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+\documentclass{article}
+\usepackage{axiom}
+\usepackage{amssymb}
+\input{diagrams.tex}
+\begin{document}
+\title{\$SPAD/src/algebra clifford.spad}
+\author{Stephen M. Watt}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{domain QFORM QuadraticForm}
+<<domain QFORM QuadraticForm>>=
+)abbrev domain QFORM QuadraticForm
+++ Author: Stephen M. Watt
+++ Date Created: August 1988
+++ Date Last Updated: May 17, 1991
+++ Basic Operations: quadraticForm, elt
+++ Related Domains: Matrix, SquareMatrix
+++ Also See:
+++ AMS Classifications:
+++ Keywords: quadratic form
+++ Examples:
+++ References:
+++
+++ Description:
+++   This domain provides modest support for quadratic forms.
+QuadraticForm(n, K): T == Impl where
+    n: PositiveInteger
+    K: Field
+    SM ==> SquareMatrix
+    V  ==> DirectProduct
+ 
+    T ==> AbelianGroup with
+        quadraticForm: SM(n, K) -> %
+            ++ quadraticForm(m) creates a quadratic form from a symmetric,
+            ++ square matrix m.
+        matrix: % -> SM(n, K)
+            ++ matrix(qf) creates a square matrix from the quadratic form qf.
+        elt: (%, V(n, K)) -> K
+            ++ elt(qf,v) evaluates the quadratic form qf on the vector v, 
+            ++ producing a scalar.
+ 
+    Impl ==> SM(n,K) add
+        Rep := SM(n,K)
+ 
+        quadraticForm m ==
+            not symmetric? m =>
+                error "quadraticForm requires a symmetric matrix"
+            m::%
+        matrix q == q pretend SM(n,K)
+        elt(q,v) == dot(v, (matrix q * v))
+
+@
+\section{domain CLIF CliffordAlgebra\cite{7,12}}
+\subsection{Vector (linear) spaces}
+This information is originally from Paul Leopardi's presentation on
+the {\sl Introduction to Clifford Algebras} and is included here as
+an outline with his permission. Further details are based on the book
+by Doran and Lasenby called {\sl Geometric Algebra for Physicists}.
+
+Consider the various kinds of products that can occur between vectors.
+There are scalar and vector products from 3D geometry. There are the
+complex and quaterion products. There is also
+the {\sl outer} or {\sl exterior} product.
+
+Vector addition commutes:
+\[a + b = b + a\]
+Vector addtion is associative:
+\[a + (b + c) = (a + b) + c\]
+The identity vector exists:
+\[a + 0 = a\]
+Every vector has an inverse:
+\[a + (-a) = 0\]
+
+If we consider vectors to be directed line segments, thus establishing
+a geometric meaning for a vector, then each of these properties has a
+geometric meaning.
+
+A multiplication operator exists between scalars and vectors with
+the properties:
+\[\lambda(a + b) = \lambda a + \lambda b\]
+\[(\lambda + \mu)a = \lambda a + \mu a\]
+\[(\lambda\mu)a = \lambda(\mu a)\]
+\[{\rm If\ }1\lambda = \lambda{\rm\ for\ all\ scalars\ }\lambda
+{\rm\ then\ }1a=a{\rm\ for\ all\ vectors\ }a\]
+
+These properties completely define a vector (linear) space. The
+$+$ operation for scalar arithmetic is not the same as the $+$
+operation for vectors.
+
+{\bf Definition: Isomorphic} The vector space $A$ is isomorphic to
+the vector space $B$ if their exists a one-to-one correspondence
+between their elements which preserves sums and there is a one-to-one
+correspondence between the scalars which preserves sums and products.
+
+{\bf Definition: Subspace} Vector space $B$ is a subspace of vector
+space $A$ if all of the elements of $B$ are contained in $A$ and
+they share the same scalars.
+
+{\bf Definition: Linear Combination} Given vectors $a_1,\ldots,a_n$
+the vector $b$ is a linear combination of the vectors if we can find
+scalars $\lambda_i$ such that
+\[b = \lambda_1 a_1+\ldots+\lambda_n a_n = \sum_{k=1}^n \lambda_i a_i\]
+
+{\bf Definition: Linearly Independent} If there exists scalars $\lambda_i$
+such that
+\[\lambda_1 a_1 + \ldots + \lambda_n a_n = 0\]
+and at least one of the $\lambda_i$ is not zero
+then the vectors $a_1,\ldots,a_n$ are linearly dependent. If no such
+scalars exist then the vectors are linearly independent.
+
+{\bf Definition: Span} If every vector can be written as a linear
+combination of a fixed set of vectors $a_1,\ldots,a_n$ then this set
+of vectors is said to span the vector space.
+
+{\bf Definition: Basis} If a set of vectors $a_1,\ldots,a_n$ is linearly
+independent and spans a vector space $A$ then the vectors form a basis
+for $A$.
+
+{\bf Definition: Dimension} The dimension of a vector space is the 
+number of basis elements, which is unique since all bases of a 
+vector space have the same number of elements.
+\subsection{Quadratic Forms\cite{1}}
+For vector space $\mathbb{V}$ over field $\mathbb{F}$, characteristic 
+$\ne 2$:
+\begin{list}{}
+\item Map $f:\mathbb{V} \rightarrow \mathbb{F}$, with
+$$f(\lambda x)=\lambda^2f(x),\forall \lambda \in \mathbb{F}, x \in \mathbb{V}$$
+\item $f(x) = b(x,x)$, where
+$$b:\mathbb{V}{\rm\ x\ }\mathbb{V} \rightarrow \mathbb{F}{\rm\ ,given\ by\ }$$
+$$b(x,y):=\frac{1}{2}(f(x+y)-f(x)=f(y))$$
+is a symmetric bilinear form
+\end{list}
+\subsection{Quadratic spaces, Clifford Maps\cite{1,2}}
+\begin{list}{}
+\item A quadratic space is the pair($\mathbb{V}$,$f$), where $f$ is a 
+quadratic form on $\mathbb{V}$
+\item A Clifford map is a vector space homomorphism
+$$\rho : \mathbb{V} \rightarrow \mathbb{A}$$
+where $\mathbb{A}$ is an associated algebra, and
+$$(\rho v)^2 = f(v),{\rm\ \ \ } \forall v \in \mathbb{V}$$
+\end{list}
+\subsection{Universal Clifford algebras\cite{1}}
+\begin{list}{}
+\item The {\sl universal Clifford algebra} $Cl(f)$ for the quadratic space
+$(\mathbb{V},f)$ is the algebra generated by the image of the Clifford
+map $\phi_f$ such that $Cl(f)$ is the universal initial object such
+that $\forall$ suitable algebra $\mathbb{A}$ with Clifford map
+$\phi_{\mathbb{A}} \exists$ a homomorphism
+$$P_\mathbb{A}:Cl(f) \rightarrow \mathbb{A}$$
+$$\rho_\mathbb{A} = P_\mathbb{A}\circ\rho_f$$
+\end{list}
+\subsection{Real Clifford algebras $\mathbb{R}_{p,q}$\cite{2}}
+\begin{list}{}
+\item The real quadratic space $\mathbb{R}^{p,q}$ is $\mathbb{R}^{p+q}$ with
+$$\phi(x):=-\sum_{k:=-q}^{-1}{x_k^2}+\sum_{k=1}^p{x_k^2}$$
+\item For each $p,q \in \mathbb{N}$, the real universal Clifford algebra
+for $\mathbb{R}^{p,q}$ is called $\mathbb{R}_{p,q}$
+\item $\mathbb{R}_{p,q}$ is isomorphic to some matrix algebra over one of:
+$\mathbb{R}$,$\mathbb{R}\oplus\mathbb{R}$,$\mathbb{C}$,
+$\mathbb{H}$,$\mathbb{H}\oplus\mathbb{H}$ 
+\item For example, $\mathbb{R}_{1,1} \cong \mathbb{R}(2)$ 
+\end{list}
+\subsection{Notation for integer sets}
+\begin{list}{}
+\item For $S \subseteq \mathbb{Z}$, define
+$$\sum_{k \in S}{f_k}:=\sum_{k={\rm min\ }S, k \in S}^{{\rm max\ } S}{f_k}$$
+$$\prod_{k \in S}{f_k}:=\prod_{k={\rm min\ }S, k \in S}^{{\rm max\ } S}{f_k}$$
+$$\mathbb{P}(S):={\rm\ the\ }\ power\ set\ {\rm\ of\ }S$$
+\item For $m \le n \in \mathbb{Z}$, define
+$$\zeta(m,n):=\{m,m+1,\ldots,n-1,n\}\backslash\{0\}$$
+\end{list}
+\subsection{Frames for Clifford algebras\cite{9,10,11}}
+\begin{list}{}
+\item A {\sl frame} is an ordered basis $(\gamma_{-q},\ldots,\gamma_p)$
+for $\mathbb{R}^{p,q}$ which puts a quadratic form into the canonical
+form $\phi$
+\item For $p,q \in \mathbb{N}$, embed the frame for $\mathbb{R}^{p,q}$
+into $\mathbb{R}_{p,q}$ via the maps
+$$\gamma:\zeta(-q,p) \rightarrow \mathbb{R}^{p,q}$$
+$$\rho:\mathbb{R}^{p,q} \rightarrow \mathbb{R}_{p,q}$$
+$$(\rho\gamma k)^2 = \phi\gamma k = {\rm\ sgn\ }k$$
+\end{list}
+\subsection{Real frame groups\cite{5,6}}
+\begin{list}{}
+\item For $p,q \in \mathbb{N}$, define the real {\sl frame group} $\mathbb{G}_{p,q}$
+via the map
+$$g:\zeta(-q,p) \rightarrow \mathbb{G}_{p,q}$$
+with generators and relations
+$$\langle \mu,g_k | \mu g_k = g_k \mu,{\rm\ \ \ }\mu^2 = 1,$$
+$$(g_k)^2 = 
+\left\{
+\begin{array}{lcc}
+\mu,&{\rm\ \ }&{\rm\ if\ }k < 0\\
+1&{\rm\ \ }&{\rm\ if\ }k > 0
+\end{array}
+\right.$$
+$$g_kg_m = \mu g_mg_k{\rm\ \ \ }\forall k \ne m\rangle$$
+\end{list}
+\subsection{Canonical products\cite{1,3,4}}
+\begin{list}{}
+\item The real frame group $\mathbb{G}_{p,q}$ has order $2^{p+q+1}$
+\item Each member $w$ can be expressed as the canonically ordered product
+$$w=\mu^a\prod_{k \in T}{g_k}$$
+$$\ =\mu^a\prod_{k=-q,k\ne0}^p{g_k^{b_k}}$$
+where $T \subseteq \zeta(-q,p),a,b_k \in \{0,1\}$
+\end{list}
+\subsection{Clifford algebra of frame group\cite{1,4,5,6}}
+\begin{list}{}
+\item For $p,q \in \mathbb{N}$ embed $\mathbb{G}_{p,q}$ into 
+$\mathbb{R}_{p,q}$ via the map
+$$\alpha  \mathbb{G}_{p,q} \rightarrow \mathbb{R}_{p,q}$$
+$$\alpha 1 := 1,{\rm\ \ \ \ \ } \alpha\mu := -1$$
+$$\alpha g_k := \rho\gamma_k, {\rm \ \ \ \ \ }
+\alpha(gh) := (\alpha g)(\alpha h)$$
+\item Define {\sl basis elements} via the map
+$$e:\mathbb{P}\zeta(-q,p) \rightarrow \mathbb{R}_{p,q}, 
+{\rm \ \ \ \ \ }e_T := \alpha \prod_{k \in T}{g_k}$$
+\item Each $a \in \mathbb{R}_{p,q}$ can be expressed as
+$$a = \sum_{T \subseteq \zeta(-q,p)}{a_T e_T}$$
+\end{list} 
+\subsection{Neutral matrix representations\cite{1,2,8}}
+The {\sl representation map} $P_m$ and {\sl representation matrix} $R_m$
+make the following diagram commute:
+\begin{diagram}
+\mathbb{R}_{m,m} & \rTo^{coord} & \mathbb{R}^{4^m}\\
+\dTo^{P_m} & & \dTo_{R_m}\\
+\mathbb{R}(2^m) & \rTo_{reshape} &  \mathbb{R}^{4^m}\\
+\end{diagram}
+<<domain CLIF CliffordAlgebra>>=
+)abbrev domain CLIF CliffordAlgebra
+++ Author: Stephen M. Watt
+++ Date Created: August 1988
+++ Date Last Updated: May 17, 1991
+++ Basic Operations: wholeRadix, fractRadix, wholeRagits, fractRagits
+++ Related Domains: QuadraticForm, Quaternion, Complex
+++ Also See:
+++ AMS Classifications:
+++ Keywords: clifford algebra, grassman algebra, spin algebra
+++ Examples:
+++ References:
+++
+++ Description:
+++  CliffordAlgebra(n, K, Q) defines a vector space of dimension \spad{2**n}
+++  over K, given a quadratic form Q on \spad{K**n}.
+++
+++  If \spad{e[i]}, \spad{1<=i<=n} is a basis for \spad{K**n} then
+++     1, \spad{e[i]} (\spad{1<=i<=n}), \spad{e[i1]*e[i2]} 
+++  (\spad{1<=i1<i2<=n}),...,\spad{e[1]*e[2]*..*e[n]}
+++  is a basis for the Clifford Algebra.
+++
+++  The algebra is defined by the relations
+++     \spad{e[i]*e[j] = -e[j]*e[i]}  (\spad{i \~~= j}),
+++     \spad{e[i]*e[i] = Q(e[i])}
+++
+++  Examples of Clifford Algebras are: gaussians, quaternions, exterior 
+++  algebras and spin algebras.
+ 
+CliffordAlgebra(n, K, Q): T == Impl where
+    n: PositiveInteger
+    K: Field
+    Q: QuadraticForm(n, K)
+ 
+    PI ==> PositiveInteger
+    NNI==> NonNegativeInteger
+ 
+    T ==> Join(Ring, Algebra(K), VectorSpace(K)) with
+        e: PI -> %
+          ++ e(n) produces the appropriate unit element.
+        monomial: (K, List PI) -> %
+          ++ monomial(c,[i1,i2,...,iN]) produces the value given by
+          ++ \spad{c*e(i1)*e(i2)*...*e(iN)}.
+        coefficient:  (%, List PI) -> K
+          ++ coefficient(x,[i1,i2,...,iN])  extracts the coefficient of
+          ++ \spad{e(i1)*e(i2)*...*e(iN)} in x.
+        recip: % -> Union(%, "failed")
+          ++ recip(x) computes the multiplicative inverse of x or "failed"
+          ++ if x is not invertible.
+ 
+    Impl ==> add
+        Qeelist :=  [Q unitVector(i::PositiveInteger) for i in 1..n]
+        dim     :=  2**n
+ 
+        Rep     := PrimitiveArray K
+ 
+        New     ==> new(dim, 0$K)$Rep
+ 
+        x, y, z: %
+        c: K
+        m: Integer
+ 
+        characteristic() == characteristic()$K
+        dimension()      == dim::CardinalNumber
+ 
+        x = y ==
+            for i in 0..dim-1 repeat
+                if x.i ^= y.i then return false
+            true
+ 
+        x + y == (z := New; for i in 0..dim-1 repeat z.i := x.i + y.i; z)
+        x - y == (z := New; for i in 0..dim-1 repeat z.i := x.i - y.i; z)
+        - x   == (z := New; for i in 0..dim-1 repeat z.i := - x.i; z)
+        m * x == (z := New; for i in 0..dim-1 repeat z.i := m*x.i; z)
+        c * x == (z := New; for i in 0..dim-1 repeat z.i := c*x.i; z)
+ 
+        0            == New
+        1            == (z := New; z.0 := 1; z)
+        coerce(m): % == (z := New; z.0 := m::K; z)
+        coerce(c): % == (z := New; z.0 := c; z)
+ 
+        e b ==
+            b::NNI > n => error "No such basis element"
+            iz := 2**((b-1)::NNI)
+            z := New; z.iz := 1; z
+ 
+        -- The ei*ej products could instead be precomputed in
+        -- a (2**n)**2 multiplication table.
+        addMonomProd(c1: K, b1: NNI, c2: K, b2: NNI, z: %): % ==
+            c  := c1 * c2
+            bz := b2
+            for i in 0..n-1 | bit?(b1,i) repeat
+                -- Apply rule  ei*ej = -ej*ei for i^=j
+                k := 0
+                for j in i+1..n-1 | bit?(b1, j) repeat k := k+1
+                for j in 0..i-1   | bit?(bz, j) repeat k := k+1
+                if odd? k then c := -c
+                -- Apply rule  ei**2 = Q(ei)
+                if bit?(bz,i) then
+                    c := c * Qeelist.(i+1)
+                    bz:= (bz - 2**i)::NNI
+                else
+                    bz:= bz + 2**i
+            z.bz := z.bz + c
+            z
+ 
+        x * y ==
+            z := New
+            for ix in 0..dim-1 repeat
+                if x.ix ^= 0 then for iy in 0..dim-1 repeat
+                    if y.iy ^= 0 then addMonomProd(x.ix,ix,y.iy,iy,z)
+            z
+ 
+        canonMonom(c: K, lb: List PI): Record(coef: K, basel: NNI) ==
+            -- 0. Check input
+            for b in lb repeat b > n => error "No such basis element"
+ 
+            -- 1. Apply identity ei*ej = -ej*ei, i^=j.
+            -- The Rep assumes n is small so bubble sort is ok.
+            -- Using bubble sort keeps the exchange info obvious.
+            wasordered   := false
+            exchanges := 0
+            while not wasordered repeat
+                wasordered := true
+                for i in 1..#lb-1 repeat
+                    if lb.i > lb.(i+1) then
+                        t := lb.i; lb.i := lb.(i+1); lb.(i+1) := t
+                        exchanges := exchanges + 1
+                        wasordered := false
+            if odd? exchanges then c := -c
+ 
+            -- 2. Prepare the basis element
+            -- Apply identity ei*ei = Q(ei).
+            bz := 0
+            for b in lb repeat
+                bn := (b-1)::NNI
+                if bit?(bz, bn) then
+                    c := c * Qeelist bn
+                    bz:= ( bz - 2**bn )::NNI
+                else
+                    bz:= bz + 2**bn
+            [c, bz::NNI]
+ 
+        monomial(c, lb) ==
+            r := canonMonom(c, lb)
+            z := New
+            z r.basel := r.coef
+            z
+        coefficient(z, lb) ==
+            r := canonMonom(1, lb)
+            r.coef = 0 => error "Cannot take coef of 0"
+            z r.basel/r.coef
+ 
+        Ex ==> OutputForm
+ 
+        coerceMonom(c: K, b: NNI): Ex ==
+            b = 0 => c::Ex
+            ml := [sub("e"::Ex, i::Ex) for i in 1..n | bit?(b,i-1)]
+            be := reduce("*", ml)
+            c = 1 => be
+            c::Ex * be
+        coerce(x): Ex ==
+            tl := [coerceMonom(x.i,i) for i in 0..dim-1 | x.i^=0]
+            null tl => "0"::Ex
+            reduce("+", tl)
+
+
+	localPowerSets(j:NNI): List(List(PI)) ==
+	  l: List List PI := list []
+	  j = 0 => l
+	  Sm := localPowerSets((j-1)::NNI)
+          Sn: List List PI := []
+	  for x in Sm repeat Sn := cons(cons(j pretend PI, x),Sn)
+	  append(Sn, Sm)
+
+	powerSets(j:NNI):List List PI == map(reverse, localPowerSets j)
+
+	Pn:List List PI := powerSets(n)
+
+	recip(x: %): Union(%, "failed") ==
+	  one:% := 1
+	  -- tmp:c := x*yC - 1$C
+	  rhsEqs : List K := []
+	  lhsEqs: List List K := []
+	  lhsEqi: List K
+	  for pi in Pn repeat
+	    rhsEqs := cons(coefficient(one, pi), rhsEqs)
+
+	    lhsEqi := []
+	    for pj in Pn repeat
+		lhsEqi := cons(coefficient(x*monomial(1,pj),pi),lhsEqi)
+	    lhsEqs := cons(reverse(lhsEqi),lhsEqs)
+	  ans := particularSolution(matrix(lhsEqs),
+            vector(rhsEqs))$LinearSystemMatrixPackage(K, Vector K, Vector K, Matrix K)
+          ans case "failed" => "failed"
+	  ansP := parts(ans)
+	  ansC:% := 0
+	  for pj in Pn repeat
+	    cj:= first ansP
+	    ansP := rest ansP
+	    ansC := ansC + cj*monomial(1,pj)
+	  ansC
+
+@
+\section{License}
+<<license>>=
+--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
+--All rights reserved.
+--
+--Redistribution and use in source and binary forms, with or without
+--modification, are permitted provided that the following conditions are
+--met:
+--
+--    - Redistributions of source code must retain the above copyright
+--      notice, this list of conditions and the following disclaimer.
+--
+--    - Redistributions in binary form must reproduce the above copyright
+--      notice, this list of conditions and the following disclaimer in
+--      the documentation and/or other materials provided with the
+--      distribution.
+--
+--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
+--      names of its contributors may be used to endorse or promote products
+--      derived from this software without specific prior written permission.
+--
+--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
+--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
+--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
+--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+@
+<<*>>=
+<<license>>
+
+<<domain QFORM QuadraticForm>> 
+<<domain CLIF CliffordAlgebra>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Lounesto, P.
+"Clifford algebras and spinors",
+2nd edition, Cambridge University Press (2001)
+\bibitem{2} Porteous, I.,
+"Clifford algebras and the classical groups",
+Cambridge University Press (1995)
+Van Nostrand Reinhold, (1969)
+\bibitem{3} Bergdolt, G.
+"Orthonormal basis sets in Clifford algebras",
+in \cite{16} (1996)
+\bibitem{4} Dorst, Leo, 
+"Honing geometric algebra for its use in the computer sciences",
+pp127-152 from \cite{15} (2001)
+\bibitem{5} Braden, H.W., 
+"N-dimensional spinors: Their properties in terms of finite groups",
+American Institute of Physics,
+J. Math. Phys. 26(4), April 1985
+\bibitem{6} Lam, T.Y. and Smith, Tara L.,
+"On the Clifford-Littlewood-Eckmann groups: a new look at periodicity mod 8",
+Rocky Mountains Journal of Mathematics, vol 19, no. 3, (Summer 1989)
+\bibitem{7} Leopardi, Paul "Quick Introduction to Clifford Algebras"\\
+{\bf http://web.maths.unsw.edu.au/~leopardi/clifford-2003-06-05.pdf}
+\bibitem{8} Cartan, Elie and Study, Eduard
+"Nombres Complexes",
+Encyclopaedia Sciences Math\'ematique, \'edition fran\c caise, 15, (1908),
+d'apr\`es l'article allemand de Eduard Study, pp329-468. Reproduced as
+pp107-246 of \cite{17} 
+\bibitem{9} Hestenes, David and Sobczyck, Garret
+"Clifford algebra to geometric calculus: a unified language for 
+mathematics and physics", D. Reidel, (1984)
+\bibitem{10} Wene, G.P.,
+"The Idempotent structure of an infinite dimensional Clifford algebra",
+pp161-164 of \cite{13} (1995)
+\bibitem{11} Ashdown, M. 
+"GA Package for Maple V",\\
+http://www.mrao.cam.ac.uk/~clifford/software/GA/GAhelp5.html
+\bibitem{12} Doran, Chris and Lasenby, Anthony, 
+"Geometric Algebra for Physicists" 
+Cambridge University Press (2003) ISBN 0-521-48022-1
+\bibitem{13} Micali, A., Boudet, R., Helmstetter, J. (eds),
+"Clifford algebras and their applications in mathematical physics:
+proceedings of second workshop held at Montpellier, France, 1989",
+Kluwer Academic Publishers (1992)
+\bibitem{14} Porteous, I.,
+"Topological geometry"
+Van Nostrand Reinhold, (1969)
+\bibitem{15} Sommer, G. (editor),
+"Geometric Computing with Clifford Algebras",
+Springer, (2001)
+\bibitem{16} Ablamowicz, R., Lounesto, P., Parra, J.M. (eds)
+"Clifford algebras with numeric and symbolic computations",
+Birkh\"auser (1996)
+\bibitem{17} Cartan, Elie and Montel, P. (eds), 
+"\OE uvres Compl\`etes" Gauthier-Villars, (1953)
+\end{thebibliography}
+\end{document}