people couldn't understand even after a month. Maybe the learning style didn't fit those specific people, or do we all really have some sort of cap that we cannot break thorough?

Is there a way for me to figure out what I want to prioritize my time on when it comes to learning some subjects or is there no method that can tell me this? I have a lot of things I'm interested in, yet humans have such a small amount of time.

i have a 76% grade average on college roast me

What's considered a good GPA in STEM?

>>11446186
Anything below 4.0 means youre never gonna make it

>>11446186
good for what purpose?

im having problems in finding 4.999...'s fraction form. so far i have that it equals to 5 but i dont know what im doing wrong.

x = a.bc - ab
_____________
10^(B+C) - 10^B

x = 499 - 49
________
10^2 - 10
x = 5

wah ;;

>>11446355
4.999...=5=5/1

>>11446360
im clearly not a math major but i don't think that's right lol

>>11446369
4.999... is an integer. There is no "fraction form."

>>11446355
btw same happens to 3.999..., etc. apparently the way i'm doing it works for other numbers so i believe it's not 100 wrong. but it doesn't seem to be right either

>>11446393
Fuck off schizo.

>>11446392
how come? can you drop me some keywords so i can look up and learn more on that?

>>11446399
[math] 0.999...=\sum_{i=1}^\infty\frac{9}{10^i}=1 [/math]. This is a geometric series. Look that up.

>>11446399
nvm i found this: https://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html

>>11446410
thanks a lot!

>>11446186
depends on the university and your goals
if you just want a job then 2.8+ should even be fine, preferably 3.0+. if you want to get into grad school then maybe 3.3+, and if you want to get into a top phd program then probably 3.8-3.9+. some jobs might say 3.5 or 3.75+ if you want to go to a top tier company that receives a lot of applications, but after your first job your gpa will not matter at all
some universities with a good brand name like ivy leagues or other top universities might let you get into jobs or grad school with lower gpa. also some universities grade inflate or grade deflate

>>11446355

>>11445470
>notices the issue five hours later
Nevermind, the identity annihilates because it's similar to it's sum with itself

>>11446410
minding being asked a last stupid question? I'm quite convinced .9 repeating equals 1 now, but how would you represent the upper bound of S = any x in R that is less than 1?

>>11446496
mind*

>>11446496
I'm ESL so maybe I should rephrase that just in case it's too confusing. what's the real number right to 1's left in a number line?

>>11446496
There is no largest real number less than one (I think that's what you are asking).

>>11446506
>what's the real number right to 1's left in a number line
No such thing.

>>11446507
>>11446513
sugoi. thanks again. weird thing that we weren't taught that in high school

>>11446484
Oh I should be totally fine then, was just a bit worried since this semester was going a bit mediocre. I think I'm just too used to highschool grade inflation

>>11446521
internships for jobs or research experience for grad school will mean much more than gpa, you will be fine

I feel like I am stupid and want to be smarter. How do I get smarter? I'm trying to relearn maths again, but I feel like just memorizing how to do certain problems won't make me smarter necessarily.

>>11446617
I think what makes me feel smarter is reevaluate what I take by granted. specially when it comes to science

I'm a fucking retard. I thought I knew how line integrals worked, but apparently not.
I'm not looking for a final answer, just verification, but I need to go through the whole problem so you know where I'm at; please bear with my dumb ass.

I have an Electric Field, E=<3xz,2xy,-x^2>
I'm supposed to find the potential difference V between points A=(1,-1,1) and B=(3,2,-5).
I thought this was straight forward and I knew how to do it: Vab = [math]-\int E \cdot dl[/math]
That brings me to V=-(12z+3x+6x^2).

I thought I could do this by just finding Vb from that equation (using the z and x values for point B) and subtracting Va (using the z and x values for point A), which gave me 18 Volts.
But then I went to check by parameterizing from the beginning, but that gave me 12 V.

Does my process check out and I've forgotten how to do parameterization, and I should just ignore my check since I'm retarded? Or have I badly bungled everything from the get-go?

>>11446738
line integral of vector field.....

take the vector field, paramatrize it in terms of the path, dot them and then integrate that in terms of the paramater of the path

each piece of the path is dotted with the associated piece of vector field, at each given paramater T. so for each T you have a certain amount of "interaction" picked up, integrating (summing) into the total interaction between path and field. summed up electric field passed through for total voltage potential difference difference along a path

although isnt voltage supposed to be path independent since its a potential difference? should work out in the shape of the field anywho

>>11446764
>although isnt voltage supposed to be path independent since its a potential difference?
Yeah, that's how I caught on that I was an idiot; the more different methods I checked myself with the more different answers I got.

It's been a while since I learned about parameterization (I left college for a few years and just came back), but when I did it here I did
V=<1,-1,1>+t<3,2,-5> which gave me
x=1+3t
y=2t-1
z=1-5t

and then bringing those back to my original relationship for V and E, I had three integrals with 0<t<1 of
-([math]- \int 3-6t-15t^2 dt[/math]+[math]- \int -2-2t+12t^2 dt[/math]-[math]- \int 1+6t+9t^2 dz[/math])

from substitution and dot products

but that got me -(-5+1-7) = 12 V, which finally spurred me to come here.
I thought I knew how both of these worked, but they gave me different answers where I should be getting the same answer and I'm unsure which to trust.

>>11446780
mangled the formatting a little bit there. Should be more like:
[math]- ( \int 3-6t-15t^2 dt + \int -2-2t+12^2 dt - \int 1+6t+9t^2 dt)[/math]

>>11446780
i think we forgot to mention that you have to differentiate your path function relative to the paramater first. because its dPath not path, the path function just points vectors at your path, the dpath shows you the actual direction of change and rate

>>11446800
I've forgotten quite a bit about vector calculus, it seems.
Thanks, anon(s). I'm gonna go steal my old textbook off of zLibrary and see if I can work my way through this.
The instructor hates it when we take a mathematical approach like this, but I care more about figuring out how the vector calculus works at this point than I do about applying the Maxwell Equation.

>>11446810
Do you still need help? Ill type up a nice solution unless you got it.

>>11446810
the maxwell equations come from vector calculus. it was quite beautiful to see how they actually fit together, have u learned divergence, stokes, etc

just go on wikipedia and pauls online notes

>>11446823
I may, but I'm running through it the problem again right now; I'll post again shortly with what I've done.
But if you don't mind typing up a solution, I would appreciate having something correct that I can check myself against.

>>11446825
I did, but I'm incredibly rusty. I'm in an electromagnetics course, but I took all of my calculus courses back in 2011-2013. I do enjoy it, but my foggy math base is really hurting me.

>>11446836
well keep practicing, it really isnt too hard. just get an intuition for vector algebra and paramatrizations and youll be fine

>>11446836
[math] \mathbf{E}=(3xz,2xy,-x^2)\ \ ;\ \ A=(1,-1,1)\ \ ;\ \ B=(3,2,-5) [/math]. So then you parameter the path from A to B as [math] \mathbf{r}(t)=(1+2t,-1+3t,1-6t) [/math]. Notice that r is at A for t=0, and B for t=1. Then [eqn]\Delta V=-\int\mathbf{E}(\mathbf{r})\cdot\text{d}\mathbf{r}=-\int_0^1\mathbf{E}(\mathbf{r})\cdot \mathbf{r}'\text{ d}t=-\int_0^1 \begin{pmatrix} 3(1+2t)(1-6t)\\ 2(1+2t)(-1+3t)\\ -(-1+3t)^2\end{pmatrix}\cdot\begin{pmatrix}2\\ 3\\ -6\end{pmatrix}\text{d}t=-(-15)=15\ \text{volts} [/eqn]
I may have well made a simple mistake, but this is definitely the proper procedure. You should be able to fill in the (small) gaps.

are infrared saunas safe? isn't infrared radiation bad and shit?

>>11446888
Thanks, anon. That's almost exactly what I worked out this time, except I ended up at 7 volts, which was totally different than my previous attempts.
Checking against yours, I fucked up parameterization of x and wrote x=1-2t instead of 1+2t. Fixing that brings it in line.
Thanks a ton - you've been really helpful.

>>11446836
[math] \mathbf{E}=(3xz,2xy,-x^2)\ \ ;\ \ A=(1,-1,1)\ \ ;\ \ B=(3,2,-5) [/math]. So then you parameter the path from A to B as [math] \mathbf{r}(t)=(1+2t,-1+3t,1-6t) [/math]. Notice that r is at A for t=0, and B for t=1. Then [eqn]\Delta V=-\int\mathbf{E}(\mathbf{r})\cdot\text{d}\mathbf{r}=-\int_0^1\mathbf{E}(\mathbf{r})\cdot \mathbf{r}'\text{ d}t=-\int_0^1 \begin{pmatrix} 3(1+2t)(1-6t)\\ 2(1+2t)(-1+3t)\\ -(1+2t)^2\end{pmatrix}\cdot\begin{pmatrix}2\\ 3\\ -6\end{pmatrix}\text{d}t=-5\ \text{volts} [/eqn]
I may have made a small error somewhere, but this is definitely the correct procedure. You should be able to fill in the (small) gaps.

>>11442281
I lived.

>>11446894
yw~

can I make hand sanitizer by mixing ethanol and body lotion, or might the lotion contain compounds that react poorly with ethanol?

retarded here...
>cortisol is an anti-inflammation hormone
>aging is some kind of inflammation
>cortisol is an anti-aging hormone
what's wrong here? why doesn't it work and why do doctors call it the death hormone ?!!!

>>11445470
Thanks for the effort, but I've realized earlier today that trace-class operators are compact, so their operator indices vanish and hence their [math]K[/math]-homology class coincides with that of the identity. There is no way the inclusion can be a quasi-isomorphism in this case.
I believe the trace-class constraint is way too strong. I will need to regularize the trace so it's defined on just the smooth subalgebra [math]\mathcal{R}[/math] instead.

If I scaled up the Earth to the size of the sun what would happen? Let's say I got a planet mover and just lumped a bunch of Earth's together until they equaled 1 solar mass. It's not going to collapse into a star, what the hell would it fuse? What the hell else would happen?

>>11447157
>>aging is some kind of inflammation
It's not. Aging is cell deformities and gene defects caused by depletion of telomeres.
Inflammation is a defense mechanism used by leukocytes.

>>11447389
Are you saying that inflammaging is a hoax?

>>11446399
>can you drop me some keywords so i can look up and learn more on that?
The cauchy completion of the rational numbers will clear up any of these issues.
If you don't want to dive into "deep math" just accept that if you can not find a finite difference between two numbers they are the same.

Is this proof good?
>If n is even, then (-1)^n = 1
Proof:
>Suppose n is even, then n = 2a
>So (-1)^2a = (-1)(-1)(-1)(-1)...(-1)(-1) 2a times
>Since n is even, there's a pairs of (-1)
>We already know that (-1)(-1) = 1
>So we end up with (1)(1)(1)...(1)(1) a times
>(1)(1)(1)...(1)(1) = 1
>Therefore, if n is even, then (-1)^n = 1

>>11445607
Absolute brainchad, my average is only 74%.
>>11447255
>I spent a few hours studying operator k-theory because Yukarifag forgot the single most basic result about trace class operators
Ok.
>>11447778
Not really. A half decent proof would look like this.
[math](-1)^n = (-1)^{2a} = ((-1)^2 )^a = 1^a = 1[/math]
The step where you "arrange them into pairs" and cancel them out there is unecessarily informal.

>>11447778
>>11447984
It's literally the same proof.

>>11447778
>>11447993
And yes, both are perfectly fine proofs.

I'm feeling like an 100% retard

>>11448007
100% probability that shit wont fly in court

>>11448007
0.6/(0.6 + 0.18*0.4)

>>11448022
I didn't understood the 0.18 * 0.4

>>11448032
LH = event that X is left handed
G = event that X is guilty
[math]P(G|LH) = \frac{P(LH|G) P(G)}{P(LH)}= \frac{P(G)}{P(LH|G) P(G) + P(LH|\neg G) P(\neg G)}[/math]

>>11448033
why is P(LH|~G) = 0.18?

>>11447993
Is it, tho? Are you quite sure about that?

>>11448171
not him but the prob of anyone being left handed is pointed out to be .18, so the whole left handed population - 1 (LH - ~G) is very close to the whole left handed population still

>>11448185
Yes. Your
[math](-1)^{2a} = ((-1)^2 )^a[/math]
step corresponds to his step of grouping (-1)'s in pairs.

>>11448192
this.

Is it difficult to get a student loan deferment? Figure /sci/ would be the best place to ask.

>>11437575
Anyone got an invite to the new /sci/ discord?

>>11447778
It's a bit informal, but I still would consider it correct.

>>11448229
Really?
And here I was thinking that it was a perfectly simple and formal application of [math](a^b)^c=a^{bc}[/math], a basic theorem about powers, instead of a heuristic, extremely informal grouping of the [math]-1[/math]s into pairs that should cancel out, whose formal proof would probably look something like [math](-1)^n = \Pi _{i=1} ^n (-1) = [ \Pi _{i ~ odd, ~ i \leq n}(-1) ][ \Pi_{i ~ even, ~ i \leq n} (-1) ] = [ \Pi_{i ~ odd, ~ i \leq n} (-1)][ \Pi_{i+1, ~ i ~ odd, ~ i \leq n} (-1) ]= \Pi_{i ~ odd, i \leq n} (-1)^2 = \Pi_{i ~ odd, ~ i \leq n} 1 = 1[/math]
>>11448267
Same tbqh.
>>11448263
>discord
OH NO NO NO NO

>>11447993
It's the "same proof" but one is much more concise and clear and the other is a fucking trainwreck.

for the integral sqrt(4-x^2)
can't you just do the square root to get 2-x and then take the integral?....
probably am retarded

>>11448341
>2-x
Are you sure that's the sqrt of 4-x^2

>>11448350
no but my brain thinks that for some reason

>>11447389
Depletion of telomeres isn't even the full story though. There's many factors that cause aging and telomeres are just one part of the puzzle. There's a website with all sorts of current studies going on into different aging mechanisms and their current progress. I forget the name though.

>>11448246
>taking out student loans ever
You already fucked up

Hey /sqt/bros it's me again with a question from Fomenko-Fuchs.
He says that homeomorphism between S^n/ (S^n - Ua) and S^n/(S^n - U) is a homeomorphism between two spheres. Although each space is homeomorphic to a sphere, isn't the induced homeomorphism of spheres dependent on which homeomorphism you choose? Or does he mean the homeomorphisms which are homotopic to the canonical projection maps S^n S^n/(S^n - U) and S^n S^n/ (S^n - Ua) ?

>>11448702
>on which homeomorphism you choose?
Yes. That's why he says [math]f[/math] determines the homeomorphism, and that's why he uses the induced homeomorphism for the degree.

>>11448702
>>11448708
Also, [math]f[/math] isn't necessarily a homeomorphism.

>>11448708
I don't think you understood my question. I'm not talking about the homeomorphism determined by f between the quotient spaces, but rather the homeomorphism which arises once you identify the quotient spaces with spheres. I'm saying that this could depend on the different ways in which you identify the quotient spaces with spheres.

>>11448717
What do you mean?
That's like asking if an abstract linear transformation depends on the vector space's choice of base.
Both sides are homeomorphic to S^n. That's that.

>>11448726
They're homeomorphic to S^n, but not in a canonical way. The degree could change by a factor of +-1 if you choose a different identification. So my thought was that the author left out that the identification is any homemorphism which is homotopic to the canonical projection.

>>11448731
Oh, my bad, I forgot about orientations.
Basically, IIRC the quotients inherit orientations from S^n, and then the homeomorphism only needs to have degree one.

>>11448735
>then the homeomorphism only needs to have degree one.
Which homeomorphism?

>>11448753
The two black ones.
The loop is an automorphism of S^n, and thus has well defined degree.

>>11447984
Thanks. I didn't think of that.

>>11448764
Yes the degree of the loop arrow is the degree we want, which could be +- 1.
>then the homeomorphism only needs to have degree one.
Are you saying we require black homeomorphism S^n -> S^n to have degree 1?

>>11448781
No, no, the two straight arrows are orientation preserving (or homotopic to the projection, same same).
Then, whether or not the loop preserves orientation (degree one) or not (-1) only depends on f.

>>11448790
Ok, so that's what I said in my first post. Because since the straight arrows are between different spaces, it doesn't make much sense to say they're orientation preserving, especially when they're not differentiable. But yeah, thanks.

what's a good book on the story of modern mathematics? specifically one that will explain me leibniz and newton's calculus and how different they are in comparison to modern calculus. i want to grasp better concepts like infinitesimals and the like, but have no idea how to go about it besides hopping between Quora pages

how do i find the other 2 angles in a trapezium picrelated given 2 adjacent 90* angles and the 3 lengths touching the known right angles

>>11448835
Rudin, Principles of Mathematical Analysis

>>11448835
>I want to grasp better concepts like infinitesimals and the like
You should first try to get a solid understanding of phlogiston and the lumeniferous aether.

>>11448851
[math]\alpha=90^\circ+\arctan\frac{3.6-2.8}{12.2}[/math] ; [math] \beta=180^\circ-\alpha [/math]

>>11448865
thank you very much :D

>>11437575
how can i compute the killing vectors for a sphere: [math]ds^2 = d\theta^2 + \sin ^2 \theta d\phi ^2[/math]?
I immediately can tell that [math]\xi ^1 = \partial _{\phi}[/math] since there's no [math]\phi[/math] dependence on the metric, but how do i get the other two? I need at least one then i can use the commutator to get the third..

>>11437575
I have pain in my right eye and I cant figure out what it is online what is it

Do squirrels play in my yard or, am I witnessing rape?

https://youtu.be/3d6DsjIBzJ4?t=592
9:52 if the video doesn't automatically start there

why is it that, in 3b1b's animations for a taylor polynomial, the graph of his polynomial continues to "hug"/approximate the parent graph (cos(x) in this case) when he shifts the function via (x-π), but when I try to do it with graphing software, it just translates the polynomial by π units right (as it should)?

What is he doing to the actual polynomial behind the scenes to get it to slide over the parent function instead of translating along the x-axis?

Dudes, in [math]\int_ \frac{dt}{\sqrt{t}} = 2 \sqrt{t} + c[/math], why is dt appearing as a numerator rather than a symbol?

>>11448982
It's supposed to be int(dt/sqrt(t)), Idk why it's so tiny.

>>11448969
Did you try just assuming that it has the form [math]f( \theta , \phi ) \partial \theta [/math] and solving for [math]f[/math]?
There's always the [math]SO(3)[/math] action on the sphere, as well.
>>11448982
Abuse of notation.
Formally, it should be [math] \int \frac{1}{ \sqrt{2} } dt[/math].

>>11448987
Ah, I thought so. Thank you very much!

For anyone studying science outside of the school/laboratory environment, how do you "perform" basic experiments without having to set up some $500 equipment set for one experiment?

>>11437575
I really wonder what fun knowers like you take from visiting /sqt/ threads, t2h2anon

>>11449018
with pen and paper
t. math neet

with frictionless surfaces and vacuum tubes and shit
t. physics neet

with cookware and explosive dangerous fuming volatile chemicals and poor ventilation
(just kidding i just burn stuff on my countertop)
t. chem neet

with masturbation meditation and drugs and pain
t. psychology neet

>>11443128
There are objects so far away their parallax cannot be measured. These are 'Background' stars. They don't appear to move even as the Earth rotates around the sun. You photograph it in June, you photograph it again in January. You measure the distance it traveled against the background stars on your photographic plate. The rest is basic geometry.
https://starguyspeaks.files.wordpress.com/2013/09/proxima_2010.gif

>>11443128
Oh, I just realized the question you're asking is how did we measure the distance to the sun.

First, we calculated the distance to the moon. This was done in ancient times using the known distance between two cities, then a person in each city calculating the position of the moon at the same time on the same day. Then using geometry to work out the triangle formed between the two cities and the angle of the moon. It's been refined since then but that was how it began.

That's step 1.

Step 2 was using the known distance from the moon to the Sun to calculate a new angle from Earth to the Sun but it turned out too hard in ancient times and the distance measured was fucked. The Sun is just too damn far away and its so bright it blots out any background stars to use for measurement. The key was a transit of Venus back in the 18th century. It was a massive undertaking of several astronomers traveling to known locations around the world and taking precise measurements of the angle of Venus against the background of the Sun all at the exact same time. This allowed them to use the same method used 2000 years earlier to calculate the distance to the Moon.

Bam, distance to the Sun.
93 million miles. Now we get to use that measurement for the rest of the galaxy close enough to measure.

>>11449520
dude.... science. its really weird that we are just smart chimp and zees

>>11447984
>Absolute brainchad, my average is only 74%.
oof, for both of us
apparently the average for people at college is 85%

Does addition on R^n constitute a groupoid or am I way off?

>>11449823
>A groupoid is a Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory.[1] A groupoid where there is only one object is a usual group.
well youre at least not way off

Is this the most god tier snack to refuel your brain after a hard study session?

>>11449842
What I meant to ask was if it's just a trivial groupoid with one object or if it in fact has n objects.
Seems like it is just a regular group though, you'll have to excuse me, all this nLab junk is making my brain hang.

>>11449823
Yes.
>>11449891
It has one object. It is a group. You can add any two vectors in R^n.

>>11449891
the + binary operator is a single object, but if you count each +n as a single operator for all n then it has infinite operators

>>11449886
I like sour gummy but for Haribo plain gummy bears are better than the sour ones.

If g is differentiable and g(0,0)=(0,0), and f is continuous prove that [math] \frac{\partial f g}{\partial x}(0,0)=f(0,0)\frac{\partial g(0,0)}{\partial x} [\math] and [math] \frac{\partial f g}{\partial y}(0,0)=f(0,0)\frac{\partial g(0,0)}{\partial y} [\math].
Note that f g is not a function composition. I guess I have to prove it by definition for it's limit, but you could do that if f is not differentiable, how would you do it?
I also need to prove that f g is differentiable in the (0,0).

>>11449978
damn [math] \frac{\partial f g}{\partial x} (0,0) = f(0,0) \frac{\partial g(0,0)}{\partial x} [\math] and [math] \frac{\partial f g}{\partial y} (0,0) =f(0,0) \frac{\partial g(0,0)}{\partial y} [\math].

>>11449980
>>11449978
[math] \frac{\partial f g}{\partial x}(0,0)=f(0,0)\frac{\partial g(0,0)}{\partial x} [\math] and [math] \frac{\partial f g}{\partial y}(0,0)=f(0,0)\frac{\partial g(0,0)}{\partial y} [/math]
Here ya go bby
[/math] not [\math]

>>11450026
>>11449891
Btw, the solution would just be to prove the product rule. [math] (fg)_x=fg_x+f_xg [/math] and [math] (fg)_y=fg_y+f_yg [/math]. Now let [math] g(0,0)=(0,0) [/math] and evaluate. The product of differentiable functions is differentiable.
>>11449906
You have good taste, Jon

>>11450040
Yeah I was doing that in my professor told me that I can't do that cause I don't now if f is continuous or not.

>>11450144
>If g is differentiable and g(0,0)=(0,0), and f is continuous
Are you sure thats not supposed to read f is differentiable? Differentiable implies continuous. If not, the thing you are supposed to prove isnt true

>>11450169
I have a final yesterday, and I could do that exercise. I mean I did something but it was probably wrong. I basically used the definition of partial derivative of f in the product rule. I said that I could write the limit because f was continuous, and that every limit multiplied by 0 (g(0,0)) was 0 so I "prove" it like that. At first I just wrote the partial derivative of f as you (or whoever) did but the professor told me that I should think it again cause I couldn't write the partial derivative of f because i didn't know if it was differentiable or not, that I had to use that it was continuous.

How do I systematically solve congruence such as
[eqn]130x \equiv 150 \mod 232[/eqn]
where the modulus and coefficient at [math]x[/math] are not coprime? Bezout for 130 and 232 gives
[eqn]130*25 \equiv 2 \mod 232[/eqn]
so I can multiply byy 75 to get
[eqn]130*75*25 \equiv 150 \mod 232[/eqn]
which gives x = 75*25 mod 232. But this is only one solution and there should be two of them.

>>11448987
>f(θ,ϕ)∂θ and solving for f

i see what you trying to tell but where am i suppose to put this in and solve for f? The killing equation?

>>11448835
If you are German there is a pretty good book on the History of Analysis by Sonar.

>i want to grasp better concepts like infinitesimals and the like
Except for historical interest there is literally no point.

>>11450379
130x = 150 mod 232
is equivalent to (dividing everything by 2)
65x = 75 mod 116
and now the solution x is unique because 65 is a unit mod 116
once you get x, the solutions mod 232 are x and x+116

Can you make soaps and salts out of any group 1 element?

>>11449978
I'll prove it for function in [math]\mathbb{R}[/math] instead of [math]\mathbb{R}^2[/math] for general convenience.
We have that [math] \frac{dfg}{dx} (0) = lim_{ \epsilon \rightarrow 0} \frac{fg( \epsilon) - fg(0)}{ \epsilon} = lim_{ \epsilon \rightarrow 0} \frac{fg( \epsilon)}{ \epsilon} [/math], where [math]g(0)=0[/math] by hypothesis.
We also have that [math]f(0) = \lim_{ \epsilon \rightarrow 0} f( \epsilon)[/math], by continuity, so [math] f(0) \frac{dg}{dx} (0) = \lim_{ \epsilon \rightarrow 0} f( \epsilon ) \frac{g( \epsilon) - g(0) }{ \epsilon } = \lim_{ \epsilon \rightarrow 0} \frac{ fg( \epsilon) }{ \epsilon} [/math].
>>11450386
Yes.

>>11449345
I understand the star appearing to move relative to other background stars. I want to know how the angle is found. The purpose of finding the angle is to find the distance from the Earth to the star but any geometrical arguments I can make don't need the angle to do that.
>>11449520
>Oh, I just realized the question you're asking is how did we measure the distance to the sun.
No. Not at all. How did you infer that from my post?

>>11450550
Not him, but why does [math]\lim_{ \epsilon \rightarrow 0} \frac{fg( \epsilon)}{ \epsilon} [/math] or [math]\lim_{ \epsilon \rightarrow 0} \frac{fg( \epsilon) - fg(0)}{ \epsilon}[/math] exist?
And you HAVE to use that f(0)=0, eg. take f=x+1, g = abs, then the limit definitely doesn't exist.

I am actually unsure whether the result is correct. What you would need to prove is that if f is continuous with f(0)=0 and g is differentiable, then f*g is differentiable at zero, which is definitely stronger than what the standard product rule gives you.
I haven't thought about it all that much and maybe I am being stupid, but I think there is still something to prove.

>>11450682
>And you HAVE to use that f(0)=0, eg. take f=x+1, g = abs, then the limit definitely doesn't exist.
I switched f and g there, but that shouldn't matter.

> f is continuous with f(0)=0 and g is differentiable
not f(0)=0, but g(0)=0

>>11450682
>>11450693
>why does it exist
The second one necessarily exists by definition, that is, it's a multiplication of two numbers which exist.
The first limit exists because it's the exact same as the second one.
>And you HAVE to use that g(0)=0
I used it, tho. That's why [math]\frac{dfg}{dx} (0) = lim_{ \epsilon \rightarrow 0} \frac{fg( \epsilon) - fg(0)}{ \epsilon} = lim_{ \epsilon \rightarrow 0} \frac{fg( \epsilon)}{ \epsilon}[/math], else [math]fg(0)=f(0)g(0)[/math] wouldn't vanish.
Also in the last line, same thing, [math]fg(0)[/math] vanishes.

I used literally everything given, lad.

>>11450700
>Also in the last line, same thing, fg(0) vanishes.
[math]f( \epsilon)g(0)[/math], actually.

>>11450700
>The second one necessarily exists by definition
For that you have to ASSUME that f*g is differentiable at zero, but why would that be the case?

>[math]\frac{dfg}{dx} (0)[/math]
Why is that even defined? Why would the product of a continuous and a differnetiable function be differentiable?
I even gave you a counter example, why that isn't the case in general.

>>11450716
>For that you have to ASSUME that f*g is differentiable at zero
No, no, you aren't following.
[math]\frac{dg}{dx}(0)[/math] exists by hypothesis, right?
[math]f(0)[/math] exists trivially too, right?
So [math]f(0) \frac{dg}{dx}(0)[/math] also exists, since the real numbers are closed under multiplication.
Just in case that's what's confusing you, I also used [math]f(0)\frac{dg}{dx}(0)=[ \lim_{ \epsilon \rightarrow 0}f( \epsilon)][ \lim_{\epsilon \rightarrow 0} \frac{g( \epsilon) - g(0)}{\epsilon} ] = \lim_{ \epsilon \rightarrow 0} [f( \epsilon) \frac{g( \epsilon) - g(0)}{\epsilon} ][/math], which is a classical property of limits.

>>11450736
But that is your LAST line.
In your last line you show that the first line is actually well defined, which it apriori isn't.

Starting your proof with the derivative of f*g at zero can not work, since you don't even know it exists.

>>11450744
Are you seriously getting on my ass because of the order I wrote the proof in?

>>11450746
>Are you seriously getting on my ass because of the order I wrote the proof in?
No, I just couldn't be bothered to read past the first line where you claimed something existed, which didn't exist.

And yes. A reordering of the lines in a proof in general doesn't result in a valid proof so the order of lines in a proof does indeed matter.

>>11450753
>imagine being this schizophrenic

>>11450758
>schizophrenic
What do you mean?

Schizophrenia refers to being unable to interpret reality normally. I clearly interpreted everything I read in your proof normally. Also I think my claim about the ordering of a proof is consistent with reality.

how did i spend 10 hours doing this simple homework?

why is it okay to run a gas stove or gas oven or gas fireplace in your house, but you can't run a gasoline generator, due to carbon monoxide?

>>11450040
Cease to exist.

I have to calculate the probability of this system stopping. It stops if it's impossible to connect IN to OUT. They give us a tip to solve this, which is splitting the problem in two different parts. One being where C is working, the other being where C isn't working.

>differential equations professor skipped the section on Boundary-Value problems

ok now I'm starting to get a bit nervous
how fucked am I for future EE classes if I don't learn boundary value problems
I can always self-teach but I expected something so important to be covered

>>11450836
What homework?
>>11451109
Assume [math]C[/math] isn't working. Then, at least one of B or E needs to be working, and if only B, A too, if only E, D too,and if B and E, at least one of A and D.
Assume [math]C[/math] is working. Then at least one of [math]B[/math] or [math]E[/math] needs to be working, and independent of which one, at least one of [math]A[/math] or [math]D[/math] also needs to be working.

>>11451109
C (A∨D)∧(B∨E)
¬C (A∧B)∨(D∧E)

P(X∧Y) = P(X)*P(Y)
P(X∨Y) = P(X)+P(Y)-P(X)*P(Y)

>>11451289
Well, it ate the implies sign, so
C -> (A∨D)∧(B∨E)
¬C -> (A∧B)∨(D∧E)

>>11451205
>skipped the section on Boundary-Value problems
Boundary value problems are like 90% of Differential equations...
What is there left to do after that?

>>11447984
Yeah sorry, that's what happens when I try to multitask and do research while at a conference; thanks for your ideas though. I did manage to demonstrate an instance in which Freed-Hopkins's conjecture seems to be true though.
I'll have to think more about how to regularize the trace; for now my only idea is to leverage the finite-dim formula [math]\ln \operatorname{det} = \operatorname{tr}\ln[/math] and [math]\zeta[/math]-regularize, but I'll have to flesh out the details later.

>>11451289
Ok I understood that. But I can't get an answer

>>11450550
Thanks a lot anon, that really makes sense. How about proving the differentiability in (0,0)? So far I just have that its partial derivatives exist, and that f g(0,0)=(0,0) but it really seems like a mess to prove that
[math] \lim_{(x,y) \to (0,0)} \frac{|f(x,y)-f(0,0) \cdot (\frac{\partial g (0,0)}{\partial x} x + \frac{ \partial g(0,0)}{\partial y} y|}{||x,y||} = 0 [/math}

>>11451498
> [math] \lim_{(x,y) \to (0,0)} \frac{|f(x,y)-f(0,0) \cdot (\frac{\partial g (0,0)}{\partial x} x + \frac{ \partial g(0,0)}{\partial y} y|}{||x,y||} = 0 [/math]

The decimal system literally shits itself when you divide by three

Is the above a scientifically accurate statement?

Why do lecturers feel the need to hand out 40+ pages of 'notes' for one section that are longer than the textbook chapter they're based on? It happens a lot in Electrical Technology, I've found

>>11451432
>I did manage to demonstrate an instance in which Freed-Hopkins's conjecture seems to be true though.
Nice.
>>11451498
>>11451502
>take a moment to check
>the argument doesn't really copy word for word for the general case, even if it kinda resembles it
My bad.
We set [math]Dg(0) = G[/math] (the total differential operator at 0) (technically the linear functional of the inner product with the gradient, but whatever).
For our first estimate, we start with [math]\lim_{||v|| \rightarrow 0} || \frac{g(v) - G(v)}{||v||} || = 0[/math], and we use an absolute bullshit, high illegal trick, known as "passing [math]f(v)[/math] inside because it's continuous to obtain" [math]\lim_{||v|| \rightarrow 0} || \frac{f(v)g(v) - f(v)G(v)}{||v||} || = 0[/math] , which is estimate number one.
Estimate number two is [math]\lim _{||v|| \rightarrow 0} || \frac{f(v)G(v) - f(0)G(v)}{||v||} || = \lim _{||v|| \rightarrow 0} ||\frac{[f(v)-f(0)]G(v)}{||v||} || \leq \lim _{||v|| \rightarrow 0} || \frac{[f(v)-f(0)] ~ ||G|| ~ ||v||}{||v||} || = \lim _{||v|| \rightarrow 0} [f(v)-f(0)]||G|| = 0[/math].
Then you throw in a triangle inequality: [math]\lim _{||v|| \rightarrow 0} || \frac{fg(v) - f(0)G(v)}{||v||} || \leq \lim _{||v|| \rightarrow 0} [ ||\frac{f(v)g(v) - f(v)G(v)}{||v||} || + || \frac{f(v)G(v) - f(0)G(v)}{||v||} ||] = 0[/math].

I think the only weird thing I used is the operator norm. For a reasonable reference, see https://en.wikipedia.org/wiki/Operator_norm
>>11451823
Your professors do that?
A few professors I had literally wrote ~100 pages long books on the subject and had us print them out and use as material for the semester.

>>11451841
Yeah, our lecturers write up notes that are sometimes legitimately longer than the textbook chapter they're based on.

I checked. I went to the library and opened the relevant textbook chapter and compared the number of pages.

Why is Pythagoras' Theorem framed as sqrt(a^2 + b^2)? Isn't the effect of the square root in this case to eliminate the squaring of the variables? So, why not just write it as a + b = c?

>>11451909
Because a+b=c isnt remotely true

>>11451912
Yeah, but how is sqrt(a^2 + b^2) different to a + b = c?

>>11451916
Circles

>>11451916
(a+b)^2 = a^2 + 2ab + b^2 != a^2 + b^2

>>11451996
But that's true in [math]\mathbb{Z}_2[/math] :))))))))))))))

>>11452015
Z2 is not C1, Z2 is vectors with dot products while C1 has the imaginary multiplier. i still dont understand imaginary numbers. can someone please help

is pic related right??

>>11452509
i mean i dont agree with the pic or anyone who says stuff like "math is distinct from reality" bc reality is just stuff and math is just thoughts that we can map to occurences, anything can be mapped to anything else if you select the right pieces of it for example the string ------- can be mapped to schrodingers equation if you so please, math is like that

but i guess what does imaginary numbers map to? i guess it can make sense sometimes. but why oh why do they make a perfect circle? aaaaHHHahahah

>>11451841
Sorry anon, I can't fully understand what you are doing. I will try to read it again a couple more times. I will write the prove that I though after you post >>11450550. It doesn't seem right to me to be quite honest, "forgetting" about the absolute value seems kinda retarded, even though the limit is equal to zero.
[math]\lim_{(x,y) \to (0,0)} \frac{fg(x,y)}{||x,y||} = \lim_{(x,y) \to (0,0)} \frac{f(0,0)\left(\frac{\partial g (0,0)}{\partial x} x + \frac{ \partial g(0,0)}{\partial y} y \right)}{||x,y||}[/math]
[math]\lim_{(x,y) \to (0,0)}f(x,y) \lim_{(x,y) \to (0,0)} \frac{g(x,y)}{||x,y||} =f(0,0) \lim_{(x,y) \to (0,0)} \frac{\left(\frac{\partial g (0,0)}{\partial x} x + \frac{ \partial g(0,0)}{\partial y} y \right)}{||x,y||}[/math]
And since f is continuous [math]lim{(x,y) \to (0,0)} \frac{f(x,y)}{f(0,0)} = 1[/math].

[math]\lim_{(x,y) \to (0,0)} \frac{g(x,y)}{||x,y||} = \lim_{(x,y) \to (0,0)} \frac{\left(\frac{\partial g (0,0)}{\partial x} x + \frac{ \partial g(0,0)}{\partial y} y \right)}{||x,y||}[/math]
And then
[math] \lim_{(x,y) \to (0,0)} \frac{|g(x,y)-\left(\frac{\partial g (0,0)}{\partial x} x + \frac{ \partial g(0,0)}{\partial y} y \right)|}{||x,y||} = 0[/math]
is true since g is differentiable.

Ok, I'm a second year EE and I have come to realize I hate the subject. I hate fucking soldering and programming and having to design circuits and print them and do stuff with my hands. I should have gone the physics route but i was afraid i was too low iq. in any case, since switching to physics is not possible (i would need to start in year 1) i was thinking of switching to another, less hands-on engineering. i look around and chemical seems a right fit. a lot of physics and maths (fluid,heat, mass transfer, etc.) and very little working with my hands (it's all simulation or jusyt going to the process lab and watching the pipe, &c.) would you reccomend i change subjects? is cheme a good major in your opinion? how's the job market when compared to EE's? or should i just push trough EE and then do a master's in math or physics? please help.

>>11451302
wrote-solving differential equations of various form, and endless initial value problems

Ok, so I have:
>taken notes during a lecture
>now have several pages of concise notes that is basically all the shit I need to know for the exam
What now? How do I 'study' these notes so I remember everything on them?

>>11439156
Extremely correctpilled

Retard here. Taking intro to university mechanics despite being in a non-science major. I think I'm in over my head. We've just started going over forces.
Can anyone give guidance as to how I know that my manipulation of algebraic equations is going to find me the correct force that is a part of the net force?
For example, a block being pushed down a 30° slide. I am given the force applied on the block, and need to find the kinetic friction force against the block. I keep messing up when to use cos/sin for the horizontal/vertical force, plugging in one equation to another, etc. Am I just a brainlet, or will it come with practice? My highest math level is calc 1, but that was 4 years ago.

the lab i want to research in during phd is run by a chinaman and out of his current students and past 20+ he has only had chinese phd students

>>11453426
It's literally just practice.
Keep in mind Newton's laws to get your equations and do manipulations correctly, that's basically it.

>>11452804
You're either going to have to program/code (physics) or do hands-on experiments (chemE) or both. You really can't avoid it anon, that's what science is. You can go do math or circlejerk with "mathematical physics" if you see yourself going into academia, but otherwise you'll need to pick up some sort of practical skillset.
t. theoretical physics phd student, I hate doing numerics but it's what pays the bills since I have no plans of being a professor

>>11452800
Not him. The first line doesn't make sense. Firstly, you haven't even shown the limit exists in the first place (that is what you do in the next line), secondly you should say at least a word why you can exchange g with it's gradient, yes I know why it's true, but is it obvious to you?
The third line is redundant, you never use f after that.
The fourth line is a restatement of what you didn't show in the first line.
The line below that clearly isn't what you wanted to show in the first place.

>>11452545
>i dont agree with the pic or anyone who says stuff like "math is distinct from reality"
Then you are pretty dumb.
The map is not the territory, a mathematical model isn't reality and it can never be.

>bc reality is just stuff and math is just thoughts that we can map to occurences, anything can be mapped to anything else
What an abysmal argument.

>>11453426
>I keep messing up when to use cos/sin for the horizontal/vertical force
You need to better internalize what cos and sin actually mean in terms of the geometry of triangles. So many students just blindly throw in one or the other because they memorize formulas and don't actually understand why the particular trig function appears.
>plugging in one equation to another
Keep your work organized. Make your steps linear and logical, at least for yourself. I also see lots of first-time physics students struggle with this, they start attacking like 5 different angles of the problem at the start and literally just get confused about what they've written down because it's a mess and there's absolutely nothing written in words describing the logic to go from line to line. USE WORDS as you write your answers, don't just string together equations.

>>11453238
what kind of course?

>>11453536
you completely misinterpreted me. stop being a retard just because youre annoyed im cooler than you

>mathematical model isnt reality
Well, its a part of reality. But sure, the diff eqs describing fluid flow arent fluid flow themselves. However, the fact that the fluid can be described in terms of those diff eqs means it has that trait for at least some aspects of its whole nature. The mapping just constructs a description, proving that reality falls under mathematical description and both are consistent alongside eachother, if perception matches model

>>11452509
btw guys! i finally figured out how come it works like a circle. (a+ib)(c+id) is just "building" a new vector ontop of c+id, and thatvector is similar to a+ib so the angles are added, and the things are multiplied

=====DDDDD im so happy i finally understand this

now the next questions are, how does this help with the intuition of eulers formula? ANNND
what exactly is i? i understand what i is in algebra, its just a definition, and since its 4 partite loopy it works out well for 90 degrees (which as seen above can fit to making angles add). however, i need some better intuition for what it is in itself, where it arises. can someone provide real life exampels? ive seen it used in quantum mechanics (or just diff eq in general?) but i need some BACKING

is it just about 4s?

>>11454079
i know theres the common EE thing with impedance, but ive never worked with AC, only DC. maybe someone could explain?

>>11454079
Complex numbers are equivalent to orthogonal matrices. (a+bi)(c+di) = (ac-bd)+(bc+ad)i
[a -b][c]
[b a][d]

Note that if R(x) is the 2x2 matrix which represents a rotation by angle x, then R(a+b)=R(a).R(b) and R(kx)=R(x)^k. Cf e^(a+b)=e^a.e^b and e^kx = (e^x)^k, but also note that this is true for any base, not just e.

As for what i "is". Well, you can treat at as a free variable about which nothing is known except that i^2=-1.

>>11454106
>As for what i "is". Well, you can treat at as a free variable about which nothing is known except that i^2=-1.
see, i tried that. But there's so much richness to the theory of complex numbers that i feel more understanding ought to exist, a broader perspective that i fits naturally into. One thing i noticed was the 4 exponentiations of i to return to i, that fits well with 4 90 degree rotations. And it fits well with the shape of the e and trig taylor series (but that ones a little weird)

i just wanna see where it arises on its own in reality, at least in a very simple mapping.. btw what constitutes the simplicity of a mapping? why is mapping my fingers to the number 5 easier than mapping my fingers to the cube rt of 125? why does the universe make my brain do this?

>>11454071
>im cooler than you
???

>However, the fact that the fluid can be described in terms of those diff eqs
They can't. They can be approximated, but nothing more.
The mathematical description says nothing about the object, instead it creates an idealized version of that object.
In fact reality can not be investigated mathematically, there are no perfect models.

> it has that trait for at least some aspects of its whole nature.
Wrong. The PDEs used to describe fluids are based on Newtons laws. Newtons laws are known to be wrong, so what you are saying is false.

>The mapping just constructs a description
It doesn't describe reality.

>if perception matches model
If at ANY point in time a model matched the perception perfectly physics would be over.
There is no known model which describes reality beyond approximation.

>>11454129
so consider this: your whole argument is based off the fact that math doesnt model it approximates

well if it approximates, that means its a function with an error bound. it describes the error bounded version of an object, so its still mapped to the object , with a second layer of indirectness. but thats not so bad. 25 maps to the number of fingers i have with a layer of indirectness, because its the square of my finger count. indirectness is still fine

second of all. the natural numbers are a perfect model for cardinality of discrete objects. unless you want to talk about objects not being discrete due to quantum mechanics. then reality is imperfect sure, but then that begs the question, how can we have perfect perceptions of numbers? that doesnt make sense. so its an imperfect perception (due to qm in neurons and in written symbols of math) approximating an imperfect thing aiming at but never hitting a platonic ideal? then what is it aiming at? im confused.

Why are (matrix) representations of SO(2) limited to those of determinant +1 specifically? I was told that +1 keeps right handed coord systems right handed (and likewise -1 for lefthanded systems), but it isn‘t immediately obvious to me why and I‘m not really quite sure how to show this to myself either.

>>11454180
>I was told that +1 keeps right handed coord systems right handed (and likewise -1 for lefthanded systems), but it isn‘t immediately obvious to me why and I‘m not really quite sure how to show this to myself either.
That's wrong. +1 preserves the orientation (left handed remains left handed and right handed remains right handed) while -1 switches the orientation.
You want your maps to preserve orientation (no reflections), hence +1.

>>11454191
ah, ok, that makes way more sense. thanks.

>>11454083
Impedance is just a way of generalizing the concept of resistance to any passive circuit element. This way, we can solve a circuit with complex algebra instead of pig disgusting coupled ODEs. We are after a modified form of Ohm's law that looks like [math]
V=ZI [/math] where Z is the impedance of the element. In an AC circuit at frequency ω, we have the voltage over a capacitor as [math] V=V_p\exp(j\omega t) [/math]. So [math] I=CV'=j\omega CV_p\exp(j\omega t)=j\omega CV [/math]. For an inductor, suppose a current [math] I=I_p\exp(j\omega t)[/math]. Then [math] V=LI'=j\omega L I_p\exp(j\omega t)=j\omega L I[/math]. So for a capacitor and inductor we have the "resistance" as a function of frequency: [math] Z_C=(j\omega C)^{-1} [/math] and [math] Z_L=j\omega L [/math]. The impedance of a resistor is just its resistance. Now you can use Ohm's law for impedance and avoid solving ODEs entirely when figuring out the steady state or frequency response of an RLC circuit.

>>11454262
i dont.. i dont see where the complex algebra comes in though. all those equations you listed could just as well be with Z subset R, couldnt it?

Can someone check if this makes sense.

1. As the speed of an object increases, time as experienced by the object slows down
2. This time dilation is caused by the object's increase in mass
3. Since mass is equivalent to energy, as time dilates, the object's energy also increases
4. The rate of the object's energy increase is asymptotically related to the object's speed
5. Therefore there exist a maximum speed c at which objects can travel, since moving at c requires infinite energy for a massive object
6. Photons can move at c because they are massless

>>11454262
>>11454295
OH wait? does it have to do with the fact that complex exponent is needed to make the exp function a sine and cosine thing? and exp allows you to do algebra but sine and cos force ODE? can you show me some algebra comparing, im confused still and have low RAM

>>11453487
>>11453547
Thanks anons. I'll keep at it and hope to do manipulations in a concise manner.
>they start attacking like 5 different angles of the problem at the start and literally just get confused about what they've written down
I think this is my issue, so thanks for mentioning it.

What is [math]U_n[math] in group context?
What is [math]D_n[math] in group context?
What is [math]S_n[math] in group context?

What is [math]U_n[/math] in group context?
What is [math]D_n[/math] in group context?
What is [math]S_n[/math] in group context?

>>11454298
>1
time dilates
>2
no
>3
no. energy increases because kinetic energy increases with velocity
>4
sure
>5
kinda. c is the velocity of light by defn. it is emperically true that the speed of light is a constant to all observers. you cannot derive this fact within SR.
>6
photons move at c by definition.
>>11454300
>does it have to do with the fact that complex exponent is needed to make the exp function a sine and cosine thing
Complex exponents are just another way of representing a sinusoidal signal. The ODEs are converted to complex algebra by something called the Laplace transform.

>>11454298
>time as experienced by object
the object experiences objects going fast relative to it changing less per one unit of its change than it would if other objects were going slow relative to it, if thats what you mean

>2 and 3
can be composed into one statement. also, time DILATION? what a stupid phrase, its not showing which is relative to which. bugs me. and i think its inconsistent with length contractions basis/relativity, afaik. but anyway yes

>4
yes, as the object makes other objects move faster and faster relative to it, its energy begins to approach infinity before the other objects reach a certain speed

>5
infinite energy, massive... its the same. and yes

>6
the logical implication here is a bit weird relating to the laws of logic, but the universe has its own rules and defines operators around infinity in a strange (but eventually comprehensible) way. yes.

although i am curious about the photons having energy/momentum but not mass, can some one shed some light on this for me? (pun subconsciously intended)

>>11454326
actually anon, if youre reading this, only take my answer half seriously. its kind of confusing

>>11454323
>it is emperically true that the speed of light is a constant to all observers. you cannot derive this fact within SR.
So SR doesn't imply that kinetic energy approaches infinity at at some value? I'm not asking if that value is c, I use c to mean constant.

>>11454326
I don't understand why there are two distinct principles here. Isn't time dilation and length contraction the same thing, given the unary nature of spacetime?

>>11454346
well they sure as shit go together, i tell ya that. im just bitching about the nomenclature

but how are they equivalent? i mean in the sense that theyre simultaneous occurences and imply eachother, yes, but they effect different aspects of perception.. or do they? what do you mean by unary anyway

>>11454350
I mean that on Wikipedia for example, you have two pages
https://en.wikipedia.org/wiki/Time_dilation
https://en.wikipedia.org/wiki/Length_contraction
Which imply some qualitatively different concept, but they're really the same thing.

Imagine you're moving towards an object, I can either say "you're 1 meter away" or "you're 1 second away".

>>11454355
>>11454355
>>11454355
>>11454355
>>11454355
>>11454355

ignore schizophrenics
>>11454375
>>11454375
>>11454375
>>11454375

>>11450855
It is a different kinda burn. From a stove, you are burning the fuel, straight from the source in and engine, you need those expanding gasses to keep things moving. Anyhow, run an engine in a closed environment and you die. Unless you use a cleaner burning gas like propane or natural gas. You can probably get away with that.
>>11454361
>>11454361

>>11454159
>well if it approximates, that means its a function with an error bound.
That is a RIDICULOUS thing to say.
There is no error bound, it could be totally off and IT WILL BE.
Just name a model for fluid dynamics with an error bound, a rigorous error bound that guarantees that some result will be in some sense close to reality. Just the idea seems absurd, it would be the greatest discovery in the history of physics.

So, no, you are again wrong.
Consider a fluid moving closer and closer to the speed of light, the Newtonian model will be off arbitrarily badly.

>the natural numbers are a perfect model for cardinality of discrete objects
Discrete objects do not really exist in reality, well maybe discrete particles might exist, but imagine modelling logistics based on elementary particles...
So, for all practical purposes they don't exist.

>approximating an imperfect thing
Mathematics is pretending the imperfections aren't there. It is about idealization, but an idealized version of something can never be the object itself and if you go further into the unknowns of the object the further idealization and the object will grow apart.