Hello Everyone...

Hello @JohnRennie Sir. I am waiting for you since 3 hours. Need your knowledge.

@123 If you want the help of JohnRennie then visit Problem-Solving-Strategies chat room, he remains very active there.

Thanks

hi

@LuckyChouhan now that would be rude :P

@LuckyChouhan what subjects r u interested in

@RyderRude math and programming(specially c++ and python). What about you?

Hello @RyderRude

@LuckyChouhan physics, math and philosophy mainly

Why negative sign in change in potential energy? If we take care of direction between force and displacement equation gives us valid results.

@123 hello

@LuckyChouhan nice

@RyderRude can you tell me something new in math?

I mean like a concept or a problem which you found interesting

@LuckyChouhan bolbteppa told me about this thing called the Langlad's program

it's not a well defined problem. more like figuring out a big zigsaw puzzle

numberphile has a video on it

it is mysterious unkown relations between broad branches of math

According to Wikipedia In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and consequential conjectures about connections between number theory and geometry.

@LuckyChouhan i found algebraic topology from Nakahara interesting

the book covers it superficially tho. i havent checked detailed books yet

@RyderRude nice, I haven't studied topology. I like number theory and analysis.

i am interested in number theory a lot..

but i havent studied it

what book would u recommend

So what are you reading or learning nowadays.

@RyderRude I have a little problem for you...

@LuckyChouhan i have taken a long break from studying...

@LuckyChouhan great

@LuckyChouhan i cant focus for long...but i hope to get back to studying now

If $p \equiv 3\pmod{4}$ then $ \left( \frac{p-1}{2} \right)! \equiv \pm 1 \pmod{p} $ where $p$ is a prime.

@RyderRude ADHD??

@LuckyChouhan maybe ... idk

@RyderRude Why? Are you a university student?

@LuckyChouhan no, i just study as a hobby

thats y im not consistent

@LuckyChouhan do we hav to prove this

@RyderRude do you watch a lots of short, reels or tiktok. I think you have decreased your attention span.

@RyderRude yeah, or just convince yourself that is true.

@LuckyChouhan yeah.. i do sometimes watch shorts in quick succession

@LuckyChouhan okay

@RyderRude So you have a job right?

@LuckyChouhan yeah...

@RyderRude do you like english literature?

@LuckyChouhan idk how to approach this

@LuckyChouhan i have read Harry Potter and some other novels

but i havent been reading novels for long

@RyderRude It is related to the solution of $x^2 \equiv -1 \pmod{p}$

Hint: Try proving it using Wilson's theorem,

@LuckyChouhan oh

@LuckyChouhan idk this theorem :P

lemme search

@RyderRude great, do you know about The Fault in Our Stars.

@RyderRude sure, very intriguing theorem :(

great theorem

@RyderRude does mathjax work in your device in chat room?

@LuckyChouhan yeah. i havent seen it though

@LuckyChouhan yeah

@RyderRude Oh, It is very famous novel by John Green. Elder brothe at Vlogbrothere, Nerfighteria. haha

@RyderRude then what novel you liked much so far?

@LuckyChouhan nice

@LuckyChouhan Harry Potter and the Prisoner of Azakaban

i also read philosophy sometimes, but not whole books

i am interested in the nature of consciousness somewhat

@RyderRude I haven't read any Harry Potter novels. But yeah Harry Potter is very famous.

@RyderRude great, then would you like to tell me about it?

@LuckyChouhan u will love it

Then please...

@LuckyChouhan there are several schools of thought : illusionism, idealism, materialism

an interesting idealism philosophy that i was interested in was this : physics describes what u can measure about an entity, and not the entity's internal essence

Metaphysics

let's take it to the age old idea : u can never know if other people have subjective experiences and what those expetiences r like

but u can sure measure electric signals of their brain

yeah,

so what the philosophy does is to extrapolate this idea to every entity

like, an electron. what u can measure is its energy, momentum and stuff

but there r internal qualities to the electront too

I'm not getting the point connected to brain signals and entity.

and these qualities of particles come together and ur subjective experience emerges

@LuckyChouhan an entity is any physical thing : a brain, a chair, a wall, a bird

or a particle

@RyderRude Yeah, so you're saying that behavior of people changes according to their past experience and what they experience.

@RyderRude yeah,

@LuckyChouhan no, that would be psychology :P

im bad at explaining...

@RyderRude so here we're going to talk about soul?

@LuckyChouhan i prefer to call it "internal qualities"

@RyderRude No no don't worry, it happens because it was not planned that we would be discussing.

i dont believe in an indivisible soul inside a body.. i believe the internal TV of the mind is emergent out of particles

@RyderRude yeah, then what after it...

ok so take a brain (an entity) : neuroscience can make statements about what u can measure about the brain. electric signals and stuff. but we know that those signals correspond to something inside us that is the experience

so let's extrapolate this idea to every entity. sciences make sttements about the measurements of things. so science doesnt capture the complete nature of reality

@RyderRude can you give me bird eye view of what you're trying to say? Then frog eye view.

so we r hypothesizing that, just like the brain has an internal immesurable experience, an eelctron does too

Are you talking about Cognition?

@LuckyChouhan im talking about qualia..

have u read about the hard problem of consciousness

maybe u r lacking the context of the discussion

No, but I found this article by Quanta Magzine quantamagazine.org/what-is-the-nature-of-consciousness-20230531 should I read it?

also, im not an expert on the subject.. so im maybe explaining poorly

@LuckyChouhan idk about that. i will get the Ted Talk of Chalmers

@RyderRude excatly,

I'm gonna watch this

also this en.m.wikipedia.org/wiki/Hard_problem_of_consciousness

@LuckyChouhan ok :)

Sure, I'll ping you after watching the vide and reading that wiki article.

Thank you :)

@123 hi. sorry for the late reply.. the work done by an internal force is $-\nabla V$. this is because V is defined by the equation : $F=-\frac{dV}{dx}$

intuitively, when the internal force does positive work.. the body gains kinetic energy, so the potential energy must decrease.. hence the change in potential energy is of the opposite sign of the work done by the internal force

@RyderRude now I got little bit. We want to understand that what is there inside us which makes us conscious. This makes the fancy term nature of consciousness clear to me.

Because of consciousness we're different from not only from each other but from every other species on earth. Am I correct?

@LuckyChouhan yes!!

@LuckyChouhan this is called the "Problem of other minds". u cant even know for sure if any humna else except u is consciousness, let alone know if a bat is conscious

like, does a virus or a spider have an inner experience or is it just a bot

and u cant know if i have an inner experience. maybe everyone except u is a bot

this is what i meant by "immesaurable properties of the brain" above. these r the properties that science doesnt deal with

and hr own consciousness is the evidence that these properties exist at least within u, and at most in everything

the philosophy is wrote about above says that these properties exist in everhthing ... and ur inner TV is emergent out of these properties of the particles in ur brain

this is a philosophy of idealism and panpsychism. some other philosophies r illusionism and materialism

en.m.wikipedia.org/wiki/Illusionism_(philosophy) this is on illusionism

some others r dualism, dual aspect monism, physicalism.. i would say the philosophy i gave is a dual aspect monism

sorry i linked the wrong illusionism. this is the one i meant en.m.wikipedia.org/w/…

@naturallyInconsistent Hi!

Would you have time later on to discuss stuff regarding scattering theory ?

Non-relativistic*

whatcha wanna know about that?

I'm making a program that displays the GP (geographical position) of a Solar System body on a map of the Earth. And I need some feedback.

The GP of a celestial body is the longitude & latitude where the body is at the zenith at a given moment of time. Thus the latitude equals the body's declination, and the local hour angle of the body is zero, so the local sidereal time equals the body's right ascension.

My diagram uses a contour plot to show the altitude angles of the body, in 5° steps. It also plots contour bands for the Sun, showing the "twilight zones". Civil twilight begins/ends when the centre of the Sun is on the horizon (0° altitude). Nautical twilight begins/ends at -6°, astronomical twilight at -12°, and full night is when the Sun is below -18°.

I've been tweaking the this diagram for several hours, so it's hard for me to tell how to improve it. ;) Here's an example:

@naturallyInconsistent Are you there?

no, it is 5am

oh

Are the twilight zones & night region dark enough, or too dark? If I make it much darker, it can be hard to see the coloured altitude contours.

it looks good

Is the altitudes palette ok? I want to make it easy to follow a contour, so I think I need a broad range of bright colours. But some of those yellowish lines can be hard to see, especially in the dark zone.

yes, the yellows are bad

"mathematical logic" is so frustrating :(

@naturallyInconsistent Thanks. I'll remove the yellows.

Since these maps would usually be used at night, I was thinking if it makes more sense to invert day v.s. night

And maybe the yellows can be made bright and put where the darks are

@Obliv what happened now?

Is the background map ok? I can make it a it brighter or darker. But it's layered on top of the twilight zones, acting as an alpha (transparency) mask. So if I make it too pale, the dark regions get more intense. I can't directly control the alpha of the contour lines or bands, I can only fiddle with their palettes.

@naturallyInconsistent Trying to understand a couple theorems and am not making much progress. I wanted to understand the diagonal lemma since it's used in the proofs of tarski's undefinability theorem & godel's incompleteness theorems. I understand the proofs of uncountability of certain sets but I wanted to learn about these other meta theorems.

@naturallyInconsistent That wouldn't really help, in general. It'd work for that diagram, but not if the relative position of the Sun is different.

@PM2Ring it looks ok to meow

I wanna understand these limitations of n-th order formal theories

All the resources I've found about this subject of mathematical logic are extremely unappealing to me, but I don't want to just hand wave and move on :\

I found it much easier to learn each specific case, i.e. the Cantor's diagonal argument, the Gödel's incompleteness, Tarski's, and Turing's Halting problem, than the meta argument.

I also find it weird that one can prove anything at all about the limitations of the theory in which you are working in.

Unless the proof is just inherently contradictions that one inevitably gets if they do try to reason completeness/decidability/consistency

@Obliv Well, they're all just variations on the theme of Cantor's original diagonal argument about the reals.

@PM2Ring Yeah that's the vibe I'm getting. So maybe if I just study that proof more, the others will kind of just click

I just haven't found any good resources on the other theorems that aren't super dense and hard to read.

The Turing Halting theorem is probably equally easy to understand. But I've never bothered reading a formal proof of it.

Guys, given $U = a_0\mathbb{I} + i\mathbf{a} \cdot \mathbf{\sigma}, a_0, a_i \in \mathbb{C}, \text{ for all }i$ and $\sigma_i$ are the Pauli matrices, is it true that $UU^{\dagger} = U^{\dagger}U $ ?

Thanks for your feedback, @naturallyInconsistent. FWIW, that diagram is for when Apophis reaches its maximum brightness (at it's next close approach), magnitude : 2.816. It occurs about an hour and 15 minutes before the closest approach.

So if you want to see an asteroid with your naked eyes, you can use that diagram to figure out where to go. ;)

I get $$UU^{\dagger} = |a_0|^2\mathbb{I} + i\sigma_k(a_0a_k-a_0a_k^{\ast}) +i\epsilon_{ijk}\sigma_ka_ia_j^{\ast}= U^{\dagger}U$$

@Claudio Finite dimensional. This is a theorem.

but my notes say that the 2nd term $i\sigma_k(...)$ should have opposite signs so that when you sum these results the middle term cancels out

@Obliv Isnt there a good book on it? Escher Bach Gödel?

@Claudio wait, is $a_i\in\mathbb C$?

yep it's complex valued

Like, we usually want all 4 $a_\mu\in\mathbb R$

No my professor wants to show that upon imposing unitarity

you get the conditions: $$a_0,a_i \in \mathbb{R}$$

and $$a_0^2 + \sum_i a_i^2 = 1 $$

Ah, yes, you get that.

can you open this drive.google.com/file/d/1lpelfuD1UzYVLgk74yOOk9GQzG4o912k/…

@naturallyInconsistent he gets a minus sign out of nowhere hahaha

@Claudio no, the notes are correct.

but there's no minus sign I swear

hmm, let meow see

When you get the $i\sigma_k$ term to have the same sign, do you get $i\varepsilon_{ijk}\sigma_k$ term to have a different sign?

nope the same sign for the last term

wait

now that I think about it

no it should be the same

$$ \sigma \cdot \mathbf{a} \sigma \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{b}+i\sigma\cdot(\mathbf{a} \times \mathbf{b})$$

so whether you consider $(\sigma \cdot \mathbf{a})(\sigma \cdot \mathbf{a}^{\ast})$ or $(\sigma \cdot \mathbf{a}^{\ast})(\sigma \cdot \mathbf{a})$

it should stay the same

$$\begin{align}[a_i\sigma_i,a_j^*\sigma_j]&=i\varepsilon_{ijk}\sigma_ka_ia_j^*=i\varepsilon_{ijk}\sigma_k\dfrac{a_ia_j^*-a_j^*a_i}2\\ [a_i^*\sigma_i,a_j\sigma_j]&=i\varepsilon_{ijk}\sigma_ka_i^*a_j=i\varepsilon_{ijk}\sigma_k\dfrac{a_i^*a_j-a_ja_i^*}2\end {align}$$

@Claudio hello, the cross product term obviously changes sign

wait let me see

yep you're right

dang

Anyway, the argument is that we want $U^\dagger U=\mathbb I$ and $UU^\dagger=\mathbb I$

subtracting the two, the $\varepsilon_{ijk}$ terms are the only thing left, and thus is equal to zero.

I see I see

I got stuck in the computations unfortunately

This leaves the $|a_0|^2+\sum_i|a_i|^2=1$ and the other term, and then that other term has to zero

Then you can use the argument in the notes

slightly different, but no big deal

yep I'll go with this

anyway, imma snooze

@naturallyInconsistent thanks :P

good night