user avatar
user avatar
user avatar
Simply Beautiful Art
  • Member for 5 years, 10 months
  • Last seen more than a week ago
Stats
2,389
reputation
52k
reached
21
answers
7
questions
Loading…
About

10/22/2020: Recently I've taken a liking to bracketing methods for root-finding and have even written my own code. It would seem most of the well-known bracketing methods suffer from myriad problems, including very suboptimal orders of convergence and insufficiently intelligent conditions for using bisection.

8/24/2019: I defined a neat ordinal collapsing function:

S(A) ⇔ ∀ f : sup A ↦ sup A, ∃ α ∈ A, ∀ η ∈ α (f(η) ∈ α)

B(α, κ, 0) = κ ∪ {0, K}

B(α, κ, n+1) = {γ + δ | γ, δ ∈ B(α, κ, n)}
               ∪ {Ψ_η(μ) | μ ∈ B(α, κ, n) ∧ η ∈ α ∩ B(α, κ, n)}

B(α, κ) = ⋃ {B(α, κ, n) | n ∈ N}

Ξ(α) = {κ, K ∈ K′ | κ ∉ B(α, κ) ∧ α ∈ cl(B(α, κ)) ∧ S(⋂ {Ξ(η) ∩ κ | η ∈ B(α, κ) ∩ α})}

Ψ_α = enum(Ξ(α))

C(α, κ, 0) = κ ∪ {0, K}

C(α, κ, n+1) = {γ + δ | γ, δ ∈ C(α, κ, n)}
               ∪ {ψ^η_ξ(μ) | μ, ξ, η ∈ C(α, κ, n) ∧ η ∈ α}

C(α, κ) = ⋃ {C(α, κ, n) | n ∈ N}

ψ^α_π = enum{κ, K ∈ Ξ(π) | κ ∉ C(α, κ) ∧ α ∈ cl(C(α, κ))}

where K is a weakly compact cardinal and K' is the (K+1)th hyper-Mahlo or alternatively, the smallest ordinal larger than K closed under γ ↦ M(γ), where M(γ) is the first γ-Mahlo. On its own this doesn't make a notation for large countable ordinals, but it can be used with another ordinal collapsing function for such purpose.

If you need me, you can find me here:

This is the realm of SBA

or on Discord.

My favorite topics include , , , , , , and on math.SE.

Some of my favorite posts:

Golf a number bigger than TREE(3)

Largest Number Printable

Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula

How to prove this $\pi$ formula?

Visually stunning math concepts which are easy to explain

1
gold badge
17
silver badges
35
bronze badges
75
Score
9
Posts
32
Posts %
74
Score
17
Posts
61
Posts %
67
Score
20
Posts
71
Posts %
60
Score
5
Posts
18
Posts %
35
Score
8
Posts
29
Posts %
10
Score
4
Posts
14
Posts %
Top posts
View all questions and answers
answer
43
Oct 20, 2017
question
22
Oct 17, 2017
question
18
Jan 1, 2018
question
16
Nov 23, 2017
question
13
Dec 4, 2017
question
11
Nov 23, 2017
answer
11
Dec 2, 2017
answer
8
May 16, 2019
answer
8
Nov 24, 2017