Post History
#6: Post edited
by
r~~
·
2024-08-09T05:14:13Z (5 months ago)
Be more precise about the n = 0 case
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
- 1 &\text{if \(m = 0\) or \(\lfloor \log_2 m\rfloor = \lfloor \log_2 n\rfloor\)}\\
- (-1)^{\lfloor n / 2^{\lfloor \log_2 m \rfloor}\rfloor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- \sigma_<(m, n) & \text{if \(m < n\)}\\
- -1 & \text{if \(m = n \neq 0\)}\\
-\sigma_<(n, m) & \text{if \(m > n\)}- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
- 1 &\text{if \(m = 0\) or \(\lfloor \log_2 m\rfloor = \lfloor \log_2 n\rfloor\)}\\
- (-1)^{\lfloor n / 2^{\lfloor \log_2 m \rfloor}\rfloor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- 1 & \text{if \(n = 0\)}\\
- \sigma_<(m, n) & \text{if \(m < n\)}\\
- -1 & \text{if \(m = n \neq 0\)}\\
- -\sigma_<(n, m) & \text{if \(m > n \neq 0\)}
- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
#5: Post edited
by
r~~
·
2024-08-09T04:05:07Z (5 months ago)
Yikes, missed an important minus sign!
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
- 1 &\text{if \(m = 0\) or \(\lfloor \log_2 m\rfloor = \lfloor \log_2 n\rfloor\)}\\
- (-1)^{\lfloor n / 2^{\lfloor \log_2 m \rfloor}\rfloor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- \sigma_<(m, n) & \text{if \(m < n\)}\\
- -1 & \text{if \(m = n \neq 0\)}\\
\sigma_<(n, m) & \text{if \(m > n\)}- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
- 1 &\text{if \(m = 0\) or \(\lfloor \log_2 m\rfloor = \lfloor \log_2 n\rfloor\)}\\
- (-1)^{\lfloor n / 2^{\lfloor \log_2 m \rfloor}\rfloor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- \sigma_<(m, n) & \text{if \(m < n\)}\\
- -1 & \text{if \(m = n \neq 0\)}\\
- -\sigma_<(n, m) & \text{if \(m > n\)}
- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
#4: Post edited
by
r~~
·
2024-08-06T03:50:54Z (5 months ago)
Update log notation
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
1 &\text{if \(m = 0\) or \(\lfloor \lg m floor = \lfloor \lg n floor\)}\\(-1)^{\lfloor n / 2^{\lfloor \lg m floor} floor}&\text{otherwise}- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- \sigma_<(m, n) & \text{if \(m < n\)}\\
- -1 & \text{if \(m = n \neq 0\)}\\
- \sigma_<(n, m) & \text{if \(m > n\)}
- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
- 1 &\text{if \(m = 0\) or \(\lfloor \log_2 m floor = \lfloor \log_2 n floor\)}\\
- (-1)^{\lfloor n / 2^{\lfloor \log_2 m floor} floor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- \sigma_<(m, n) & \text{if \(m < n\)}\\
- -1 & \text{if \(m = n \neq 0\)}\\
- \sigma_<(n, m) & \text{if \(m > n\)}
- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
#3: Post edited
by
r~~
·
2024-08-06T01:52:23Z (5 months ago)
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
- 1 &\text{if \(m = 0\) or \(\lfloor \lg m\rfloor = \lfloor \lg n\rfloor\)}\\
- (-1)^{\lfloor n / 2^{\lfloor \lg m \rfloor}\rfloor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- \sigma_<(m, n) & \text{if \(m < n\)}\\
-1 & \text{if \(m = n\)}\\- \sigma_<(n, m) & \text{if \(m > n\)}
- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
- 1 &\text{if \(m = 0\) or \(\lfloor \lg m\rfloor = \lfloor \lg n\rfloor\)}\\
- (-1)^{\lfloor n / 2^{\lfloor \lg m \rfloor}\rfloor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- \sigma_<(m, n) & \text{if \(m < n\)}\\
- -1 & \text{if \(m = n \neq 0\)}\\
- \sigma_<(n, m) & \text{if \(m > n\)}
- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
#2: Post edited
by
r~~
·
2024-08-06T01:51:22Z (5 months ago)
Simplify sigma formulas
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
\sigma_\leq(m, n) &= \begin{cases}1 &\text{if \(m = 0\) or (\(m \ne n\) and \(\lfloor \lg m floor = \lfloor \lg n floor\))}\\-1 &\text{if \(m = n \neq 0\)}\\- (-1)^{\lfloor n / 2^{\lfloor \lg m \rfloor}\rfloor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
\sigma_\leq(m, n) & \text{if \(m \leq n\)}\\\sigma_\leq(n, m) & \text{if \(m > n\)}- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
- You might find this paper helpful: [
- An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
- The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
- The preferred formula in that paper is
- \[
- (a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
- \]
- and using this formula the desired \(\sigma\) is
- \[
- \begin{align}
- \sigma_<(m, n) &= \begin{cases}
- 1 &\text{if \(m = 0\) or \(\lfloor \lg m floor = \lfloor \lg n floor\)}\\
- (-1)^{\lfloor n / 2^{\lfloor \lg m \rfloor}\rfloor}&\text{otherwise}
- \end{cases}\\
- \sigma(m, n) &= \begin{cases}
- \sigma_<(m, n) & \text{if \(m < n\)}\\
- -1 & \text{if \(m = n\)}\\
- \sigma_<(n, m) & \text{if \(m > n\)}
- \end{cases}
- \end{align}
- \]
- Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.
#1: Initial revision
by
r~~
·
2024-08-06T01:08:32Z (5 months ago)
You might find this paper helpful: [
An Alternate Cayley-Dickson Product](https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-28/issue-1/An-Alternate-Cayley-Dickson-Product/10.35834/mjms/1474295358.full).
The author observes that while the standard product formula for the Cayley-Dickson construction leads to precisely this difficulty with finding a closed form, there are alternatives for the product that produce isomorphic algebras, and one of these has a nice closed-form formula on basis elements.
The preferred formula in that paper is
\[
(a, b) \cdot (c, d) = (ac - b^*d, da^* + bc)
\]
and using this formula the desired \(\sigma\) is
\[
\begin{align}
\sigma_\leq(m, n) &= \begin{cases}
1 &\text{if \(m = 0\) or (\(m \ne n\) and \(\lfloor \lg m\rfloor = \lfloor \lg n\rfloor\))}\\
-1 &\text{if \(m = n \neq 0\)}\\
(-1)^{\lfloor n / 2^{\lfloor \lg m \rfloor}\rfloor}&\text{otherwise}
\end{cases}\\
\sigma(m, n) &= \begin{cases}
\sigma_\leq(m, n) & \text{if \(m \leq n\)}\\
\sigma_\leq(n, m) & \text{if \(m > n\)}
\end{cases}
\end{align}
\]
Note that this formula assumes the author's preferred basis numbering, which also differs from the usual presentation. In the usual presentation, at the \(n\)th step of the construction, the pair \((i_k, 0)\) corresponds to the basis element \(i_k\) and the pair \((0, i_k)\) to the basis element \(i_{2^n + k}\). In the author's ‘shuffle basis’, the pair \((i_k, 0)\) is identified with \(i_{2k}\), and the pair \((0, i_k)\) with \(i_{2k + 1}\). Fortunately, if you prefer the usual basis numbering, converting between the two is easily done by representing the basis number as a fixed-width binary string and reversing the digits; in the direct limit, it suffices to choose a width for each use of \(\sigma\) that is wide enough to cover the larger argument.