/sci/ - Science & Math

>>9505009
* The book "Topoi" by Goldblatt is simple and very logic oriented (but odd in terms of the order of things)-
* That "Rosetta Stone" article by Beaz mentiones those judgement rules, but doesn't go in-depth at all.
But maybe the references in those two.
* There is "toposes theories triples" by Baar, which might be relevant.
Maybe there's hints in the study guide for logic in "Teach Yourself Logic" by Smith.
* There's abook "Non-classical logic" by Priest which I had in my hands at one point and which may cover that logic (although hardly the category part)
* There's also "Categorical Logic and Type Theory" by Jacobs, which is hard and it's been a while and I don't know how much linear logic is in it.

Not if you keep all other axioms, see

http://planetmath.org/definitionofvectorspaceneedsnocommutativity

Now you could invest some time and think about which axioms to drop.
I'd be interested to see what happens when you drop
(a+b)· v = a · v + b · v

Not if you keep all other axioms, see

http://planetmath.org/definitionofvectorspaceneedsnocommutativity

Now you could invest some time and thing about which axioms to drop.
It would probably make sense to drop

(a+b)· v = a · v + b · v

or stop working with fields.

I'd like to characterize maps [math] (a,b,c) \mapsto M(x,y)z [/math], that are linear in z, with
[math] M(x,y)x=y [/math]

More concretely:
Let a,b be vectors in a vector space like R^n.
I'd like to characterize matrices, with arbitrary dependency on a and b, so that
[math] Ma=b [/math]

----

Background/Example:
I came across this situation with the angular velocity tensor
https://en.wikipedia.org/wiki/Angular_velocity#Angular_velocity_tensor
There you have a curve r (a map from [math] \mathbb R [/math] to [math] \mathbb R^3 [/math]) and if its such that it rotates around the origin, then
[math] \frac{d}{dt} r = \omega \times r [/math]
where the vector [math] \omega [/math] is a function of [math] r [/math] and [math] \frac{d}{dt} r [/math] and the map [math] b \mapsto \omega \times b [/math] is linear.

What I go so far:
Why this works here, algebraically speaking, can be traced back to the vector triple product
https://en.wikipedia.org/wiki/Triple_product#Vector_triple_product
You have
[math] a\times(b\times c) = (a\cdot c)\,b - (a\cdot b)\,c [/math]
so
[math] a\times(a\times b) = (a\cdot b)\,a - (a\cdot a)\,b [/math]
so
[math] M(a,b)\,a = b [/math]
with
[math] M(a,b) = \frac {1 } { ||a|| } \left[ ( \frac{a} {||a||} \times b) \times + ( \frac {a} {||a||} \cdot b)\, 1_3 \right] [/math]

This map [math] (a,b) \mapsto M(a,b) [/math] actually is linear in b and "|a|^{-1}-linear" in a.

Can all matrices with all those properties be written like that?
Can all matrices with [math] Ma=b [/math] be composed with a cross product and a unity matrix?

All subfields of mathematics are listed here

http://www.ams.org/msc/pdfs/classifications2010.pdf

A comprehensive modern list and review of introduction to logic (including set theory and such) is covered in

http://www.logicmatters.net/resources/pdfs/TeachYourselfLogic2017.pdf

I have a project where I try to map out the relationships between different topics in math, and graph it, here

https://graph.axiomsofchoice.org/?to=set_theory

but I update it rarely.

I'm not gonna TeX a whole bunch just to have this end up being a troll thread and OP throwing random QFT textbook formulas around.

>Basically, how do I compute it?
To outline the general principle, one starts with looking at what's approachable, namely integrals like

[math] I_a:=\int_{-\infty}^\infty { e}^{-a \phi^2/2} d \phi =(2\pi)^{1/2}a^{-1/2} [/math]

and

[math] \int_{-\infty}^\infty e^{-a \phi^2/2+i\,\phi J} d\phi =I_a e^{ - J^2 / (2a)} [/math]

A long list of higher dimensional generalizations of this "Gaussian with an extra source term" can be found here
https://en.wikipedia.org/wiki/Common_integrals_in_quantum_field_theory

Now looking up your definition of the bracket
[math] \langle 0 | ... | 0 \rangle [/math]
For four phi's as in your example (me dropping the arguments of the phi's here as well a indicess), you have something like

[math] \langle | \phi \phi \phi \phi | \rangle = \int (\phi \phi \phi \phi) e^L D\phi[/math]

with L the Lagranguan you posted and Lagrangian D\phi some measure over a field/function space, and for which you at least postulate some properties like D const=0.

Your L is not approachable, it has the form (neglecting indicess ... of course you have sever)

[math] L = (\phi, (A+m^2) \phi) + B(\phi)[/math]
and then
[math] e^L = e^{ (\phi, (A+m^2) \phi) } e^{B(\phi)} [/math]

As

[math] \frac{d}{dJ} e^{x\phi} = \phi e^{J\phi} [/math]

there is some expandable function f so that the above can be written as

[math] e^L = f(\frac{d}{dJ} ) \phi e^{J\phi} e^{ (\phi, (A+m^2) \phi) } [/math]

The idea is that we can now pull out f of the (path) integral over it and the integrant is the free one. We can now use the "easy" integrals I mentioned in the beginning. This still involved inverting A, which is in general (and in your case) a differential operator.
The solution will the e to the power of a quadratic term with the propagator as matrix (the "1/(2a)" above). You need that matrix now. Any more concrete questions?

I saw this and it was really disappointing. Herzog is a big name director, but really alsoan old men who is completely distant and unknowning of the topic and maybe here doesn't even realize that all his interviewees are simply spouting pure Silicon Valley ideology. If you read the tech section of a basic newspaper in the last 2 years, you'll not even get a new perspective out of this.

>>8428451
https://en.wikipedia.org/wiki/Particle_physics_and_representation_theory

I'm sure hard math is everywhere. It's more a matter of how much the people working on a field comprehend and thus use. Look at this shit
https://en.wikipedia.org/wiki/Standard_RAID_levels#RAID_6

>>8429184
If you want to be the one creating new stuff, you better not only look at the things that are already there and used in physics.