/sci/ - Science & Math

>>12522366
i think it's fair to say that mandelbrot set and related topics are about "single variable quadratic polynomials over C" and I suspect you could find some open questions there

>>12409199
A, B connected, this implies [math]A \times B[/math] connected
the function [math]A \times B \ni (a, b) \mapsto ab \in AB[/math] is continuous
image of connected set via a continuous function is connected
end of proof

[math]\mathbb{Q}[/math] and [math]\mathbb{Q}^2 [/math] are homeomorphic as topological spaces (with the topologies induced from [math]\mathbb{R}[/math] and [math]\mathbb{R}^2 [/math], respectively).

>>11917144
to check that the problem is well-defined, you just need to check that [math]f(x,y) = f(y,x)[/math] and [math]f(f(x,y),z) = f(x, f(y,z))[/math]
>>11917115
>>11917156
Indeed, there is a way to solve this without a computer.
Let [math]F(x_1, \dots x_n)[/math] be the result of the operation described in the problem, on numbers [math]x_1, \dots x_n[/math]. Let's also denote by [math]\sigma_n, \sigma_{n-1}[/math] the nth and (n-1)th symmetric polynomials of those variables; i.e. [math]\sigma_n[/math] is the product of all x_n's, [math]\sigma_{n-1}[/math] is the symmetric polynomial of degree n-1.
Then, it is easily verified that for any [math]n \geq 2[/math],
[eqn]F(x_1 - 1, \dots x_{n} - 1) = - \frac{(n+1)\sigma_n - 2\sigma_{n-1}}{(n-1)\sigma_n - 2\sigma_{n-1}} [/eqn]
The problem asks to evaluate [math]F(2, 3, \dots 1000)[/math], which the interested reader can easily do using the formula above.

>>11863323
Removed the background for the sake of it.