1: Definition. Examples
- Page ID
- 204819
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A time scale is an arbitrary nonempty closed subset of the real numbers.
We will denote a time scale by the symbol \(\mathbb{T}\). We suppose that a time scale \(\mathbb{T}\) has the topology that inherits from the real numbers with the standard topology.
The sets \([-1,4], \mathbb{R}, \mathbb{Z}, \mathbb{N}\),
\[\left\{\begin{array}{lllllllll}
-2, & -1, & -\dfrac{1}{2}, & 0, & \dfrac{1}{4}, & \dfrac{1}{3}, & 2, & 3, & 6 \end{array}\right\}, \notag \]
and
\[ \{1\} \bigcup\left\{\dfrac{1}{n}+1\right\}_{n \in \mathbb{N}} \bigcup\{3\} \bigcup\left\{\dfrac{4}{n^2}+3\right\} \bigcup\{9\} \bigcup\left\{\dfrac{7}{n^4}+9\right\}_{n \in \mathbb{N}}\notag \]
are time scales.
The sets \((-3,7),[0,5),(1,7]\) and \(\left\{\dfrac{5}{n}+3\right\}_{n \in \mathbb{N}}\) are not time scales.
Let \(a, b>0\). The sets
\[P_{a, b}=\bigcup_{k=0}^{\infty}[k(a+b), k(a+b)+a] \notag \]
are time scales.
The set of harmonic numbers
\[\begin{aligned}
H_0 & =0 \\
H_n & =\sum_{k=1}^n \dfrac{1}{k}, \quad n \in \mathbb{N}
\end{aligned}\notag \]
is a time scale.
Let \(\left\{\alpha_n\right\}_{n \in \mathbb{N}_0}\) be a sequence of real numbers with \(\alpha_n>0, n \in \mathbb{N}_0\). Define
\[t_n=\sum_{k=0}^{n-1} \alpha_k, \quad n \in \mathbb{N} .\notag \]
Then the set
\[\mathbb{T}=\left\{t_n: n \in \mathbb{N}\right\}\notag \]
is a time scale.
(Cantor Set) Consider the interval \(K_0=[0,1]\). We obtain a subset \(K_1\) of \(K_0\) by removing the open "middle third" of \(K_0\), i.e., the open interval \(\left(\dfrac{1}{3}, \dfrac{2}{3}\right)\) from \(K_0\). The set \(K_2\) is obtained by removing the two open middle thirds of \(K_1\), i.e., the two open inetrvals \(\left(\dfrac{1}{9}, \dfrac{2}{9}\right)\) and \(\left(\dfrac{7}{9}, \dfrac{8}{9}\right)\) from \(K_1\). Proceeding in this manner, we obtain a sequence \(\left\{K_n\right\}_{n \in \mathbb{N}_0}\) of sub sets of the interval [0,1]. In Fig. 0.1 are shown the sets \(K_0, K_1, K_2, K_3\) and so forth. The Cantor set \(C\) is now defined as follows
\[C=\bigcap_{n=0}^{\infty} K_n .\notag \]
The Cantor set is a time scale.
Any its element \(x\) can be represented in its ternary expansion as follows
\[x=\sum_{j=1}^{\infty} \dfrac{a_j}{3^j}, \quad \text { where } \quad a_j \in\{1,2,3\}, \quad j \in \mathbb{N} .\notag \]
This expansion is unique unless \(x\) is of the form \(p 3^{-k}\) for some integers \(p\) and \(k\). In this case, \(x\) has two expansions
1. \(a_j=0\) for \(j>k\).
2. \(a_j=2\) for \(j>k\).
Assume that \(p\) is not divisible by 3 . One of these expansions will have \(a_k=1\) and the other wiill \(a_k=0\) or \(a_k=2\). We have that
\[a_1=1 \quad \text { if and only if } \quad \dfrac{1}{3}<x<\dfrac{2}{3}\notag \]
and
\[a_1 \neq 1 \quad \text { and } \quad a_2=1 \quad \text { if and only if } \quad \dfrac{1}{9}<x<\dfrac{2}{9} \quad \text { or } \quad \dfrac{7}{9}<x<\dfrac{8}{9}\notag \]
and so forth. If
\[x=\sum_{j=1}^{\infty} \dfrac{a_j}{3^j} \quad \text { and } \quad y=\sum_{j=1}^{\infty} \dfrac{b_j}{3^j},\notag \]
then \(x<y\) if and only if there exists an \(n \in \mathbb{N}\) such that \(a_n<b_n\) and \(a_j=b_j\) for \(j<n\). Thus, the Cantor set \(C\) is the set of all \(0 \leq x \leq 1\) that have a base- 3 expansion
\[x=\sum_{j=1}^{\infty} \dfrac{a_j}{3^j} \quad \text { with } \quad a_j \neq 1 \quad \text { for any } \quad j .\notag \]
The set
\[\mathbb{T}=\left\{t_n=-\dfrac{1}{n}: n \in \mathbb{N}\right\} \cup \mathbb{N}_0\notag \]
is a time scale.
The set
\[[0,1] \cup\left\{1+\dfrac{1}{n}\right\}_{n \in \mathbb{N}} \cup(2,3] \cup\left\{3+\dfrac{1}{n}\right\}_{n \in \mathbb{N}}\notag \]
is a time scale.
Let
\[U=\left\{\dfrac{1}{2^n}: n \in \mathbb{N}_0\right\} .\notag \]
Then the set
\[\{0\} \cup U \cup(1-u) \cup(1+U) \cup(2-U) \cup(2+U) \cup(3-U) \cup(3+U) \cup\{1,2,3,4\}\notag \]
is a time scale.
Check if the following sets are time scales
1. \(2^{\mathbb{N}_0}\).
2. \((-1,1] \cup[2,3] \cup[4,8]\).
3. \(\left\{-\dfrac{1}{2 n}: n \in \mathbb{N}\right\} \cup 2 \mathbb{N}_0\).
4. \(U \cup(2-U) \cup(2+U), U=\left\{\dfrac{1}{4^n}: n \in \mathbb{N}_0\right\}\).
5. \([0,2] \cup\left\{2+\dfrac{1}{n}\right\}_{n \in \mathbb{N}} \cup(3,5] \cup 7^{\mathbb{N}_0}\).
- Answer
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