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00:00
Continuity ?
Not even with that.
Isn't codomain(f)=codomain(g)=R ?
Consider e^x.
That's increasing everywhere. But what is it's range?
$\Bbb R^+_*$
Right. So there's an entire half of the real line it'll never visit.
Can you find a decreasing example that never takes positive values?
00:04
But $\Bbb R^+_*\not=\Bbb R$
$h(x)=-e^x$
Right. Does this ever intersect e^x?
@Semiclassical Yup, but it isn't defined on all $\Bbb R$
Eh, only if you insist on more than just increasing/decreasing
@Semiclassical Yes,
I meant to do.
It's not a subjection to R, but it is a well-defined map
If you insist on that, then sure
00:12
Now this is theoretically right :P
Thinking about the Math $\not=$ Proving the Math $\not=$ Notating the Math.
my god don't put that last bit there
Better ? @MikeMiller
god forbid math without words! Matteus the third chapter
It's like different subjects.
(Thinking about the Math $=$ Proving the Math $=$ Notating the Math) $\implies$ Perfect Symmetry.
But that happens ever so rarely.
Anyone ? :D
hi
im back
00:25
WB
okay done, I think, unless anyone has suggested improvements
behold the graph of |W(z)|, Re(W(z)), and Im(W(z))
@dhmo
the Im one surprised me
it's a spiral staircase
Well, note that the Re plot isn't so different than the Abs plot
So if you were to plot arg(W_0), I think you'd get something rather like Im but even more of a staircase.
For instance, in your Im plot you've got contour lines that start on the branch cut and move away.
I suspect that in the Arg plot, you wouldn't even see that: all lines would emanate from the origin.
complex geometry is so interesting
:o
SPIDER! KILL IT! KILL IT WITH FIRE! KILL IT WITH THE FIRE OF MULTIPLICATION BY ZERO!!!!
interesting. i'm a little surprised to see those lines curling back onto the negative real axis.
00:40
:( no it's a spiral staircase again, just a melted one
it's a spider
lol
it's rather similar to the Im plot, as I indicated it would.
TheGreatDuck stare at the Re(W) one it will calm you down
so explain to me how you knew that, please
enlighten me :)
lol
i am already calm
just kidding with you guys
:)
Nothing too complicated. The Re plot looked similar to the Abs plot, and the Im/Arg contour lines will be orthogonal to these.
so, therefore, the Im and Arg plots should also be similar.
00:43
I got a brand new hat
why isn't it changing...
I hear why Im and Re are orthogonal, why are mod and Arg orthogonal?
I'm not sure how long it takes to update that.
there we go
anyone want some yummy bacon!
@GFauxPas If you take $\log$ of a complex number, then its real and imaginary parts will be the modulus and the argument respectively of said complex number.
right
00:46
so it's just the orthogonality of real and imaginary contour lines again.
no he means why is the graph of modulo = constant orthogonal everywhere to the graph of arg = constant
I mean
...and that's what I just explained. \log z=|z|+i\arg(z)
@GFauxPas argument is theta. modulo is radius. Let me ask you. Is polar coordinates an orthogonal coordinate system?
I hope so
I've been assuming it is
00:47
locally, yes.
@Semiclassical the two questions are equivalent.
in the sense of differentials
exactly
the bit about 'locally' is just that $e_\theta$ means a different thing when $\theta=0$ than when $\theta=\pi/2$
00:49
|z| = c and arg(z) = d is the same as r = c and theta = d
is that a unit vector?
yeah.
okay, back later.
thanks Semiclassical and Mr Duck!
I need to go grocery shoppingt anyway
@Semiclassical you're thing about logartihm returning mod and arg in real and imaginary made no sense.
as I could derive a function where they are both added together in imaginary
:)
it's really a question of looking at derivatives and polar coordinates
0
Q: brun's method and primitive roots

alex morfilet be $N(x) = \sum\limits_{d|p - 1} {\mu \left( d \right)} {F_x}(d)$ that $p$ is a prime number and ${F_x}(d) = \frac{{{x^2}}}{{4d}} + O(x{p^{\frac{1}{2}}})$ that $x+1=g(p)$ and $g(p)$ is the least primitive root modulo $p$. Applying Brun's method to $N(x)$ in conjunction with ${F_x}(d)$ in orde...

please look at this question maybe you can solve it
00:56
lol
you've only reached 275 people?
i thought Euclid's geometric ideas have reached millions?
(if not billions)
im not enough great
but you're Euclid!
01:19
Hello everyone.
01:33
@TheGreatDuck Analytic functions are conformal i.e. preserve angles. So if the Re and Im contours of $f(z)$ are orthogonal, then so are the Re and Im contours of $\log f(z)$.
hi @JessyCat. Done with finals?
oooh
log is conformal?
away from the branch cut, sure.
it's analytic on a cut domain.
(never heard that term before to be honest)
conformal mappings are fun stuff.
(never done complex analysis either or real analysis)
I just noticed it was a trivial thing to show is true.
:p
or rather
it's equivalent to polar coordinates. :)
01:35
Conformal mappings give a way to solve Laplace's equation in 2D for a lot of different boundary conditions.
i don't know what laplace's equation is either
XD
i know laplace transform though
what?
please don't tell me it's something obvious...
$\partial_{xx}u+\partial_{yy}u=0$ in 2D.
oh
that's neat
we didn't learn PDE's in any of my classes beyond the obvious examples in calculus
01:37
Have you done any physics?
at least other than basic examples
I'm a math/computer science major
It shows up when you ask what the electric potential will be in a region that doesn't contain any charge.
though I am taking physics as my science elective this coming semester
im sure if I stared at that a while I'd understand
01:38
heh.
by computer science I mean programming
Were you the robot from yesterday @TheGreatDuck?
not circuitry
@JackDon lolwat?
Now you tell me you are a math/computer science major, it seems more reasonable
It's simple enough once you do intro E&M stuff. Not anything too complicated.
01:39
ah fair enough
it probably is simple
It also shows up when you talk about streamflows. In that vein you get pictures like this
to be honest, there have been many times where I'd claim "I know nothing about that" and then after looking a few minutes the old high school archives kick in and I remember a bunch of things I didn't even realize I knew. XD
though I don't think I've ever learned of electric potential. I would assume it is the potential for a jump of static electricity?
Well, have you heard of potential energy?
it sounds familiar but I cannot seem to remember exactly what it was
that's like the energy stored by height (as example) right?
Right. Gravitational potential energy in that case.
01:42
like if you hold a bowling ball it has potential energy and it becomes kinetic energy when you drop it?
It trades from one kind of energy to the other.
oh I see. It's the potential force that an electrical discharge could produce.
In electrostatics, you also talk about potential energy. Namely, the potential energy of two charges will be $k\frac{q_1q_2}{r}$
where $k$ is a dimensionful constant, $q_1,q_2$ are the charges of the two particles, and $r$ is their separation.
01:44
ah
so it's inversely proportional to their distance?
like gravity
Exactly.
makes sense I suppose
Similarly, one has Coulomb's force law $F=kq_1q_2/r^2$.
which is again similar to gravity.
in a way, all those electrical attractions have to eventually add up to the force we call gravity
Um, no. Electric force != gravitational force.
01:45
well do particles have gravity between them as well?
I did not know that
It's just that the individual attraction between two particles is tiny tiny tiny
I thought particles only have magnetic and kinetic forces between them
i thought gravity was the collective magnetism of all the atoms in a substance
But once you have a whole heck of a lot of tiny particles, the net effect is huge.
no.
01:46
fair enough.
electric and magnetic forces are distinct from gravitational forces.
for one, the former tend to be a lot stronger than the latter. but you have both attraction and repulsion with electric force, whereas you only have attraction for gravitation.
explains why my thing never worked quite right.
I tried to make a 2D electron/proton/neutron simulator type thingy. Never actually wound up with 2D atoms.
aaanyways
granted, that was probably me just weird anyway.
:p
In electrodynamics, it's typical to focus on one of the particles as a probe, and all the others as just an overall background.
So you'd have a charge q, subject to a whole bunch of other charges.
And you'll note that both the force law and the electric potential energy scale with the strength of the charge.
Hence it's typical to define the electric field E by E=F/q, and the electric potential by V=U/q.
It's the force per unit charge, as they would say.
01:50
sounds like relativity in the non-Einstein sense
is intelligence constant?
everything is moving around the particle rather than the other way around
@Null go stay up 4 nights in a row while not eating. You'll have your answer.
I think you're generalizing in a vague and hasty way.
01:51
Quite.
The point is just that you could put two different charges, one with twice the charge of the other, into the same environment.
fair enough
intelligent would be, to not do what you just suggested
One would experience twice the electric force of the other.
how do you prove that all groups of order 9 are abelian? Other than bashing all the $9!^9$ possibilities
The electric field is defined so as to get rid of that charge-dependence.
01:52
@Null true, but I'm saying one's intelligence varies over time. If you become inebriated your intelligence decreased.
but, what is intelligence, that has to be defined prior
@Sophie somehow prove all groups are abelian by definition somehow?
because the mere ability to solve problems is imo not it
You want to care about the strengths of the charges which produce the background, but not the strength of the charge that you're using to probe it. That's why electric potential / electric fields are nice.
Anyways. I'm getting distracted.
@TheGreatDuck if a person ever proves something it is "by definition somehow"
01:54
@Null idk, but I know that if it has anything to do with the performance of the brain then there are many ways to decrease it through actions accidental or otherwise.
@Sophie Theorems are 'just' applied definitions, after all
@Sophie no need to be snarky. I'm just saying that maybe groups are defined as abelian.
NO
theorems are applied axioms in conjunction with definitions
mmh, then a way to increase intelligence should be goal number 1
@TheGreatDuck I incidentally just read that being shelled with depleted uranium lowers your intelligence test scores, and that's scary
01:55
@Sophie :D
For the purposes of what I'm saying, the distinction between axiom and definition isn't particularly interesting.
The group axiom comprise the definition of a group, after all.
can we treat axioms and definitions as the same mathematical object?
not neccessarily
think of geometry
I don't really see a distinction there.
euclid had five postulates or axioms
but he also had 20 some definitions of various terms
01:57
Yes, and those axioms define Euclidean geometry.
yes but the definitions of the terms do not define euclidean geometry
maybe from a strictly logical point of view the distinction doesn't matter much, but from a philosophical point of view it does
Eh. That strikes me as nitpicking.
imo: definitions:"what we talk about", axioms:"what we assume"
the reason for that is that in my eyes the terms only serve as an extra tool in expression
01:58
I guess the way I'd put it: Not all definitions are axioms, but all axioms are definitions.
whereas you can still do everything in geometry without the definitions
you can still make a 3 sided figure without naming it a triangle
you'd just have to be incredibly wordy all the time
sure. but you can't do euclidean geometry without having something that's equivalent to "parallel means this"
no
parallel is not even in the postulates

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