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…
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…
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…
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\[ \label{eq1} \tag{eq1} a = b+c \]
\[ \begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \]
\[ \label{eq2} A = \begin{cases} B & \text{if $P$} \\ C & \text{if $Q$} \end{cases} \]