Googology Wiki
Googology Wiki

Natsugoh Natsugoh 22 days ago
0

Discriminating whether a function of a real variable is an iterated function

Abstract

Suppose \(a\) be a sequence and there exists a map \(f\) such that \(a_n = f^n(a_0)\) for all \(n \in \mathbb{N}\). Then for any \(l \in \mathbb{N}\), \(a_m = a_n\) implies \(a_{l+m} = f^l(a_m) = f^l(a_n) = a_{l+n}\). We will generalize the idea to \(a\) its domain is an linearly ordered group.


  • 1 Main text
    • 1.1 Definition 1.1.
    • 1.2 Definition 1.2.
    • 1.3 Lemma 1.3.
    • 1.4 Proposition 1.4.
    • 1.5 Lemma 1.5.
    • 1.6 Corollary 1.6.
    • 1.7 Lemma 1.7.
    • 1.8 Lemma 1.8.
    • 1.9 Lemma 1.9.
    • 1.10 Proposition 1.10.
    • 1.11 Proposition 1.11.
    • 1.12 Example 1.12.
  • 2 Postscript
  • 3 References


\[ \newcommand{\ang}[1]{\langle #1 \rangle} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\1}{^{-1}} \newcommand{\e}{\epsilon} \newcommand{\oa}{\text{☆}} \newcommand{\P…



Read Full Post
Natsugoh Natsugoh 14 March
0

Solving the functional equation a◦f = g◦a for f and for g

Abstract

The functional equation \(a \circ f = g \circ a\) with \(f: X \to X\) and \(g: Y \to Y\) is an Abel equation when \(f = s\), the successor function. That is generalization of the iterate of \(g\).

We provide an expression for \(g\) in terms of \(a\) and \(f\). That is an "unitetate" of \(g\) when \(f = s\).


  • 1 Main text
    • 1.1 Lemma 1.1.
    • 1.2 Definition 1.2.
    • 1.3 Lemmma 1.3.
    • 1.4 Lemma 1.4.
    • 1.5 Proposition 1.5.
    • 1.6 Corollary 1.6.
    • 1.7 Example 1.7.


PDF: https://drive.google.com/file/d/1D1QFIPzsM0I8kiK8m9SLL4cFzhPlQQV6/view?usp=sharing

Old PDF: https://drive.google.com/file/d/1dgi1s4isEMUAmoQQcA22oszKpPGmc-F4/view?usp=sharing

\[ \newcommand{\U}[1]{\mathcal{U}_{#1}} \]




From here, let \(\sim_f\) denotes the equivalence kernel of a function \(f\); \(a \sim_f b :\if…






Read Full Post
Natsugoh Natsugoh 26 July 2023
0

Negative rank hyperoperations

Abstract

We construct the bi-infinite sequence of binary operations where the next element is an iteration of the previous one, such that \[ \dots , +_{r = 1}, \times_{r = 2}, \verb|^|_{r = 3}, \dots \]


  • 1 1. Previous research
    • 1.1 Hyperoperations (Goodstein's hyperoperations)
    • 1.2 Rubtsov and Trapmann's zeration
    • 1.3 Lower hyperoperations
    • 1.4 Tropical semiring
    • 1.5 Commutative hyperoperations (Bennett's hyperoperations)
    • 1.6 Pisa hyperoperations
    • 1.7 Hyperoperations we construct here
  • 2 2. The general solution of first order recurrence relations
    • 2.1 Lemma 2.1.
    • 2.2 Lemma 2.2.
    • 2.3 Lemma 2.3.
    • 2.4 Lemma 2.4.
    • 2.5 Definition 2.5.
    • 2.6 Lemma 2.6.
    • 2.7 Lemma 2.7.
  • 3 3. Definition of multiplicative and additive
    • 3.1 Definition 3.1.
    • 3.2 Definition 3.2.
    • 3.3 Definition 3.3.
    • 3.4 Definition 3.4.
  • 4 4. Properties…


Read Full Post
Natsugoh Natsugoh 26 July 2023
0

Test

\[ \newcommand{\a}{\alpha} \]




test

\(\a\)

no indent

indent1
indent1
indent1
indent2
indent1

no indent

{abc}

abcdef

textbf



\[ \label{eq1} \tag{eq1} a = b+c \]

\[ \begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \]

\[ \label{eq2} A = \begin{cases} B & \text{if $P$} \\ C & \text{if $Q$} \end{cases} \]

Read Full Post