diff --git a/books/bookvol10.1.pamphlet b/books/bookvol10.1.pamphlet index 803ce2b..ffd6276 100644 --- a/books/bookvol10.1.pamphlet +++ b/books/bookvol10.1.pamphlet @@ -7586,6 +7586,760 @@ i = \right] $$ +\chapter{Clifford Algebra\cite{39}} + +This is quoted from John Fletcher's web page\cite{39} (with permission). + +The theory of Clifford Algebra includes a statement that each Clifford +Algebra is isomorphic to a matrix representation. Several authors +discuss this and in particular Ablamowicz\cite{41} gives examples of +derivation of the matrix representation. A matrix will itself satisfy +the characteristic polynomial equation obeyed by its own +eigenvalues. This relationship can be used to calculate the inverse of +a matrix from powers of the matrix itself. It is demonstrated that the +matrix basis of a Clifford number can be used to calculate the inverse +of a Clifford number using the characteristic equation of the matrix +and powers of the Clifford number. Examples are given for the algebras +Clifford(2), Clifford(3) and Clifford(2,2). + +\section{Introduction} + +Introductory texts on Clifford algebra state that for any chosen +Clifford Algebra there is a matrix representation which is equivalent. +Several authors discuss this in more detail and in particular, +Ablamowicz\cite{41} shows that the matrices can be derived for each algebra +from a choice of idempotent, a member of the algebra which when +squared gives itself. The idea of this paper is that any matrix obeys +the characteristic equation of its own eigenvalues, and that therefore +the equivalent Clifford number will also obey the same characteristic +equation. This relationship can be exploited to calculate the inverse +of a Clifford number. This result can be used symbolically to find the +general form of the inverse in a particular algebra, and also in +numerical work to calculate the inverse of a particular member. This +latter approach needs the knowledge of the matrices. Ablamowicz has +provided a method for generating them in the form of a Maple +implementation. This knowledge is not believed to be new, but the +theory is distributed in the literature and the purpose of this paper +is to make it clear. The examples have been first developed using a +system of symbolic algebra described in another paper by this +author\cite{40}. + +\section{Clifford Basis Matrix Theory} + +The theory of the matrix basis is discussed extensively by +Ablamowicz. This theory will be illustrated here following the +notation of Ablamowicz by reference to Clifford(2) algebra and can +be applied to other Clifford Algebras. For most Clifford algebras +there is at least one primitive idempotent, such that it squares to +itself. For Clifford (2), which has two basis members $e_1$ and $e_2$, one +such idempotent involves only one of the basis members, $e_1$, i.e. + +\[f_1 = f = \frac{1}{2} (1 + e_1)\] + +If the idempotent is mutiplied by the other basis function $e_2$, other +functions can be generated: + +\[f_2 = e_2 f = \left(\frac{1}{2}-\frac{1}{2}e_1\right)e_2\] + +\[f_3 = f e_2 = \left(\frac{1}{2}+\frac{1}{2}e_1\right)e_2\] + +\[f_4 = e_2 f e_2 = \frac{1}{2}-\frac{1}{2}e_1\] + +Note that $fe_22f = 0$. These four functions provide a means of +representing any member of the space, so that if a general member c is +given in terms of the basis members of the algebra + +\[ c = a_0 + a_1e_1 + a_2e_2 + a_3e_1e_2\] + +it can also be represented by a series of terms in the idempotent and +the other functions. + +\[ +\begin{array}{rcl} +c&=&a_{11}f_1 + a_{21}f_2 + a_{12}f_3 + a_{22}f_4\\ +&&\\ + &=&\frac{1}{2}a_{11} + \frac{1}{2}a_{11}e_1 + \frac{1}{2}a_{21}e_2 +-\frac{1}{2}a_{21}e_1e_2 +\\ +&&\\ +&&\frac{1}{2}a_{12}e_2 + \frac{1}{2}a_{12}e_1e_2 + \frac{1}{2}a_{22} +-\frac{1}{2}a_{22}e_1 +\end{array} +\] + + +Equating coefficients it is clear that the following equations apply. +\[ +\begin{array}{rcl} +a_0 &=& \frac{1}{2}a_{11} + \frac{1}{2}a_{22}\\ +&&\\ +a_1 &=& \frac{1}{2}a_{11} - \frac{1}{2}a_{22}\\ +&&\\ +a_2 &=& \frac{1}{2}a_{12} + \frac{1}{2}a_{21}\\ +&&\\ +a_3 &=& \frac{1}{2}a_{12} - \frac{1}{2}a_{21} +\end{array} +\] + +The reverse equations can be recovered by multiplying the two forms of +c by different combinations of the functions $f_1$, $f_2$ and $f_3$. +The equation + +\[ +\begin{array}{rcl} +f_1cf_1 &=& f_1(a_{11}f_1 + a_{21}f_2 + a_{12}f_3 + a_{22}f_4)f_1\\ +&&\\ + &=& f_1(a_0 + a_1e_1 + a_2e_2 + a_3e_1e_2)f_1 +\end{array} +\] + +reduces to the equation + +\[a_{11}f = (a_0 + a_1)f\] + +and similar equations can be deduced from other combinations of the +functions as follows. + +\[ +\begin{array}{rcl} +f_1cf_2 : a_{12}f &=& (a_2 + a_3)f\\ +&&\\ +f_2cf_1 : a_{21}f &=& (a_2 - a_3)f\\ +&&\\ +f_3cf_2 : a_{22}f &=& (a_0 - a_1)f +\end{array} +\] + +If a matrix is defined as + +\[ +A = \left( +\begin{array}{cc} +a_{11} & a_{12} \\ +a_{21} & a_{22} +\end{array} +\right) +\] + +so that + +\[ +Af = \left( +\begin{array}{cc} +a_{11}f & a_{12}f \\ +a_{21}f & a_{22}f +\end{array} +\right) += +\left( +\begin{array}{cc} +a_0+a_1 & a_2+a_3 \\ +a_2-a_3 & a_0-a_1 +\end{array} +\right) f +\] + +then the expression + +\[ +\left( +\begin{array}{cc} +1 & e_2 +\end{array} +\right) +\left( +\begin{array}{cc} +a_{11}f & a_{12}f \\ +a_{21}f & a_{22}f +\end{array} +\right) +\left( +\begin{array}{c} +1\\ +e_2 +\end{array} +\right) += +a_{11}f_1 + a_{21}f_2 + a_{12}f_3 + a_{22}f_4 = c +\] + +generates the general Clifford object c. All that remains to form the +basis matrices is to make c each basis member in turn, and named as +shown. + +\[ +\begin{array}{lrclcr} +c=1: & Af & = & +\left( +\begin{array}{cc} +f & 0\\ +0 & f +\end{array} +\right) +& = & E_0f\\ +c=e_1 & Af & = & +\left( +\begin{array}{cc} +f & 0\\ +0 & -f +\end{array} +\right) +& = & E_1f\\ +c=e_2 & Af & = & +\left( +\begin{array}{cc} +0 & f\\ +f & 0 +\end{array} +\right) +& = & E_2f\\ +c=e_1e_2 & Af & = & +\left( +\begin{array}{cc} +0 & f\\ +-f & 0 +\end{array} +\right) +& = & E_{12}f +\end{array} +\] + +These are the usual basis matrices for Clifford (2) except that they +are multiplied by the idempotent. + +This approach provides an explanation for the basis matrices in terms +only of the Clifford Algebra itself. They are the matrix +representation of the basis objects of the algebra in terms of an +idempotent and an associated vector of basis functions. This has been +shown for Clifford (2) and it can be extended to other algebras once +the idempotent and the vector of basis functions have been identified. +This has been done in many cases by Ablamowicz. This will now be +developed to show how the inverse of a Clifford number can be obtained +from the matrix representation. + +\section{Calculation of the inverse of a Clifford number} + +The matrix basis demonstrated above can be used to calculate the +inverse of a Clifford number. In simple cases this can be used to +obtain an algebraic formulation. For other cases the algebra is too +complex to be clear, but the method can still be used to obtain the +numerical value of the inverse. To apply the method it is necessary +to know a basis matrix representation of the algebra being used. + +The idea of the method is that the matrix representation will have a +characteristic polynomial obeyed by the eigenvalues of the matrix and +also by the matrix itself. There may also be a minimal polynomial +which is a factor of the characteristic polynomial, which will have +also be satisfied by the matrix. It is clear from the proceding +section that if $A$ is a matrix representation of $c$ in a Clifford +Algebra then if some function $f(A) = 0$ then the corresponding Clifford +function $f(c) = 0$ must also be zero. In particular if $f(A) = 0$ is the +characteristic or minimal polynomial of $A$, then $f(c) = 0$ implies that +$c$ also satisfies the same polynomial. Then if the inverse of the +Clifford number, $c^{-1}$ is to be found, then + +\[c^{-1}f(c)=0\] + +provides a relationship for $c^{-1}$ in terms of multiples a small number +of low powers of $c$, with the maximum power one less than the order of +the polynomial. The method suceeds unless the constant term in the +polynomial is zero, which means that the inverse does not exist. For +cases where the basis matrices are of order two, the inverse will be +shown to be a linear function of $c$. + +The method can be summed up as follows. +\begin{enumerate} +\item Find the matrix basis of the Clifford algebra. +\item Find the matrix representation of the Clifford number whose +inverse is required. +\item Compute the characteristic or minimal polynomial. +\item Check for the existence of the inverse. +\item Compute the inverse using the coefficients from the polynomial. +\end{enumerate} + +Step 1 need only be done once for any Clifford algebra, and this can +be done using the method in the previous section, where needed. + +Step 2 is trivially a matter of accumulation of the correct multiples +of the matrices. + +Step 3 may involve the use of a computer algebra system to find the +coefficients of the polynomial, if the matrix size is at all large. + +Steps 4 and 5 are then easy once the coefficients are known. + +The method will now be demonstrated using some examples. + +\subsection{Example 1: Clifford (2)} + +In this case the matrix basis for a member of the Clifford algebra + +\[c = a_0 + a_1e_1 + a_2e_2 + a_3e_1e_2\] + +was developed in the previous section as + +\[A=\left( +\begin{array}{cc} +a_0+a_1 & a_2+a_3\\ +a_2-a_3 & a_0-a_1 +\end{array} +\right)\] + +This matrix has the characteristic polynomial + +\[X^2 - 2Xa_0 + a^2_0 - a^2_1 - a^2_2 + a^2_3 = 0\] + +and therefore + +\[X^{-1}(X^2 - 2Xa_0 + a^2_0 - a^2_1 - a^2_2 + a^2_3) = 0\] + +and + +\[X^{-1} = (2a_0 - X)/(a^2_0 - a^2_1 - a^2_2 + a^2_3) = 0\] + +which provides a general solution to the inverse in this algebra. + +\[c^{-1} = (2a_0 - c)/(a^2_0 - a^2_1 - a^2_2 + a^2_3) = 0\] + +\subsection{Example 2: Clifford (3)} + +A set of basis matrices for Clifford (3) as given by Abalmowicz and +deduced are + +\[ +\begin{array}{cc} +E_0 = +\left( +\begin{array}{cc} +1&0\\ +0&1 +\end{array}\right) & +E_1 = +\left( +\begin{array}{cc} +1&0\\ +0&-1 +\end{array}\right) \\ +E_2 = +\left( +\begin{array}{cc} +0&1\\ +1&0 +\end{array}\right) & +E_3 = +\left( +\begin{array}{cc} +0&-j\\ +j&0 +\end{array}\right) \\ +E_1E_2 = +\left( +\begin{array}{cc} +0&1\\ +-1&0 +\end{array}\right) & +E_1E_3 = +\left( +\begin{array}{cc} +0&-j\\ +-j&0 +\end{array}\right) \\ +E_2E_3 = +\left( +\begin{array}{cc} +j&0\\ +0&-j +\end{array}\right) & +E_1E_2E_3 = +\left( +\begin{array}{cc} +j&0\\ +0&j +\end{array}\right) \\ +\end{array} +\] + +for the idempotent + +\[f = \frac{(1 + e_1)}{2}, {\rm\ where\ } j^2 = -1.\] + +The general member of the algebra + +\[c_3 = a_0 +a_1e_1 + a_2e_2 + a_3e_3 + +a_{12}e_1e_2 + a_{13}e_1e_3 + a_{23}e_2e_3 + a_{123}e_1e_2e_3\] + +has the matrix representation + +\[ +\begin{array}{rcl} +A_3&=&a_0E_0 + a_1E_1 + a_2E_2 +a_3E_3 + a_{12}E_1E_2\\ +&& +a_{13}E_1E_3 + a_{23}E_2E_3 + a_{123}E_1E_2E_3\\ +&&\\ +&=&\left( +\begin{array}{cc} +a_0 + a_1 + ja_{23} + ja_{123}& a_2 -ja_3 +a_{12} -ja_{13}\\ +a_2 + ja_3- a_{12}- ja_{13}& a_0- a_1- ja_{23} + ja_{123} +\end{array} +\right) +\end{array} +\] + +This has the characteristic polynomial + +\[ +\begin{array}{rl} +&a^2_0-a^2_1-a^2_2-a^2_3+a^2_{12}+a^2_{13}+a^2_{23}-a^2_{123}\\ +&\\ ++&2j(a_0a_{123}-a_1a_{23}-a_{12}a_3+a_{13}a_2)\\ +&\\ +-&2(a_0+ja_{123})X + X^2=0 +\end{array} +\] + +and the expression for the inverse is + +\[ +\begin{array}{rcl} +X^{-1}&=&(2a_0 + 2ja_{123} -X) /\\ +&&(a^2_0-a^2_1-a^2_2-a^2_3+a^2_{12}+a^2_{13}+a^2_{23}-a^2_{123}\\ +&&+2j(a_0a_{123}-a_1a_{23}-a_{12}a_3+a_{13}a_2)) +\end{array} +\] + +Complex terms arise in two cases, + +\[a_{123} \ne 0\] + +and + +\[(a_0a_{123}-a_1a_{23}-a_{12}a_3+a_{13}a_2) \ne 0\] + +Two simple cases have real minumum polynomials: + +Zero and first grade terms only: + +\[ +\begin{array}{rcl} +A_1&=&a_0E_0 + a_1E_1 + a_2E_2 + a_3E_3\\ +&=&\left( +\begin{array}{cc} +a_0+a_1 & a_2-ja_3\\ +a_2+ja_3 & a_0-a_1 +\end{array} +\right) +\end{array} +\] + +which has the minimum polynomial + +\[a^2_0-a^2_1-a^2_2-a^2_3-2a_0X+X^2=0\] + +which gives + +\[X^{-1} = (2a_0- X) / (a^2_0- a^2_1- a^2_2 - a^2_3)\] + +Zero and second grade terms only (ie. the even subspace). + +\[ +\begin{array}{rcl} +A_2&=&a_0E_0 + a_{12}E_1E_2 + a_{13}E_1E_3 + a_{23}E_2E_3\\ +&&\left( +\begin{array}{cc} +a_0+ja_{23} & a_{12}-ja_{13}\\ +-a_{12}-ja_{13} & a_0-ja_{23} +\end{array} +\right) +\end{array} +\] + +which has minimum polynomial + +\[a^2_0+a^2_{23}+a^2_{12}+a^2_{13}-2a_0X+X^2 = 0\] + +giving + +\[X^{-1} = (2a_0- X) /(a^2_0 + a^2_{23} + a^2_{12} + a^2_{13})\] + +This provides a general solution for the inverse together with two +simple cases of wide usefulness. + +\subsection{Example 3: Clifford (2,2)} + +The following basis matrices are given by Ablamowicz\cite{41} + +\[ +\begin{array}{cc} +E_1=\left( +\begin{array}{cccc} +0 & 1 & 0 & 0\\ +1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1\\ +0 & 0 & 1 & 0 +\end{array} +\right)& +E_2=\left( +\begin{array}{cccc} +0 & 0 & 1 & 0\\ +0 & 0 & 0 & -1\\ +1 & 0 & 0 & 0\\ +0 & -1 & 0 & 0 +\end{array} +\right)\\ +E_3=\left( +\begin{array}{cccc} +0 & -1 & 0 & 0\\ +1 & 0 & 0 & 0\\ +0 & 0 & 0 & -1\\ +0 & 0 & 1 & 0 +\end{array} +\right)& +E_4=\left( +\begin{array}{cccc} +0 & 0 & -1 & 0\\ +0 & 0 & 0 & 1\\ +1 & 0 & 0 & 0\\ +0 & -1 & 0 & 0 +\end{array} +\right) +\end{array} +\] + +for the idempotent +\[f = \frac{(1 +e_1e_3) (1+ e_1e_3)}{4}.\] + + Note that this implies that the order of the basis members is such +that $e_1$ and $e_2$ have square $+1$ and $e_3$ and $e_4$ have square +$-1$. Other orderings are used by other authors. The remaining basis +matrices can be deduced to be as follows. + +Second Grade members + +\[ +\begin{array}{cc} +E_1E_2 = \left( +\begin{array}{cccc} +0 & 0 & 0 & -1\\ +0 & 0 & 1 & 0\\ +0 & -1 & 0 & 0\\ +1 & 0 & 0 & 0 +\end{array}\right)& +E_1E_3 = \left( +\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & -1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & -1 +\end{array}\right)\\ +E_1E_4 = \left( +\begin{array}{cccc} +0 & 0 & 0 & 1\\ +0 & 0 & -1 & 0\\ +0 & -1 & 0 & 0\\ +1 & 0 & 0 & 0 +\end{array}\right)& +E_2E_3 = \left( +\begin{array}{cccc} +0 & 0 & 0 & -1\\ +0 & 0 & -1 & 0\\ +0 & -1 & 0 & 0\\ +-1 & 0 & 0 & 0 +\end{array}\right)\\ +E_2E_4 = \left( +\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & -1 & 0\\ +0 & 0 & 0 & -1 +\end{array}\right)& +E_3E_4 = \left( +\begin{array}{cccc} +0 & 0 & 0 & -1\\ +0 & 0 & -1 & 0\\ +0 & 1 & 0 & 0\\ +1 & 0 & 0 & 0 +\end{array}\right)\\ +\end{array} +\] + +Third grade members + +\[ +\begin{array}{cc} +E_1E_2E_3 = \left( +\begin{array}{cccc} +0 & 0 & -1 & 0\\ +0 & 0 & 0 & -1\\ +-1 & 0 & 0 & 0\\ +0 & -1 & 0 & 0 +\end{array}\right)& +E_1E_2E_4 = \left( +\begin{array}{cccc} +0 & 1 & 0 & 0\\ +1 & 0 & 0 & 0\\ +0 & 0 & 0 & -1\\ +0 & 0 & -1 & 0 +\end{array}\right)\\ +E_1E_3E_4 = \left( +\begin{array}{cccc} +0 & 0 & -1 & 0\\ +0 & 0 & 0 & -1\\ +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0 +\end{array}\right)& +E_2E_3E_4 = \left( +\begin{array}{cccc} +0 & 1 & 0 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & -1\\ +0 & 0 & 1 & 0 +\end{array}\right) +\end{array} +\] + +Fourth grade member + +\[ +E_1E_2E_3E_4 = \left( +\begin{array}{cccc} +-1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & -1 +\end{array} +\right) +\] + +Zero grade member (identity) + +\[ +E_0 = \left( +\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array} +\right) +\] + +The general member of the Clifford (2,2) algebra can be written as follows. + +\[\begin{array}{rcl} +c_{22}&=& a_0 + a_1e_1 + a_2e_2 + a_3e_3 + a_4e_4 +\\ +&&a_{12}e_1e_2+a_{13}e_1e_3+a_{14}e_1e_4+a_{23}e_2e_3+a_{24}e_2e_4+ +a_{34}e_3e_4\\ +&&+ a_{123}e_1e_2e_3 +a_{124}e_1e_2e_4 +a_{134}e_1e_3e_4 + +a_{234}e_2e_3e_4 + a_{1234}e_1e_2e_3e_4 +\end{array} +\] + +This has the following matrix representation. + +\[ +\left( +\begin{array}{cccc} +a_0+a_{13}+ & a_1-a_3+ & a_2-a_4- & -a_{12}+a_{14}-\\ +a_{24}-a_{1234}& a_{124}+a_{234} & a_{123}-a_{134} & a_{23}-a_{34}\\ +&&&\\ +a_1+a_3+ & a_0-a_{13}+ & a_{12}-a_{14}- & -a_2+a_4-\\ +a_{124}-a_{234} & a_{24}+a_{1234} & a_{23}-a_{34} & a_{123}-a_{134}\\ +&&&\\ +a_2+a_4- & -a_{12}-a_{14}- & a_0+a_{13}- & a_1-a_3-\\ +a_{123}+a_{134} & a_{23}+a_{34} & a_{24}+a_{1234} & a_{124}-a_{234}\\ +&&&\\ +a_{12}+a_{14}- & -a_2-a_4- & a_1+a_3- & a_0-a_{13}-\\ +a_{23}+a_{34} & a_{123}+a_{134} & a_{124}+a_{234} & a_{24}-a_{1234} +\end{array} +\right) +\] + +In this case it is possible to generate the characteristic equation +using computer algebra. However, it is too complex to be of practical +use. Instead here are numerical examples of the use of the method to +calculate the inverse. For the case where + +\[n1 = 1+ e_1 + e_2 + e_3 + e_4\] + +then the matrix representation is + +\[N_1 = E_0 +E_1 + E_2 + E_3 + E_4 = +\left( +\begin{array}{cccc} +1 & 0 & 0 & 0\\ +2 & 1 & 0 & 0\\ +2 & 0 & 1 & 0\\ +0 & -2 & 2 & 1 +\end{array} +\right) +\] + +This has the minimum polynomial + +\[X^2 - 2X + 1 = 0\] + +so that + +\[X^{-1} = 2- X\] + +and + +\[n^{-1}_1= 2 - n_1 = 1 - e_1 - e_2- e_3- e_4\] + +For + +\[n_2 = 1+ e_1 + e_2 + e_3 + e_4 +e_1e_2\] + +the matrix representation is + +\[N_2 = I + E_1 + E_2 + E_3 +E_4 + E_1E_2 = +\left( +\begin{array}{cccc} +1 & 0 & 0 & -1\\ +2 & 1 & 1 & 0\\ +2 & -1 & 1 & 0\\ +1 & -2 & 2 & 1 +\end{array} +\right) +\] + +This has the minimum polynomial + +\[X^4 - 4X^3 + 8X^2 - 8X - 4 = 0\] + +so that + +\[X^{-1} = \frac{X^3 - 4X^2 + 8X - 8}{4}\] + +and + +\[n^{-1}_2 = \frac{n^3_2- 4n^2_2 + 8n_2 - 8}{4}\] + +This expression can be evaluated easily using a computer algebra +system for Clifford algebra such as described in Fletcher\cite{40}. +The result is + +\[ +\begin{array}{rcl} +n^{-1}_2 &=& -0.5 + 0.5e_1 + 0.5e_2 - 0.5e_1e_2 - 0.5e_1e_3\\ +&& - 0.5e_1e_4 + 0.5e_2e_3 + 0.5e_2e_4 - 0.5e_1e_2e_3 - 0.5e_1e_2e_4 +\end{array} +\] + + +Note that in some cases the inverse is linear in the original Clifford +number, and in others it is nonlinear. + +\subsection{Conclusion} + +The paper has demonstrated a method for the calculation of inverses of +Clifford numbers by means of the matrix representation of the +corresponding Clifford algebra. The method depends upon the +calculation of the basis matrices for the algebra. This can be done +from an idempotent for the algebra if the matrices are not already +available. The method provides an easy check on the existence of the +inverse. For simple systems a general algebraic solution can be found +and for more complex systems the algebra of the inverse can be +generated and evaluated numerically for a particular example, given a +system of computer algebra for Clifford algebra. \chapter{Groebner Basis} Groebner Basis @@ -7708,6 +8462,17 @@ Addison-Wesley Publishing Company, Inc., "On Quaternions and Octonions", A.K Peters, Natick, MA. (2003) ISBN 1-56881-134-9 \bibitem{38} http://mathworld.wolfram.com/Quaternion.html +\bibitem{39} Fletcher, John P. ``Clifford Numbers and their inverses +calculated using the matrix representation.'' Chemical Engineering and +Applied Chemistry, School of Engineering and Applied Science, Aston +University, Aston Triangle, Birmingham B4 7 ET, U. K. +\verb|http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php| +\bibitem{40} Fletcher, John P. ``Symbolic processing of Clifford +Numbers in C++'', Paper 25, AGACSE 2001. +\bibitem{41} Ablamowicz Rafal, ``Spinor Representations of Clifford +Algebras: A Symbolic Approach'', Computer Physics Communications +Vol. 115, No. 2-3, December 11, 1998, pages 510-535. + \end{thebibliography} \end{document} diff --git a/books/bookvol5.pamphlet b/books/bookvol5.pamphlet index bdae2bc..04c08f6 100644 --- a/books/bookvol5.pamphlet +++ b/books/bookvol5.pamphlet @@ -356,8 +356,9 @@ of effort. We would like to acknowledge and thank the following people: "Brian Dupee Dominique Duval" "Robert Edwards Heow Eide-Goodman Lars Erickson" "Richard Fateman Bertfried Fauser Stuart Feldman" -"Brian Ford Albrecht Fortenbacher George Frances" -"Constantine Frangos Timothy Freeman Korrinn Fu" +"John Fletcher Brian Ford Albrecht Fortenbacher" +"George Frances Constantine Frangos Timothy Freeman" +"Korrinn Fu" "Marc Gaetano Rudiger Gebauer Kathy Gerber" "Patricia Gianni Samantha Goldrich Holger Gollan" "Teresa Gomez-Diaz Laureano Gonzalez-Vega Stephen Gortler" diff --git a/changelog b/changelog index 9f2b94b..3b23599 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,8 @@ +20100216 tpd src/axiom-website/patches.html 20100216.02.jpf.patch +20100216 jpf books/bookvol10.1 add Clifford chapter +20100216 jpf books/bookvol5 add John Fletcher to credits +20100216 jpf readme added to John Fletcher credits +20100216 jpf John P. Fletcher 20100216 tpd src/axiom-website/patches.html 20100216.01.rhx.patch 20100216 tpd src/input/cachedf.input fix tests for )set break quit 20100216 tpd src/input/unittest2.input fix tests for )set break quit diff --git a/readme b/readme index bf76111..c9716cd 100644 --- a/readme +++ b/readme @@ -210,8 +210,9 @@ at the axiom command prompt will prettyprint the list. "Brian Dupee Dominique Duval" "Robert Edwards Heow Eide-Goodman Lars Erickson" "Richard Fateman Bertfried Fauser Stuart Feldman" -"Brian Ford Albrecht Fortenbacher George Frances" -"Constantine Frangos Timothy Freeman Korrinn Fu" +"John Fletcher Brian Ford Albrecht Fortenbacher" +"George Frances Constantine Frangos Timothy Freeman" +"Korrinn Fu" "Marc Gaetano Rudiger Gebauer Kathy Gerber" "Patricia Gianni Samantha Goldrich Holger Gollan" "Teresa Gomez-Diaz Laureano Gonzalez-Vega Stephen Gortler" diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index f4354cc..169650b 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -2461,5 +2461,7 @@ books/bookvol5 treeshake cparse, ptrees
books/bookvol5 treeshake cparse, ptrees, ptrop vmlisp
20100216.01.rhx.patch books/bookvol5 add )set break quit
+20100216.02.jpf.patch +books/bookvol10.1 add Clifford chapter, per John Fletcher