In the Mandelbrot set, all points of the main carodioid are asymptotic (that is, the iterations of c^2 + c approach a constant). In contrast, it seems that all bulbs have a periodicity greater than 1, that is, the iterations settle into a cycle with a certain period. There are several questions to be asked […]

I’ve recently discovered that modifying the standard Newton-Raphson iteration by “squashing” $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is $$N_f (t) = t – \tanh \left( \frac{f (t)}{\dot{f} (t)} \right)$$ results in much larger (Fatou) convergence domains, also known as (immediate) basins of attraction. The convergence theorems of the […]

Intereting Posts

Prove $\sqrt{x^2+yz+2}+\sqrt{y^2+zx+2}+\sqrt{z^2+xy+2}\ge 6$, given $x+y+z=3$ and $x,y,z\ge0$
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How can I prove the maximum number of edges?
Asymptotic divisor function / primorials
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Useless math that become useful
Maximum area of a isosceles triangle in a circle with a radius r
Example of a cogroup in $\mathsf{hTop}_{\bullet}$ which is not a suspension
Is the Fibonacci constant $0.11235813213455…$ a normal number?
Is there a name for the curve $t \mapsto (t,t^2,t^3)$?
Proof with 3D vectors
Game theory – self study
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$
Divergence of a vector tensor product/outer product: $ u \bullet \nabla u = \nabla \bullet (u \otimes u) $
Why is ring addition commutative?