Mathematics

3:00 PM

Wait, as for the case r=\sqrt{2} +\sqrt{3}. We can rearrange that polynomial and multiply both sides by r to get an identity.

Semiclassical

ugh, yes.

user107952

Maybe it really is all algebraic numbers, then.

Secret

but how do you get rid of the constants -10 and 1?

Semiclassical

And the false identification of measurement with 'telekinesis', so to speak. @secret

I'm not sure that's separate from what you said, though.

user107952

By writing x*10 as x+x+x+x+x+x+x+x+x+x.

Semiclassical

3:04 PM

'Consciousness causes collapse' interpretations irritate the hell out of me.

Secret

This case becomes a hot topic recently as a news programme in HK just used a toy turtle to debunk animal telepathy as either delussion or scam

Well... measurement on one of the entangled states will always project it into some product state (as long it is not a bell measurement), thus establishing the nonlocal correlation (under many quantum interpretations), thus it does contain the "nonlocal" element which can be easily mistaken

Semiclassical

Yeah. Nonlocal correlations are a thing, but nonlocal signaling isn't

Secret

@user107952 Ah, in that case, then you will take care of all the constants, and since every algebraic number is computable and uniquely identified by its minimal polynomial, then you have the identities, always

GFauxPas

ehem, Mathematics chat guys :P

Secret

though I must admit that x+x+x+x+x+x+x+x+x+x is very clumpsy

@GFauxPas Maths chat is currently in a superposition of physics and maths, deal with it : )

GFauxPas

3:06 PM

D:<

Secret

besides, there's a universal algebra discussion ongoing, so it is fine

Semiclassical

How one interprets those correlations is a question worth asking. But one certainly doesn't have nonlocal signalling.

@GFauxPas Eh, see today's XKCD which I linked earlier in the transcript.

GFauxPas

i was just joking

Semiclassical

heh

GFauxPas

are magnets that hard to understand tho

Semiclassical

3:08 PM

A bit, yes.

Or at least, they're not as basic as basic physics.

Secret

Acuriousmind and emilo pisanty in h bar can go forever on the intricates of each interpretation

But yeah, no communucation theorem said no, each party when they look at their own measurement outcomes, they just see random distributions, the correlation is only clear after they get together and compare the results

SteamyRoot

Wonder where gravity would fit on that graph.

Semiclassical

@SteamyRoot Eh, I'd classify that under general relativity.

Secret

unless, it is quantum gravity, which should be well inside the danger zone beyond quantum mechanics

Semiclassical

definitely

I think QFT would be on the danger boundary as well, but to the right of GR

Both require heavy lifting, but at least the math for GR is well-known and settled.

user107952

3:11 PM

Now, the only question is how many equations are needed to minimally generate the identities for particular algebraic numbers. My suspicion is that besides the commutative, associative, and distributive properties, you only need one identity, except for 0, where you need two.

Secret

I think one for each algebraic number is enough, since every algebraic number can be uniquely defined by its minimal polynomial

user107952

What about 0 itself?

GFauxPas

It's just Maxwell's eqns though @Semiclassical , that's Physics II

so I guess not Physics I and I guess you need Calc III

Secret

and your way of writing it ensures all constants ca be eliminated

As for zero, I am not sure how to calculate its minimal polynomial...

Semiclassical

It's not just Maxwell's eqns. It's also how materials give rise to magnetism.

user107952

3:13 PM

Well, for 0, it is just x.

Semiclassical

Maxwell's equations don't tell you why certain materials are paramagnetic versus diamagnetic, for instance.

GFauxPas

oh

Semiclassical

And Maxwell's equations as done in first year physics typically don't talk about the magnetic H-field, and if you're doing permanent magnets that's essential.

GFauxPas

I dont know about the magnetic H-field :(

so I guess I don't "get" magnets after all

thanks for ruining my day Semi

Semiclassical

Basically, it's the magnetic field due to external free currents

GFauxPas

3:15 PM

cf to currents running in a circuit?

Semiclassical

sure, that'd be one.

Secret

The notion of quantum fields permeating all of spacetime is very very dangerously close to the concept of chakras and other esoteric energy related topics since they both have this "everywhere and everything as part of one thing" property

I am still learning QFT thus I am not confident in not mixing them up (except as always, no nonlocal signalling)

Semiclassical

the magnetization field M, by contrast, is the field generated by the material in response to an external field.

And the magnetic field B is the combination of those two.

GFauxPas

we just worked with B

Secret

@user107952 In that case, I think then 0 being apparently an exception is because x+0=x and x0=0 has nothing to do with the algebraicness of 0 (those two identities are part of the rings, and actually, under the ring axioms, x0=0 is a theorem)

Semiclassical

3:17 PM

Yeah, that's typical.

Secret

similar to how 2 has $x+x=2x$

Semiclassical

It's also reasonable if you're not doing materials, since there's no material to respond to.

hence there's no magnetization field and B=H up to a proportionality constant

GFauxPas

ah

Semiclassical

(similar story with dielectric response to electric fields, though this usually gets touched on in intro physics w/r/t capacitors)

(i.e. why inserting a dielectric medium into a capacitor increases its capacitance)

user107952

I think you still do need x*0=0. Because there is no subtraction or additive inverse in the signature, and you need those to derive x*0=0 from the ring axioms.

Semiclassical

3:20 PM

intro physics typically excludes another bit as well. How does an electron respond to an applied magnetic field?

The intro physics answer would be to cite the Lorentz force, and that's certainly there.

Secret

@user107952 But you said you have negative elements, thus we should expect a variable (forgot the correct term in universal algebra) $x+(-x)=0$ (this is a ring axiom, the existence of additive inverse) that will give you the pathway to derive $x0=0$

Semiclassical

But the electron also has an intrinsic dipole moment. this doesn't matter for constant magnetic fields, but it matters a lot for magnetic field gradients.

and this loops back to magnets, because one thing that gets drilled into people in intro physics is that the Lorentz force can never do work.

and yet I can use one magnet to push another along a table.

the point is that the magnetic field is not doing work by virtue of the charges of the constituents in the material, but rather by virtue of the collective magnetic moment

Secret

(and really, you only need one additive inverse to complete the proof of the theorem x0=0, which is why it is very very hard to divide by zero (my personal project of a year ago))

Semiclassical

blah blah blah

user107952

@Secret Well, this is a pure universal algebra question, no existence statements allowed, only universally quantified equations. If there is a single identity that implies and is implied by both x+0=x and x*0=0 under the commutative, associative, and distributive properties, I would love to hear it.

Secret

3:27 PM

Let me just double check your structure, so you are actually not working with a ring because there is no - in the signature?

user107952

Yes, no subtraction or additive inverse.

Secret

Ah I see, in that case, yes, you will need x0=0 as a separate axiom. (And from the looks of it, you seemed to be working with a semiring)

Right, so in conclusion your structure will consists of the semiring axioms as the identities, plus one identity (the minimal polynomial of r, rearranged so that there are no constants and no - signs)

anything else should be theorems, including the fact that 2 generates the ideal of even numbers

user107952

This could make a nice undergraduate research project, my universal algebra and model theory questions.

I am only an undergraduate.

Secret

looks good. Model theory is also very cool as you seek structures that satisfy a set of logical statements under some formal language. That should train you more on seeking for identities in constructing structures

(I should also made a note on including minimal polynomials just in case I need to include algebraic numbers in any structures I am constructing that need them)

Zophikel

0

Q: Showing Abel Means of a Fourier Series Converge Uniformaly to $f$?

In the text "Fourier Analysis and Related Topics", i'm having trouble proving the following Theorem in $(3.5.5)$ utilizing Fourier Methods/Summability Methods. Also i'm not sure how to approach $(ii)$ with the use of Summability methods may I have some hints please ? $$\text{Theorem} \, (3.35)$$...

Transcript for

$Mathematics$ Mathematics