/sci/ - Science & Math

>>12696094
A claim equivalent to

[math]\log(H_n) \cdot {\mathrm e}^{H_n} + H_n \ge \sigma(n) [/math]

for all n>0,
where [math]H_n=\sum_{k=1}^n\frac{1}{k}[/math] and where [math]\sigma(n) [/math] is the sum of divisors of [math]n[/math].

I see Joel David Hamkins (set theorist and guy with the highest MO rating) just started a series on the philosphy of mathematics.
Might be basic, though, but I see he already has one or two talks of his stuff online too.

https://www.youtube.com/channel/UCeMZeXYIhdxnQvZP360uoBg

>>12241785
[math] {\mathrm e}^{x+d} = c\,{\mathrm e}^x [/math]

I'm been playing with lattices and here's a quest 4u
(haven't solved it yet, but I suppose the number is divisible by 3 or something)

We consider the commutative semirings (R, ·, +, 0, 1) which are idempotent (x·x=x), simple (x+1=1).
(The idempotent relation makes this lattice-operation like and the simple condition makes this bounded-like (1 behaves like the maximum). The distributive properties of the semiring make those into the distributive lattices of that kind)

Let's pin it down to a semiring of exactly five elements, with different elements 0, P, Q, R, 1.
The identity properties for the semiring for 0 and 1, and the simple property, imply most of the multiplication and addition table.
The question is what simultaneous assignments for the six remaining operations
P·Q, P·R, R·Q,
P+Q, P+R, R+Q
is possible.

Notes:
I've looked at the situation with four elements 0, P, Q, 1 already.
You can define an order (x<=y) := (x·y=x), which is equivalent to (x<=y) := (x+y=y).
Then if you assume 0<P,Q<1, the multiplication table is fixed. This will also work for the situation with five elements. But there's also "diamonds" of lattices and such.

Relevant links
https://en.wikipedia.org/wiki/Lattice_(order)#Bounded_lattice
https://en.wikipedia.org/wiki/Semiring

The teachers are unlikely to understand statistics.
To be fair, there's things like
https://en.wikipedia.org/wiki/James%E2%80%93Stein_estimator
that I also have a hard time grokking fully or keep in mind.
Also - unpopular opinions time - I'm not sure if decision theory is mathematics in the same way other fields are. Using Bayes rule for scenarios for which (the existence of) join probabilities aren't known and then bouncing off your experiences of natural world data is very different kind of working experience than other subjects people write papers about. To be clear, at the same time you got a lot of formal analysis and theoretical insights in the field of course, just like in any.

For some fun popular examples people dabbling into the field,
https://en.wikipedia.org/wiki/German_tank_problem
https://en.wikipedia.org/wiki/Secretary_problem

>>9667718
>I can't believe no one has ever looked at these things.

Might be that you can just re-normalize the basis factors to get a simple L^2 product and that's why. But for kernels like that, check out Fredholm anything and kernel anything.
https://en.wikipedia.org/wiki/Fredholm_theory
I mean it can't be an accident you guys also chose K as letter to represent it. Or in this direction
https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_integral_operator

What I'm also immediately reminded of is considering a positive differential operator, e.g. some Laplacian, Fourier trasnforming the vectors/functions and thus getting a function kernel. Or generally any positive observable in quantum mechanics

>>9667689
https://en.wikipedia.org/wiki/Pontryagin_duality
?

maybe Halmos biography
https://www.amazon.com/I-Want-be-Mathematician-Automathography/dp/0387960783

maybe Lockarts essay
https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

here a more logical proof book
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995

There are also books like
https://en.wikipedia.org/wiki/What_Is_Mathematics%3F

I will reluctantly also point you to Taos career section on his blog
https://terrytao.wordpress.com/career-advice/

also
>experience that comes from studying and doing exercises
:P

Mathematical pratice hardly starts from axioms in the logical sense, unless you mean writing down definitions in full.
And yes I'd argue you apply "mathematical thought in everyday life", but more the logic side of things, not so much anything that has to do with numbers per se.