10,126 Pages

# Hyp cos

My favorite wikis
• ## Are these functions well-defined?

October 21, 2017 by Hyp cos

Here are two extensions of subcubic graph function, but I don't know whether they are well-defined.

Graph A is called a graph minor of graph B if A can be obtained from B by these 3 operations:

1. Delete an edge;
2. Delete an isolated vertex;
3. Edge contraction - for an edge ab, delete this edge, and replace vertex a and b by one vertex connecting to the edges which a or b formerly connect to. (If a and b are linked by another edge, it will become a self-loop after edge contraction.)

Graph A is called a topological minor of graph B if A can be obtained from B by these 3 operations:

1. Delete an edge;
2. Delete an isolated vertex;
3. Smoothing - for a (degree=2) vertex a which links to b and c, delete vertex a, and replace edge ab and ac by one edge bc.

For (unlabeled) subcubic graphs, graph mâ€¦

• ## General fundamental sequences for OCFs

October 2, 2017 by Hyp cos

In definitions of some ordinal collapsing functions (OCFs), there is always a series of $$C_n(\text{some arguments})$$ sets. $$C_0(\text{something})$$ may contains some argument in the "something", and may contains a "large" ordinal for collapsing; $$C_{n+1}(\text{something})$$ is usually obtained from $$C_n(\text{something})$$ by applying some operations with some limitations (avoiding "loop" definition); and $$C(\text{something})$$ is the union of all $$C_n(\text{something})$$. By this structure, we can define general fundamental sequences for OCFs.

Here this "general" definition works on the notation using a weakly compact cardinal. The definition of the ordinal notation is:

Let $$K$$ denote the weakly compact cardinal, $$\Omega_0=0$$ andâ€¦

September 14, 2017 by Hyp cos

Current approximations can be further improved, and this will take large amount of edits, so I start this discussion.

BEAF is not well-defined beyond tetrational arrays. More accurately, $$X\uparrow\uparrow A\&n$$ where $$A$$ is not a single number (e.g. $$X+1$$, $$2X$$, $$X^2$$, $$X^X$$, etc.) and beyond are not well-defined. But $$X\uparrow\uparrow m\&n$$ and $$X\uparrow\uparrow X\&n$$ are still considered as well-defined.

1. Approximations of numbers defined in ill-defined BEAF should be removed.
2. Approximations in ill-defined BEAF of any numbers should be removed.
• ## Growth Rate Îµ(0)âˆ’1

July 13, 2017 by Hyp cos

What's the growth rate of Goodstein function? $$\varepsilon_0$$?

No. Unlike Hydra function and Worm function, which are comparable to $$f_{\varepsilon_0}(n)$$, Goodstein function has some "cost" - $$G(2\uparrow\uparrow n)\approx f_{\varepsilon_0}(n)$$. By the definition of "functional approximation":

$$f\ge^*g$$ if there exists k such that for all n > 0, $$f(n+k)\ge g(n)$$

and

$$f\approx g$$ if $$f\ge^*g$$ and $$g\ge^*f$$

Goodstein function is not comparable to $$f_{\varepsilon_0}(n)$$. Currently we don't have a compact notation to express the growth rate of it. However, in this blog post I won't consider this thing. Instead, I'll consider what it "costs" and extend it. Goodstein has a 3-level cost under $$\varepsilon_0$$, since $$G(f_3(n))\appâ€¦ Read more > • ## Suggestions about Size Classes of Numbers July 1, 2017 by Hyp cos This kind of things have been discussed here. Currently, we have these size classes (in ascending order): 1. Class 0 (< 6) 2. Class 1 (6 ~ 106) 3. Numbers with 7 to 21 digits 4. Numbers with 22 to 100 digits 5. Numbers with 101 to 309 digits 6. Numbers with 309 to 4933 digits 7. Numbers with 4933 to 1000000 digits (#3 ~ #7 are also called "Class 2") 8. Class 3 (\(10^{10^6}$$ ~ $$10^{10^{10^6}}$$)
9. Class 4 ($$10^{10^{10^6}}$$ ~ $$10^{10^{10^{10^6}}}$$)
10. Class 5 ($$10^{10^{10^{10^6}}}$$ ~ $$10^{10^{10^{10^{10^6}}}}$$)
11. Exponentiation level ($$10^{10^{10^{10^{10^6}}}}$$ ~ $$10\uparrow\uparrow10$$)
12. Tetration level
13. Up-arrow notation level
14. Chained arrow notation level
15. 5-6 entry linear array notation level
16. 7+ entry linear array notation level
17. Two row array notation level
18. Planar array notatioâ€¦