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ВИДЕОТЕКА |
Международная конференция по комплексному анализу памяти А. А. Гончара и А. Г. Витушкина
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Periods of Negative-regular Continued Fractions for rational numbers and idempotents in the Modular group С. В. Хрущев Satbayev University |
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Аннотация: If \begin{equation*} x=\frac{-1}{a_1}\,\underset{+}{}\frac{-1}{a_2}\,\underset{+\cdots+}{}\frac{-1}{a_k}\,,\quad \frac{1}{x}=\frac{-1}{c_1}\,\underset{+}{}\frac{-1}{c_2}\,\underset{+\cdots+}{}\frac{-1}{c_m}\,. \end{equation*} These expansions are obtained by a generalized Euclidean algorithm. Then \begin{equation}\label{minperiod2011} x\oplus s(x^{-1})\overset{def}{=}\{a_1,a_2,\ldots,a_{k-1},a_k+c_m,c_{m-1},\ldots,c_1\}\mapsto x \end{equation} is the minimal period for Theorem. Let Example 1: \begin{equation*} \begin{aligned} &\{1,3,2,\overset{\blacktriangledown}{6},2\}\mapsto -\frac{7}{4}\\ &\{1,3,2,\overset{\blacktriangledown}{4},2\}\rightarrow-\frac{17\pm i}{10}\\ &\begin{matrix} &1&3&2&4&2&\\ \frac{1}{0}&\frac{0}{1}&\frac{-1}{1}&\frac{-3}{2}&\frac{-5}{3}&\frac{-17}{10}&\frac{-\mathbf{29}}{\mathbf{17}} \end{matrix} \end{aligned}\quad \begin{aligned} \frac{7}{4}&=1+\frac{1}{1}\,\underset{+}{}\,\frac{1}{2+1}\,;\\ \frac{\mathbf{29}}{\mathbf{17}}&=1+\frac{1}{1}\,\underset{+}{}\,\frac{1}{2}\,\underset{+}{}\,\frac{1}{2}\,\underset{+}{}\,\frac{1}{1}\,\underset{+}{}\,\frac{1}{1}\,. \end{aligned} \end{equation*} Example 2: \begin{equation*} \begin{aligned} &\{1,1,\overset{\blacktriangledown}{4},1,1\}\mapsto 1\\ &\{1,1,\overset{\blacktriangledown}{2},1,1\}\rightarrow \pm i\\ &\begin{matrix} &1&1&2&1&1&\\ \frac{1}{0}&\frac{0}{1}&\frac{-1}{1}&\frac{-1}{0}&\frac{-1}{-1}&\frac{0}{-1}&\frac{\mathbf{1}}{\mathbf{0}} \end{matrix} \end{aligned}\quad \begin{aligned} 1&=0+\frac{1}{0+1}\,;\\ \frac{\mathbf{1}}{\mathbf{0}}&=0+\frac{1}{0}\,\underset{+}{}\,\frac{1}{0}\,\underset{+}{}\,\frac{1}{0}\,. \end{aligned} \end{equation*} This is a joint result with Mikhail Yu. Tyaglov. Website: https://us06web.zoom.us/j/85797534479?pwd=ODU1bkNCUk1BNXZ6UEw1ejlsbHdIQT09 Список литературы
* Идентификатор: 857 9753 4479. Пароль: 782747 |