Mathematics - 2016-06-21 (page 1 of 3)

Darek

00:51

Hey. I have some doubts about answer by Hans, to this post: math.stackexchange.com/questions/67939/…

I see some assumptions: $R_n = 1$ gives $1 = \alpha, 1=1+\beta+\gamma n$

And it's straightforward to me.

But with this: $R_n = n$ I don't get the solution: $0 = \alpha, n = (n-1) + \beta + \gamma n$ and answer: $\beta = 1, \gamma = 0$

Why it is not answered like: $0 = \alpha, n = (n-1) + \beta + \gamma n => n = (\beta - 1) + (\gamma + 1)n$?

Then, the answer would be: $\alpha = 0, \beta = 1, \gamma = -1$?

Hmmm. OK, I think I just answered my own question. If I would group $(\gamma + 1)n$ I would solve this for all $\gamma != -1$

But, if I'm missing something, will be grateful for information.

hhh

01:23

Could someone check up the example in Wikipedia on Maximal Element? I think there is a mistake:

0

A: Explain Example on Maximal Element with sets

I think there must be a mistake in the Wikipedia article on maximal element. I demonstrate it below with Hasse diagrams. I use Hasse diagram to describe a poset $(S_2,\geq)$ with a set $$S_2=\{\{d\},\{d,o\},\{d,o,a\},\{d,a\},\{d,a,g\},\{g,o,a,d\}\}$$ where $\{g,o,a,d\}$ is the maximal while $...

where Wikipedia claimns in a set $S\\\{o\}$ that

1. {d,o} is minimal
2. {g,o,a,d} is maximal
3. {d,o,g} is nor maximal nor minimal
4. {o,a,f} is both minimal and maximal

...
I think there are three maximal elemens {d,o,g}, {g,o,a,d} and {o,a,f} while two minimal elements {d,o} and {o,a,f}.

Right?

arctic tern

{d,o,g} is contained in {g,o,a,d} isn't it?

hhh

@arctictern yes

Good point, then {d,o,g} is not maximal so two maximal elements left: {g,o,a,d} and {o,a,f}.

arctic tern

why is {o} even being depicted?

hhh

@arctictern it is not needed, I could remove it.

for illustrating a minimal element

I updated accordingly:

0

A: Explain Example on Maximal Element with sets

I think there must be a mistake in the Wikipedia article on maximal element. I demonstrate it below with Hasse diagrams. Trying to get attention in chat on this. I use Hasse diagram to describe a poset $(S_2,\geq)$ with a set $$S_2=\{\{d\},\{d,o\},\{d,o,a\},\{d,a\},\{d,a,g\},\{g,o,a,d\}\}$$ w...

Actually there is no mistake in Wikipedia, horray I understood it :D

I wish I understood why Tikz puts the labels so awkward :/

Prince M

02:20

Amateur hour question over here: Trying to show that if V is the curve parameterized by (x,y,z) = (t,t^n,t^m) where n and m are greater than 2 is an affine variety....

my claim is V = V(y-x^m,z-x^n).... subbing in that parameterization values shows V is a subset of this variety... any suggestions for going the other direction?

Abhishek Bhatia

Hi guys

Prince M

I swapped n and m in the parameterization...

Abhishek Bhatia

This might be a silly question, how to decide the number of variables in an equation. Say $f(x,y)=x+y$ vs. $z-x-y=0$. Would you say the equation has 2 or 3 variables?

Prince M

what is your intuition for both expressions?

How many independent variables does each one have

?

arctic tern

just count the variables

make things easier, not harder

Prince M

02:39

sometimes recognizing one of those terms as a variable isn't obvious...

Agawa001

@ForeverMozart congrats for mathematical maturity

arctic tern

03:04

@PrinceM you can take any equation, define a new variable to be equal to an expression that occurs in the equation, and then rewrite that equation as an equivalent one using the new variable. thus the number of variables in an equation is not an invariant with respect to this procedure. just count the number of variables you see written.

user19405892

03:20

hi

Let $\phi(m)$ denote the totient of $m$. Does there exist an infinite sequence of positive integers $a_1,a_2,\ldots$ such that $\phi(a_1),\phi(a_2),\ldots$ forms an increasing arithmetic sequence?

arctic tern

yes

user19405892

what is it

arctic tern

"it" is a singular noun, there are an infinitude of such sequences

do you know the formula for phi(n) in terms of factorization of n?

user19405892

yes, we break $n$ up into relatively prime parts

arctic tern

it is quite easy to make such sequences using it

for instance p,p^2,p^3,p^4,... if p>2 is prime

the totients would be (p-1), (p-1)p, (p-1)p^2, ...

user19405892

03:24

how is that arithmetic?

arctic tern

oh, you wanted arithmetic

well that certainly makes the question less trivial :P

TheGreatDuck

03:55

0

A: Removing dicontinuity from functions involving modulo?

The first form can be instead expressed as: $$f(x) - c\lfloor \frac {f(x)}c \rfloor$$ From the form of the rewritten expression, it is clear that subtracting $-c\lfloor \frac {f(x)}c \rfloor$ will render it continuous. The second form is a bit more interesting. f(x mod c) is a function where t...

Is my question and answer pair correct?

I am not quite sure if I wrote everything right in the formulae

Agawa001

taking an analytic look at

what is the expectability (probability) of coming across a gap g(n)=2 ?

twin primes conjecture says that 2 occurs infinitly, but the gap between two consecutive 2's is perceived to be increasingly outspread

but disorderly too

user116211

04:33

Hello, folks.

user116211

I was checking Limit of Function by Convergent Sequences‌;

user116211

There they stated at first about two metric spaces $M_1$ and $M_2$; but they didn't not talk about $M_2$ in any of the following discussion; what's the purpose of $M_2$ then? a typo?

user116211

Though I've not completed reading the sufficient condition yet, I'm sure they didn't mention $M_2$ there also ;/

arctic tern

@MAFIA36790 presumably f maps into M2

user116211

@arctictern ah! Thanks.

user116211

05:10

Also, why do we need two proofs? Why is the former necessary and the later sufficient?

Federico

06:59

'morning (or 'evening, depending on your timezone)

Tobias Kildetoft

Paper finally on arXiv. Only took my slightly more than 2 years to turn my dissertation into a paper.

Federico

short question: I "discovered" that curves with a linear variation of the curvature are called euler/cornu spirals. Mathworld gives a nice reference for polynomial curvature profiles (http://mathworld.wolfram.com/CornuSpiral.html).
Is there a name for curves that have hyperbolic profiles? i.e. k(t) = -1/t^2 following wolphram notation.

@TobiasKildetoft grats

user174558

07:17

How many people in the world read a paper? Is it worth the effort to write a paper?

user174558

@arctictern Yo, you and Mike make a good couple in this chat.

Tobias Kildetoft

@JasperLoy Maybe 5? 10 if I am lucky :)

Agawa001

it looks like the first gap ||g(n),g(n-1)||=1 occurs once, then ||g(n),g(n-1)||=2 is infinite and unpredicted too (called twins)

otherwise ||g(n),g(n-1)|| is progressing in non congruent rise

Landon Carter

08:17

Hey can anyone just clear a doubt please? Suppose G is a topological group (compact, abelian) and S is NOT a dense subgroup of G. Then what does it mean? Is it that there exists an open subgroup of G with null intersection with S or does it mean that there exists just an open subset of G with null intersection with S?

Tobias Kildetoft

@LandonCarter It just means that it is not dense

there could never be an open subgroup with empty intersection with the given subgroup

Landon Carter

ah yeah, as the identity is there

Tobias Kildetoft

precisely

Landon Carter

so dense subgroup is just a dense set which happens to be a subgroup, right?

it does not have any notable property as such?

Tobias Kildetoft

right

Landon Carter

08:24

Hey thanks Tobias!

Tobias Kildetoft

the "tricky" part about topological groups is the compatibility between the operation and the topology. But once this is there, it is inherited by all subgroups.

Landon Carter

okay.

lol this just made the problem harder :P

but thanks for clearing the doubt. it was stupid and i should have figured it out on my own.

Axoren

08:46

@hhh $\underset{\{\text{Bottom Stuff}\}}{\text{Top Stuff}}$ should work

Tobias Kildetoft

09:28

@JasperLoy Well, now the number of people (apart from myself) who have read it is at least up to 1 (or at least who have noticed it), since I just got an email about the paper.

Rajesh Dachiraju

09:59

1

Q: What is the mathematics behind the random experiment which produces the data with this strange property?

I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let us call this a collection $D$. I have partitioned this randomly into $4$ equal parts $D_1,D_2,D_...

user 1618033

@pleasedeleteme Yes, it is worth to share knowledge and to have your work recognized in the mathematical community.

Say, you work for years, publish nothing, and then someone else makes the same discoveries that are published by him/her. How about your work, your struggle?

Here is a point: perhaps some are pushed to publish to exist (it's said that you cease to exist if you don't publish as a professor, say). This is not exactly my perception on publishing, I mean things should come naturally without forcing yourself to produce work only to exist.

Well, publishing is just a simple effect of a good researching process (at least here). I don't embrace the idea of putting ppl under pressure to publish only to exist (preferring anytime the good quality of the published papers than the quantity)

Cool stuff

Let $a,b,c>0$ such that $ab+bc+ca=1$. Show that $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{‌b+c}+\frac{ca}{c+a}$$

user 1618033

10:52

Time for some tutoring.

BBL

user243301

11:57

@TobiasKildetoft I was try think more about it. I don't know, if it is possible do substitutions $(M'+1)^{\delta'}$ and $(M'+1)^{\delta}$ or $(M')^\eta$ and $(M')^{\eta'}$ leaving the argument of the $\sigma(n)$ as a positive integer, to obtain by means experiements with a computer a new identity that capture the notion to be a Mersenne prime. It isn't neccesary a response of this last message. Good afternoon and thanks for your attention.

Added: I have no good abilities with computers and I believe that there is no a theory of how solve a general identity involving arithmetic functions.

George

12:35

hi

Taha Akbari

12:55

hi

Anderson Felipe Viveiros

Hello guys! Do anybody know what is the usual translation to Portuguese for "path metric"? (Given a metric space $(M,d)$, $d$ is called a "path metric" if, given any pair $(x,y)\in M\times M$, there exists a path joining $x$ to $y$ with $d(x,y)=L(\gamma)$, where $L(\gamma)$ is the length of $\gamma$...)

hhh

13:22

@Axoren It looks a bit better:

Hiroto Takahashi

13:45

Hi, someone can give an example about this: Let $\varphi: G \longrightarrow G'$ be a homomorphis group. If $N$ is a normal subgroup of $G$ then not necessarily $\varphi(N)$ is a normal subgroup of $G'$?

Mike Miller

@TobiasKildetoft Congrats on the paper. For once I know some of the words in your abstract, since they were in a friend of mine's advancement to candidacy talk.

Semiclassical

morning chat

Leaky Nun

@sem good morning

Hiroto Takahashi

Good afternoon from here!

Semiclassical

@hiroto maybe start by finding some G' with a non-normal subgroup?

I forget what the simplest example of that is

Hiroto Takahashi

13:59

$\{{1, (23)}\}$ is a non normal subgroup from $S_{3}$

Semiclassical

right

so now you want to find a homomorphism from $G$ into $S_3$ which maps a normal subgroup to $\{1,(23)\}\in S_3$

looks like N.S. posted an answer to your question, so hopefully you can use the example you gave to make that concrete

Balarka Sen

14:17

@Semiclassical So I learnt a bit about the Schroedinger's equation.

Semiclassical

oh?

Samuel Yusim

I feel like I just wasted two weeks studying something that turned out to be completely pointless due to an oversight

Leaky Nun

@sem I would like to ask you a question.

Semiclassical

Sure.

Leaky Nun

Is there anything that you don't know?

Semiclassical

14:20

yes. many, many, many (ad infinitum) things

Balarka Sen

@Semiclassical It's more or less motivated from the classical equation of a one dimensional wave. But I feel like I don't really understand it.

How does it connect to the quantum part of the story?

Semiclassical

I'm not sure I follow you. The Schrodinger equation may have certain classical motivations, but it is essentially quantum.

otherwise what would $\hbar$ be doing in there?

Balarka Sen

More specifically, it's written that it's consistent with the Heisenberg's principle. How?

Semiclassical

ahh

Balarka Sen

It feels like I am just assuming particles are waves and moving on. It's not clear where the quantum-ness comes in, to me.

Semiclassical

14:24

first off, for a long discussion of the Heisenberg uncertainty principle by experts, see here

Balarka Sen

thanks!

Semiclassical

np

one thing it mentions in there is a result which I'll sketch

Balarka Sen

great, I am interested in hearing.

Semiclassical

one thing that the wave function $\psi$ used in Schrodinger's equation lets you produce are various average values

user 1618033

Tiresome and long tutoring sessions without breaks (here).

Balarka Sen

14:27

Right, I "know" $|\psi|^2$ measures the probability density (integrate over certain regions and you get the probability of the particle in your system being there).

user 1618033

Let $a,b,c>0$ such that $ab+bc+ca=1$. Show that $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{‌‌b+c}+\frac{ca}{c+a}$$

Balarka Sen

But tbh it's not clear to me why's that the case.

user 1618033

Who is done with my beautiful inequality?

Semiclassical

right. i'll skip over the 'why' for now since that's it's own can of worms (Born's rule)

Balarka Sen

alright, fair enough

user 1618033

14:28

@BalarkaSen I've never seen you finishing inequalities. But maybe inequalities are not important to you.

Semiclassical

you can also use it to compute average values of things like momentum.

Balarka Sen

@user1618033 Because I don't want to.

Leaky Nun

@user1618033 Haven't got a clue. Care to give me one?

user 1618033

@LeakyNun Sure. I like your interest for stuff.

Semiclassical

i had an interest in inequalities at one point. right now they're not my thing, and i'd have to do some practice with them to regain that.

Leaky Nun

14:29

@user1618033 so, what is the clue?

user 1618033

@LeakyNun move the terms from the right-hand side to the left-hand side excepting $\sqrt{3}$. Then you can get something more beautiful with a bit of manipulation.

Semiclassical

more precisely, the expected value of an operator $\hat{A}$ is given by $$\langle \hat{A}\rangle = \int_{-\infty}^{\infty} \psi(x)^* \hat{A} \psi(x)\,dx$$

Balarka Sen

Hmm, yes, I have seen that while skimming an introductory book I have on quantum mechanics (which doesn't provide the right physical motivations tho).

I believe it.

user 1618033

@BalarkaSen integrals, series, limits, inequalities are all not interesting to you (as you often said). On the other hand, I think it's only a matter of interest, because if you really wanted, you'd do anything from this area.

Semiclassical

now, from there you can define the standard deviation of $\hat{A}$ as $(\Delta \hat{A})^2=\langle \hat{A}^2\rangle - \langle \hat{A}\rangle^2$

user 1618033

14:32

@BalarkaSen within yourself lies the power of Ramanujan.

(my opinion)

Semiclassical

and there's a result which can be proven (probably by Cauchy-Schwartz)

Balarka Sen

@Semiclassical I agree.

user 1618033

@Semiclassical the area of inequalities is pretty hard, or better say exceptionally hard. One needs much much training, years in a row.

Semiclassical

namely, $$(\Delta A)(\Delta B)\geq \frac{1}{2}\langle (\hat{A}\hat{B}-\hat{B}\hat{A})\rangle$$

I know I did a proof of that in a quantum course, but I don't remember the method so I'm somewhat guessing

Balarka Sen

I am believing you. I'll write it down and try it later if I find the punchline of the discussion interesting :)

Semiclassical

14:36

mmkay

well, the punchline is this. take A=x and B=p

s.harp

@Semiclassical that formula is not right, the right side need not be real

user 1618033

LeakyNun has the profile of a warrior in math, he's afraid of nothing and curious to take any challenge. This is what I like to see in mathematics.

Semiclassical

woops, you're right

should've been the absolute value of the quantity on the RHS

now, in position space, the operator $p$ has the representation $p=-i\hbar \frac{d}{dx}$

you've probably seen that in the Schrodinger equation?

Balarka Sen

Yeah, I have seen that.

user 1618033

@LeakyNun no idea who you are but it's easy to see to me you have much (math) power within yourself. ;)

Leaky Nun

14:39

@user1618033 Well, can I have another clue xd

Balarka Sen

@user1618 I don't want to be Ramanujan. I want to have fun and I have fun doing the mathematics I like and integrals, series etc is not the kind of mathematics I like.

user 1618033

@LeakyNun make use of $ab+bc+ca=1$

Semiclassical

okay. what you can show, if you act $x\hat{p}-\hat{p}x$ on a generic function $y$, is that $$(x\hat{p}-\hat{p}x)y = i\hbar y$$

Balarka Sen

Mhm, I see where this is going.

Semiclassical

which means that, when you compute the expected value of those operators, you just get $\langle i\hbar \rangle = i\hbar$

Balarka Sen

14:41

Right.

Semiclassical

which, upon accounting for the absolute value, means that the result in this case is $\Delta x\cdot \Delta p \geq \frac{\hbar}{2}$

Leaky Nun

@user1618033 Oh my God.

Balarka Sen

Weird how this works out. I half-follow the manipulation, but it's surprising to me because I have no intuition why it should.

Leaky Nun

Eureka.

user 1618033

@LeakyNun :D

Semiclassical

14:43

well, to me the intuition is simply that if two operators commute, then I can measure both of them independently

that's actually just a linear algebra statement: If two matrices commute, then they're simultanteously diagnolizable.

Leaky Nun

@user1618033 $a$, $b$, and $c$ are positive, right?

user 1618033

@LeakyNun Right.

Balarka Sen

hm

Semiclassical

but x and p don't commute.

Balarka Sen

right

Semiclassical

14:44

so i should expect that any measurement of one should influence subsequent measurement of the other---in other words, measuring one should create uncertainty for the other.

Balarka Sen

interesting way to put it

Semiclassical

Now, a warning: Even to the extent that you believe the formalism, one can reasonably wonder to what extent that is equivalent to Heisenberg's uncertainty principle

Leaky Nun

@Semiclassical Quick question: how was $h$ measured?

Semiclassical

because one has to go from the abstract meaning of standard deviation to an experimental one, and that's not entirely obvious

the article i linked has a discussion about that after eq (11)-(12) (the first being the result I quoted)

@leakynun hmmm

Balarka Sen

@LeakyNun probably looking at the spectrum of something?

Semiclassical

14:47

I'm forgetting the history, tbh

Balarka Sen

*probably by

Semiclassical

I mean, Planck was able to estimate its value from black-body radiation (sayeth Wikipedia)

Obliv

if I'm trying to show a symmetry group is homomorphic to a dihedral group do i do this set up : $\varphi(r^ns^m) = \varphi(r^n)\circ \varphi(s^m)$ ?

Semiclassical

to within 1.2%, evidently, which is pretty damn good

Balarka Sen

it's unclear to me how to measure energy radiating from a blackbody in an efficient way

Semiclassical

14:49

Wikipedia has a whole section on the value's determination, actually: en.wikipedia.org/wiki/Planck_constant#Determination

Balarka Sen

@Semiclassical what do you mean?

i do a bunch of tests, look at a bunch of values, take it's average. that's the expectation - that's what i am guessing.

Semiclassical

well, see the article i linked. it talks about it

Balarka Sen

standard deviation measures how much the real data set deviates from the expectation.

that's precisely uncertainty.

Semiclassical

well, for one, the definition of standard deviation doesn't take into account how many times you measured

whereas the experimental standard deviation is necessarily dependent on it

Leaky Nun

@Semiclassical Never mind, I don't even understand a word.

@Semiclassical Another question: so we have an uncertainty of, say, position, right?

So we have a standard deviation?

user 1618033

14:52

@BalarkaSen not Ramanujan, but like Ramanujan. To be honest, I don't think anyone would refuse his mathematical skills ... (if they could be received as a gift)

Leaky Nun

But then most of the time it would be in the middle...

Semiclassical

within a given experiment, sure

do you know where the middle is in an actual experiment, though?

Leaky Nun

Then it is not very uncertain after all

Balarka Sen

uh? if I measure $n$ times, I get the results $x_1, \cdots, x_n$. standard deviation takes $(x_i - \bar{x})^2$, sums over and divides by $n$.

that takes care of how many times you measured (i.e., $n$), no?

@LeakyNun can you guarantee 100% it'd be in the middle?

Leaky Nun

@BalarkaSen Nope.

Balarka Sen

14:54

no. you can just say it with some positive amount of accuracy. 70%, 80%.

that's uncertainty.

Semiclassical

well, consider a coin toss. if i assign coin values of 0 and 1 for tails and heads, then i do expect that as I take more and more trials, the average result is 1/2

Leaky Nun

So it's really accuracy rather than certainty...

Semiclassical

but do i expect in an actual experiment to have each average value be 1/2?

Obliv

what's the difference between proving something is a left or right group action? Don't I just do $g_1*(g_2*a) = (g_1*g_2) * a$?

Balarka Sen

@LeakyNun certainty = 100% accuracy. that's what the word "certainty" means.

Obliv

14:56

and $1 * a = a$

Leaky Nun

@BalarkaSen But the uncertainty principle suggests that there is a boundary to accuracy which cannot be surpassed...

Semiclassical

following up on the above question, the standard deviation in that case is given by 1/2. do i therefore expect that, in a given experiment, the emperical standard deviation will be exactly 1/2?

Balarka Sen

@LeakyNun yes.

so?

it's not clear to me what you are asking anymore.

Leaky Nun

Never mind.

Obliv

@semiclassical do you know much about circuits? At least the concept behind them?

Semiclassical

15:02

At the level of Kirchoff's rules and things like resistors/batteries/capacitors, sure. Not so much with things like transistors.

Mike Miller

electric goes in computer comes out

4

simple

Obliv

I'm just curious about the idea behind them. It's true that when a circuit isn't connected (positive and negative terminal on each end, right?) there is no current& voltage right?

Balarka Sen

lol

Obliv

lol mike

Samuel Yusim

@Obliv for a right group action you instead want to show $(a * g_1) * g_2 = a * (g_1*g_2)$. If you were to (for some reason) write this on the left it would look like $g_2 *' (g_1 *' a) = (g_1 *' g_2) *' a$, if that makes sense

Semiclassical

15:03

no applied voltage means no resulting current, sure.

Obliv

@samuelYusim ah okay. thanks

@semiC so as soon as you connect the circuit to a terminal, current begins to flow, right?

Semiclassical

if that results in a voltage difference between applied to the system, then yes.

Samuel Yusim

also I like to use a different symbol for the group multiplication and the action operation. usually I use a period for the action, so I'd have $g_1.a$ or something.

but notation is whatever

Obliv

My question is, how does the positive terminal know when the negative terminal is connected to the circuit? I would ask on main but I'm afraid it's a silly question and I haven't studied e&m yet >_<

Leaky Nun

@Obliv The negative terminal likes to give off electron and the positive terminal likes to receive electron

Mike Miller

15:06

the physicists have won

Semiclassical

suppose i had two volumes of water at different heights. if i don't connect them, water won't flow. if i do, it will.

Leaky Nun

@Obliv So when they are connected, the negative terminal uses the circuit to release its electrons

Mike Miller

TERRORISTS WIN

►

Semiclassical

admittedly, this is somewhat a matter of microscopics description, and that's a bit of a pain to think through at times.

macroscopic stuff like currents and voltages etc. are much simpler

Obliv

hmm

Leaky Nun

15:08

@Obliv while the positive terminal uses the circuit to receive the electrons

Semiclassical

the other thing that complicates all of this: ultimately, electrons are things that are properly described by quantum mechanics

Samuel Yusim

this reminds me that I should read about series-parallel graphs later

Semiclassical

so, for instance, what it means for an electron to conduct in a metal is a more complicated topic than one might realize.

classically, one uses the Drude model for that. but a modern treatment of it is in terms of band theory.

Obliv

@leakyNun I know that much. I'm wondering about the part where you put the circuit together. Is it an instantaneous resulting in current? The water tanks are physically constrained by a boundary, which you remove so that they can move. I suppose it's similar

Balarka Sen

@MikeMiller nice shot

Leaky Nun

15:10

@Obliv What may help your understanding, is if you understand that the electrons flow very slow in a current

as slow as honey

Obliv

@leakynun o_o really? I thought they moved near the speed of light or something

Leaky Nun

They move around their nucleus maybe near the speed of light

Balarka Sen

in vacuum.

Leaky Nun

That I am not certain

Balarka Sen

but while moving through things, not so much

Semiclassical

15:11

Speed of electricity

The word electricity refers generally to the movement of electrons (or other charge carriers) through a conductor in the presence of potential and an electric field. The speed of this flow has multiple meanings. In everyday electrical and electronic devices, the signals or energy travel as electromagnetic waves typically on the order of 50%–99% of the speed of light, while the electrons themselves move (drift) much more slowly. == Electromagnetic waves == The speed at which energy or signals travel down a cable is actually the speed of the electromagnetic wave, not the movement of electr...

Leaky Nun

> the signals or energy travel as electromagnetic waves typically on the order of 50%–99% of the speed of light, while the electrons themselves move (drift) much more slowly.

:o

Semiclassical

see also drift velocity, which includes some typical numbers for copper wire

Samuel Yusim

The Drude model: electrons bounce around like pinballs. The Darude model: youtube.com/watch?v=2HQaBWziYvY

Obliv

i knew what that link was before I clicked it @samuel

can't fool me

Samuel Yusim

as you should

Leaky Nun

15:14

@obl an analogy is likes people in the tunnel pushing each other

the people walk very slowly

Semiclassical

one point they make in that article is that, if you work with alternating current, then the current is oscillating i.e. always changing direction

Leaky Nun

but the push travels quite fast

Obliv

oh that's a cool analogy @leaky

Balarka Sen

i don't get physics. people explain florescence in cathode ray tube by electron moving close to speed of light. but i'd think they emit photons or something, and i have no idea how they would do it.

Semiclassical

which means that any given electron won't have a drift velocity, since it'll be moving forward just as much as backwards on average.

Leaky Nun

15:15

@Semiclassical but do have a root-mean-square dfirt velocity

drift*

Semiclassical

sure.

Leaky Nun

I was told, that if you could define energy, you would have won the Nobel prize

price*

Balarka Sen

trying to think of a pun related to nobel price

Obliv

A noble price of only 19.99

@leaky A lot of people would be winning that nobel prize, then. Perhaps the first to earn it might be Leibniz who defined work.

Leaky Nun

@Obliv what is work?

Obliv

15:22

$W = F\Delta{x}$

Leaky Nun

That's how you calculate work

Samuel Yusim

what's a definition

Balarka Sen

that's also the definition.

Leaky Nun

$W=\int F\ \mathrm dx$

Obliv

lol

Balarka Sen

15:23

what's a

Leaky Nun

what's

Semiclassical

what is love

Leaky Nun

Actually, $W=\int\vec{F}\ \mathrm d\vec{x}$

Samuel Yusim

well at least I know what work isn't: what I'm doing right now

Obliv

It's just a transferring of potential to kinetic energy via a force through a distance. i.e changing something's position in a field or giving something velocity without any opposing forces.

Balarka Sen

15:24

not gonna click on that

@SamuelYusim hehehe

Samuel Yusim

my favourite would be if the what is love link went to darude sandstorm

Leaky Nun

@Obliv Alright, then what is energy?

Obliv

@leakynun there's a more complicated definition that I don't know about, related to noether's theorem. The classical definition is just potential or kinetic energy which I'm sure you already know.

Balarka Sen

if someone asks what is a tree, the best thing to do is to take him to a tree.

Obliv

What if they're blind though

Leaky Nun

15:28

@Obliv so energy is potential energy

Obliv

or kinetic.

Semiclassical

The conserved quantity corresponding to the continuous symmetry of laws of motion under infinitesimal time evolution, as per Noether's theorem.

Obliv

You know the definition of acceleration right? $a = \frac{dv}{dt}$ rewrite as $a = \frac{dv}{dx}*\frac{dx}{dt}$ so then you have $\int a dx = \int v dv$ so $\int a dx = \frac{v^2}{2}$ multiply by mass and you have a definition of energy. $W = m\Delta\frac{v^2}{2}$ @leaky

Leaky Nun

Well, I have no idea who I got that from, so let's just ignore it

@Obliv that's work

Mike Miller

@SamuelY f work tho

Obliv

15:33

it's the same thing pretty much

$W = F\Delta x = ma\Delta x = m\Delta\frac{v^2}{2}$ it's kinetic energy at two points (final - initial)

Leaky Nun

energy is without the delta

Obliv

yeah if you only care about the current velocity. This is the process of gaining or losing energy.

Leaky Nun

What amazes me is that energy is always $mv^2/2$ no matter the uniformity of the acceleration

Samuel Yusim

@Mike I gotta get these algorithms written and analyzed ):

Balarka Sen

@LeakyNun you shouldn't expect it to depend on the acceleration, though, because of conservation of energy.

Mike Miller

15:40

I gotta do paperwork

Leaky Nun

@BalarkaSen sure

Samuel Yusim

that's probably worse

Mike Miller

I hate paperwork so much

Mike Miller

15:55

I realized today I don't actually know how to prove that surfaces aren't the homotopy type of a nontrivial wedge sum. You'd want to show that the fundamental group isn't a free product, but how would you ever do that?

Obliv

are all uncountable sets the same size, just as all countable sets are the same size?

Mike Miller

No.

Balarka Sen

@MikeMiller Surfaces have nonzero H_2, wedge of circles don't?

Mike Miller

$\mathcal P(X)$ always has strictly greater cardinality than $X$.

@Balarka I never said wedge of circles.

Leaky Nun

@Obliv There is an infinite amount of sizes of uncountable sets

Obliv

15:57

what is $\mathcal{P}$

Balarka Sen

Oh.

Right, that's going to be hard. Hmm.

Mike Miller

Power set.

I wanted to say something like "Well, it's not a free product of a free group and something else" (true) "and deficiency is additive under free product" (no idea), so the deficiency is too high for it to be a free product. But I get nothing about additivity of deficiency when googling.

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$Mathematics$ Mathematics