diff git a/books/bookvol10.1.pamphlet b/books/bookvol10.1.pamphlet
index ffd6276..a291a3a 100644
 a/books/bookvol10.1.pamphlet
+++ b/books/bookvol10.1.pamphlet
@@ 277,7 +277,8 @@ last century, the difficulties posed by algebraic functions caused
Hardy (1916) to state that ``there is reason to suppose that no such
method can be given''. This conjecture was eventually disproved by
Risch (1970), who described an algorithm for this problem in a series
of reports \cite{12,13,14,15}. In the past 30 years, this procedure
+of reports \cite{Ost1845,Ris68,Ris69a,Ris69b}.
+In the past 30 years, this procedure
has been repeatedly improved, extended and refined, yielding practical
algorithms that are now becoming standard and are implemented in most
of the major computer algebra systems. In this tutorial, we outline
@@ 400,7 +401,7 @@ approach is the need to factor polynomials over $\mathbb{R}$,
$\mathbb{C}$, or $\overline{K}$, thereby introducing algebraic numbers
even if the integrand and its integral are both in $\mathbb{Q}(x)$. On
the other hand, introducing algebraic numbers may be necessary, for
example it is proven in \cite{14} that any field containing an
+example it is proven in \cite{Ris69a} that any field containing an
integral of $1/(x^2+2)$ must also contain $\sqrt{2}$. Modern research
has yielded socalled ``rational'' algorithms that
\begin{itemize}
@@ 410,8 +411,8 @@ calculations being done in $K(x)$, and
express the integral
\end{itemize}
The first rational algorithms for integration date back to the
$19^{{\rm th}}$ century, when both Hermite\cite{6} and
Ostrogradsky\cite{11} invented methods for computing the $v$ of (4)
+$19^{{\rm th}}$ century, when both Hermite\cite{Her1872} and
+Ostrogradsky\cite{Ost1845} invented methods for computing the $v$ of (4)
entirely within $K(x)$. We describe here only Hermite's method, since
it is the one that has been generalized to arbitrary elementary
functions. The basic idea is that if an irreducible $p \in K[x]$
@@ 430,7 +431,7 @@ finally that
D_1=\frac{D/R}{gcd(R,D/R)}
\]
Computing recursively a squarefree factorization of $R$ completes the
one for $D$. Note that \cite{23} presents a more efficient method for
+one for $D$. Note that \cite{Yu76} presents a more efficient method for
this decomposition. Let now $f \in K(x)$ be our integrand, and write
$f=P+A/D$ where $P,A,D \in K[x]$, $gcd(A,D)=1$, and $deg(A) 1$, and that solution always has a
@@ 706,9 +707,9 @@ a factor of $FUV^{m1}$ where $F \in K[x]$ is squarefree and coprime
with $UV$. He also described an algorithm for computing an integral
basis, a necessary preprocessing for his Hermite reduction. The main
problem with that approach is that computing the integral basis,
whether by the method of \cite{20} or the local alternative \cite{21},
+whether by the method of \cite{Tr84} or the local alternative \cite{vH94},
can be in general more expansive than the rest of the reduction
process. We describe here the lazy Hermite reduction \cite{5}, which
+process. We describe here the lazy Hermite reduction \cite{Bro98}, which
avoids the precomputation of an integral basis. It is based on the
observation that if $m > 1$ and (8) does not have a solution allowing
us to perform the reduction, then either
@@ 723,7 +724,7 @@ also made up of integral elements, so that that $K[x]$module
generated by the new basis strictly contains the one generated by $w$:
\noindent
{\bf Theorem 1 (\cite{5})} {\sl Suppose that $m \ge 2$ and that
+{\bf Theorem 1 (\cite{Bro98})} {\sl Suppose that $m \ge 2$ and that
$\{S_1,\ldots,S_n\}$ as given by (9) are linearly dependent over $K(x)$,
and let $T_1,\ldots,T_n \in K[x]$ be not all 0 and such that
$\sum_{i=1}^n T_iS_i=0$. Then,
@@ 734,7 +735,7 @@ Furthermore, if $\gcd(T_1,\ldots,T_n)=1$ then
$w_0 \notin K[x]w_1+\cdots+K[x]w_n$.}
\noindent
{\bf Theorem 2 (\cite{5})} {\sl Suppose that $m \ge 2$ and that
+{\bf Theorem 2 (\cite{Bro98})} {\sl Suppose that $m \ge 2$ and that
$\{S_1,\ldots,S_n\}$ as given by (9) are linearly independent over
$K(x)$, and let $Q,T_1,\ldots,T_n \in K[x]$ be such that
\[
@@ 749,7 +750,7 @@ Furthermore,
if $\gcd(Q,T_1,\ldots,T_n)=1$ and $\deg(\gcd(V,Q)) \ge 1$, then
$w_0 \notin K[x]w_1+\cdots+K[x]w_n$.}
{\bf Theorem 3 (\cite{5})} {\sl Suppose that the denominator $F$ of
+{\bf Theorem 3 (\cite{Bro98})} {\sl Suppose that the denominator $F$ of
some $w_i$ is not squarefree, and let $F=F_1F_2^2\cdots F_k^k$ be its
squarefree factorization. Then,}
\[
@@ 929,7 +930,7 @@ integration problem by allowing only new logarithms to appear linearly
in the integral, all the other terms appearing in the integral being
already in the integrand.
{\bf Theorem 4 (Liouville \cite{8,9})} {\sl
+{\bf Theorem 4 (Liouville \cite{Lio1833a,Lio1833b})} {\sl
Let $E$ be an algebraic extension of the rational function field
$K(x)$, and $f \in E$. If $f$ has an elementary integral, then there
exist $v \in E$, constants $c_1,\ldots,c_n \in \overline{K}$ and
@@ 938,9 +939,9 @@ $u_1,\ldots,u_k \in E(c_1,\ldots,c_k)^{*}$ such that}
f=v^{'}+c_1\frac{u_1^{'}}{u_1}+\cdots+c_k\frac{u_k^{'}}{u_k}
\end{equation}
The above is a restriction to algebraic functions of the strong
Liouville Theorem, whose proof can be found in \cite{4,14}. An elegant
+Liouville Theorem, whose proof can be found in \cite{Bro97,Ris69b}. An elegant
and elementary algebraic proof of a slightly weaker version can be
found in \cite{17}. As a consequence, we can look for an integral of
+found in \cite{Ro72}. As a consequence, we can look for an integral of
the form (4), Liouville's Theorem guaranteeing that there is no
elementary integral if we cannot find one in that form. Note that the
above theorem does not say that every integral must have the above
@@ 961,7 +962,7 @@ $c_1,\ldots,c_k$. Since $D$ is squarefree, it can be shown that
$v \in {\bf O}_{K[x]}$ for any solution, and in fact $v$
corresponds to the polynomial part of the integral of rational
functions. It is however more difficult to compute than the integral
of polynomials, so Trager \cite{20} gave a change of variable that
+of polynomials, so Trager \cite{Tr84} gave a change of variable that
guarantees that either $v^{'}=0$ or $f$ has no elementary integral. In
order to describe it, we need to define the analogue for algebraic
functions of having a nontrivial polynomial part: we say that
@@ 983,7 +984,7 @@ $\alpha = \sum_{i=1}^n B_ir_ib_i/C$ where $C,B_1,\ldots,B_n \in K[x]$
and $deg(C) \ge deg(B_i)$ for each $i$. We say that the differential
$\alpha ~dx$ is integral at infinity if
$\alpha x^{1+1/r} \in {\bf O}_\infty$ where $r$ is the smallest
ramification index at infinity. Trager \cite{20} described an
+ramification index at infinity. Trager \cite{Tr84} described an
algorithm that converts an arbitrary integral basis $w_1,\ldots,w_n$
into one that is also normal at infinity, so the first part of his
integration algorithm is as follows:
@@ 1047,7 +1048,7 @@ $K(z)$, and $w$ is normal at infinity
\end{itemize}
A primitive element can be computed by considering linear combinations
of the generators of $E$ over $K(x)$ with random coefficients in
$K(x)$, and Trager \cite{20} describes an absolute factorization
+$K(x)$, and Trager \cite{Tr84} describes an absolute factorization
algorithm, so the above assumptions can be ensured, although those
steps can be computationally very expensive, except in the case of
simple radical extensions. Before describing the second part of
@@ 1107,7 +1108,7 @@ elementary, with the smallest possible number of logarithms. Steps 3
to 6 requires computing in the splitting field $K_0$ of $R$ over $K$,
but it can be proven that, as in the case of rational functions, $K_0$
is the minimal algebraic extension of $K$ necessary to express the
integral in the form (4). Trager \cite{20} describes a representation
+integral in the form (4). Trager \cite{Tr84} describes a representation
of divisors as fractional ideals and gives algorithms for the
arithmetic of divisors and for testing whether a given divisor is
principal. In order to determine whether there exists an integer $N$
@@ 1117,7 +1118,7 @@ extension to one over a finite field $\mathbb{F}_{p^q}$ for some
known that for every divisor $\delta=\sum{n_PP}$ such that
$\sum{n_P}=0$, $M\delta$ is principal for some integer
$1 \le M \le (1+\sqrt{p^q})^{2g}$, where $g$ is the genus of the curve
\cite{22}, so we compute such an $M$ by testing $M=1,2,3,\ldots$ until
+\cite{We71}, so we compute such an $M$ by testing $M=1,2,3,\ldots$ until
we find it. It can then be shown that for almost all primes $p$, if
$M\delta$ is not principal in characteristic 0, the $N\delta$ is not
principal for any integer $N \ne 0$. Since we can test whether the
@@ 1125,7 +1126,7 @@ prime $p$ is ``good'' by testing whether the image in
$\mathbb{F}_{p^q}$ of the discriminant of the discriminant of the
minimal polynomial for $y$ over $K[z]$ is 0, this yields a complete
algorithm. In the special case of hyperelliptic extensions, {\sl i.e.}
simple radical extensions of degree 2, Bertrand \cite{1} describes a
+simple radical extensions of degree 2, Bertrand \cite{Ber95} describes a
simpler representation of divisors for which the arithmetic and
principality tests are more efficient than the general methods.
@@ 1262,7 +1263,7 @@ new constant, and an exponential could in fact be algebraic, for
example $\mathbb{Q}(x)(log(x),log(2x))=\mathbb{Q}(log(2))(x)(log(x))$
and $\mathbb{Q}(x)(e^{log(x)/2})=\mathbb{Q}(x)(\sqrt{x})$. There are
however algorithms that detect all such occurences and modify the
tower accordingly \cite{16}, so we can assume that all the logarithms
+tower accordingly \cite{Ris79}, so we can assume that all the logarithms
and exponentials appearing in $E$ are monomials, and that
${\rm Const}(E)=C$. Let now $k_0$ be the largest index such that
$t_{k_0}$ is transcendental over $K=C(x)(t_1,\ldots,t_{k_01})$ and
@@ 1404,7 +1405,7 @@ special $S \in K[t]$ with $deg_t(S) > 0$, we have
R=\frac{1}{deg_t(S)}\frac{r_{d1}}{c_d}\frac{S'}{S}+\overline{R}
\]
where $\overline{R} \in K[t]$ is such that $\overline{R}=0$ or
$deg_t(\overline{R}) < e1$. Furthermore, it can be proven \cite{4}
+$deg_t(\overline{R}) < e1$. Furthermore, it can be proven \cite{Bro97}
that if $R+A/D$ has an elementary integral over $K(t)$, then
$r_{d1}/{c_d}$ is a constant, which implies that
\[
@@ 1454,7 +1455,7 @@ g=\sum_{i=1}^k\sum_{aQ_i(a)=0} a\log(\gcd{}_t(D,AaD'))
Note that the roots of each $Q_i$ must all be constants, and that the
arguments of the logarithms can be obtained directly from the
subresultant PRS of $D$ and $AzD'$ as in the rational function
case. It can then be proven \cite{4} that
+case. It can then be proven \cite{Bro97} that
\begin{itemize}
\item $fg'$ is always ``simpler'' than $f$
\item the splitting field of $Q_1\cdots Q_k$ over $K$ is the minimal
@@ 1529,7 +1530,7 @@ $z$ be a new indeterminante and compute
\begin{equation}
R(z)={\rm resultant_t}({\rm pp_z}({\rm resultant_y}(GtHD',F)),D) \in K[t]
\end{equation}
It can then be proven \cite{2} that if $f$ has an elementary integral
+It can then be proven \cite{Bro90} that if $f$ has an elementary integral
over $E$, then $R\kappa(R)$ in $K[z]$.
{\bf Example 12} {\sl
@@ 1581,7 +1582,7 @@ to $f_d$, either proving that (18) has no solution, in which case $f$
has no elementary integral, or obtaining the constant $v_{d+1}$, and
$v_d$ up to an additive constant (in fact, we apply recursively a
specialized version of the integration algorithm to equations of the
form (18), see \cite{4} for details). Write then
+form (18), see \cite{Bro97} for details). Write then
$v_d=\overline{v_d}+c_d$ where $\overline{v_d} \in K$ is known and
$c_d \in {\rm Const}(K)$ is undetermined. Equating the coefficients of
$t^{d1}$ yields
@@ 1623,13 +1624,15 @@ which is simply an integration problem for $f_0 \in K$, and
\[
f_i=v_i^{'}+ib'v_i\quad{\rm for\ }e \le i \le d, i \ne 0
\]
The above problem is called a {\sl Risch differential equation over
K}. Although solving it seems more complicated than solving $g'=f$, it
+
+The above problem is called a {\sl Risch differential equation overK}.
+Although solving it seems more complicated than solving $g'=f$, it
is actually simpler than an integration problem because we look for
the solutions $v_i$ in $K$ only rather than in an extension of
$K$. Bronstein \cite{2,3,4} and Risch \cite{12,13,14} describe
algorithms for solving this type of equation when $K$ is an elementary
extension of the rational function field.
+$K$. Bronstein \cite{Bro90,Bro91,Bro97} and Risch
+\cite{Ris68,Ris69a,Ris69b} describe algorithms for solving this type
+of equation when $K$ is an elementary extension of the rational
+function field.
\subsection{The transcendental tangent case}
Suppose now that $t=\tan(b)$ for some $b \in K$, {\sl i.e.}
@@ 1680,7 +1683,7 @@ b
where $at+b$ and $ct+d$ are the remainders module $t^2+1$ of $A$ and
$V$ respectively. The above is a coupled differential system, which
can be solved by methods similar to the ones used for Risch
differential equations \cite{4}. If it has no solution, then the
+differential equations \cite{Bro97}. If it has no solution, then the
integral is not elementary, otherwise we reduce the integrand to
$h \in K[t]$, at which point the polynomial reduction either proves
that its integral is not elementary, or reduce the integrand to an
@@ 1869,7 +1872,7 @@ whose solution is $v_2=2$, implying that $h=2y'$, hence that
In the general case when $E$ is not a radical extension of $K(t)$,
(21) is solved by bounding $deg_t(v_i)$ and comparing the Puiseux
expansions at infinity of $\sum_{i=1}^n v_iw_i$ with those of the form
(20) of $h$, see \cite{2,12} for details.
+(20) of $h$, see \cite{Bro90,Ris68} for details.
\subsection{The algebraic exponential case}
The transcendental exponential case method also generalizes to the
@@ 1992,7 +1995,7 @@ following the Hermite reduction, any solution of (13) must have
$v=\sum_{i=1}^n v_iw_i/t^m$ where $v_1,\ldots,v_m \in K[t]$. We can
compute $v$ by bounding $deg_t(v_i)$ and comparing the Puiseux
expansions at $t=0$ and at infinity of $\sum_{i=1}^n v_iw_i/t^m$ with
those of the form (20) of the integrand, see \cite{2,12} for details.
+those of the form (20) of the integrand, see \cite{Bro90,Ris68} for details.
Once we are reduced to solving (13) for $v \in K$, constants
$c_1,\ldots,c_k \in \overline{K}$ and
@@ 2002,7 +2005,7 @@ places above $t=0$ and at infinity in a manner similar to the
algebraic logarithmic case, at which point the algorithm proceeds by
constructing the divisors $\delta_j$ and the $u_j$'s as in that
case. Again, the details are quite technical and can be found in
\cite{2,12,13}.
+\cite{Bro90,Ris68,Ris69a}.
\chapter{Singular Value Decomposition}
\section{Singular Value Decomposition Tutorial}
@@ 2415,7 +2418,7 @@ are the same. We are trying to predict patterns of how words occur
in documents instead of trying to predict patterns of how players
score on holes.
\chapter{Quaternions}
from\cite{1}:
+from\cite{Alt05}:
\begin{quotation}
Quaternions are inextricably linked to rotations.
Rotations, however, are an accident of threedimensional space.
@@ 2437,7 +2440,7 @@ The Theory of Quaternions is due to Sir William Rowan Hamilton,
Royal Astronomer of Ireland, who presented his first paper on the
subject to the Royal Irish Academy in 1843. His Lectures on
Quaternions were published in 1853, and his Elements, in 1866,
shortly after his death. The Elements of Quaternions by Tait\cite{33} is
+shortly after his death. The Elements of Quaternions by Tait\cite{Ta1980} is
the accepted textbook for advanced students.
Large portions of this file are derived from a public domain version
@@ 7586,13 +7589,13 @@ i =
\right]
$$
\chapter{Clifford Algebra\cite{39}}
+\chapter{Clifford Algebra\cite{Fl09}}
This is quoted from John Fletcher's web page\cite{39} (with permission).
+This is quoted from John Fletcher's web page\cite{Fl09} (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
+discuss this and in particular Ablamowicz\cite{Ab98} 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
@@ 7607,7 +7610,7 @@ Clifford(2), Clifford(3) and Clifford(2,2).
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
+Ablamowicz\cite{Ab98} 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
@@ 7622,7 +7625,7 @@ 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}.
+author\cite{Fl01}.
\section{Clifford Basis Matrix Theory}
@@ 8064,7 +8067,7 @@ simple cases of wide usefulness.
\subsection{Example 3: Clifford (2,2)}
The following basis matrices are given by Ablamowicz\cite{41}
+The following basis matrices are given by Ablamowicz\cite{Ab98}
\[
\begin{array}{cc}
@@ 8314,7 +8317,7 @@ 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}.
+system for Clifford algebra such as described in Fletcher\cite{Fl01}.
The result is
\[
@@ 8341,6 +8344,73 @@ 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{Package for Algebraic Function Fields}
+
+PAFF is a Package for Algebraic Function Fields in one variable
+by Ga\'etan Hach\'e
+
+PAFF is a package written in Axiom and one of its many purpose is to
+construct geometric Goppa codes (also called algebraic geometric codes
+or AGcodes). This package was written as part of Ga\'etan's doctorate
+thesis on ``Effective construction of geometric codes'': this thesis was
+done at Inria in Rocquencourt at project CODES and under the direction
+of Dominique LeBrigand at Université Pierre et Marie Curie (Paris
+6). Here is a r\'esum\'e of the thesis.
+
+It is well known that the most difficult part in constructing AGcode
+is the computation of a basis of the vector space ``L(D)'' where D is a
+divisor of the function field of an irreducible curve. To compute such
+a basis, PAFF used the BrillNoether algorithm which was generalized
+to any plane curve by D. LeBrigand and J.J. Risler (see [LR88] ). In [Ha96]
+you will find more details about the algorithmic aspect of the
+BrillNoether algorithm. Also, if you prefer, as I do, a strictly
+algebraic approach, see [Ha95]. This is the approach I used in my thesis
+([Ha96]) and of course this is where you will find complete details about
+the implementation of the algorithm. The algebraic approach use the
+theory of algebraic function field in one variable : you will find in
+[St93] a very good introduction to this theory and AGcodes.
+
+It is important to notice that PAFF can be used for most computation
+related to the function field of an irreducible plane curve. For
+example, you can compute the genus, find all places above all the
+singular points, compute the adjunction divisor and of course compute
+a basis of the vector space L(D) for any divisor D of the function
+field of the curve.
+
+There is also the package PAFFFF which is especially designed to be
+used over finite fields. This package is essentially the same as PAFF,
+except that the computation are done over ``dynamic extensions'' of the
+ground field. For this, I used a simplify version of the notion of
+dynamic algebraic closure as proposed by D. Duval (see [Du95]).
+
+Example 1
+
+This example compute the genus of the projective plane curve defined by:
+\begin{verbatim}
+ 5 2 3 4
+ X + Y Z + Y Z = 0
+\end{verbatim}
+over the field GF(2).
+
+First we define the field GF(2).
+\begin{verbatim}
+K:=PF 2
+R:=DMP([X,Y,Z],K)
+P:=PAFF(K,[X,Y,Z],BLQT)
+\end{verbatim}
+
+We defined the polynomial of the curve.
+\begin{verbatim}
+C:R:=X**5 + Y**2*Z**3+Y*Z**4
+\end{verbatim}
+
+We give it to the package PAFF(K,[X,Y,Z]) which was assigned to the
+variable $P$.
+
+\begin{verbatim}
+setCurve(C)$P
+\end{verbatim}
+
\chapter{Groebner Basis}
Groebner Basis
\chapter{Greatest Common Divisor}
@@ 8359,122 +8429,186 @@ Chinese Remainder Theorem
Gaussian Elimination
\chapter{Diophantine Equations}
Diophantine Equations
+
\begin{thebibliography}{99}
\bibitem{1} Altmann, Simon L. Rotations, Quaternions, and Double Groups
+\bibitem[Ab98]{Ab98}
+Ablamowicz Rafal, ``Spinor Representations of Clifford
+Algebras: A Symbolic Approach'', Computer Physics Communications
+Vol. 115, No. 23, December 11, 1998, pages 510535.
+\bibitem[Alt05]{Alt05}
+Altmann, Simon L. Rotations, Quaternions, and Double Groups
Dover Publications, Inc. 2005 ISBN 0486445186
\bibitem{2} Laurent Bertrand. Computing a hyperelliptic integral using
+\bibitem[Ber95]{Ber95}
+Laurent Bertrand. Computing a hyperelliptic integral using
arithmetic in the jacobian of the curve. {\sl Applicable Algebra in
Engineering, Communication and Computing}, 6:275298, 1995
\bibitem{3} M. Bronstein. On the integration of elementary functions.
+\bibitem[Bro90]{Bro90}
+M. Bronstein. ``On the integration of elementary functions''
{\sl Journal of Symbolic Computation} 9(2):117173, February 1990
\bibitem{4} M. Bronstein. The Risch differential equation on an
+\bibitem[Bro91]{Bro91}
+M. Bronstein. The Risch differential equation on an
algebraic curve. In S.Watt, editor, {\sl Proceedings of ISSAC'91},
pages 241246, ACM Press, 1991.
\bibitem{5} M. Bronstein. {\sl Symbolic Integration ITranscendental
+\bibitem[Bro97]{Bro97}
+M. Bronstein. {\sl Symbolic Integration ITranscendental
Functions.} Springer, Heidelberg, 1997
\bibitem{6} M. Bronstein. The lazy hermite reduction. Rapport de
+\bibitem[Br98]{Br98}
+Bronstein, Manuel "Symbolic Integration Tutorial"
+INRIA Sophia Antipolis ISSAC 1998 Rostock
+\bibitem[Bro98]{Bro98}
+M. Bronstein. The lazy hermite reduction. Rapport de
Recherche RR3562, INRIA, 1998
\bibitem{7} E. Hermite. Sur l'int\'{e}gration des fractions
+\bibitem[CS03]{CS03}
+Conway, John H. and Smith, Derek, A., ``On Quaternions and Octonions''
+A.K Peters, Natick, MA. (2003) ISBN 1568811349
+\bibitem[Dal03]{Dal03}
+Daly, Timothy, ``The Axiom Wiki Website''
+\verbhttp://axiom.axiomdeveloper.org
+\bibitem[Dal09]{Dal09}
+Daly, Timothy, "The Axiom Literate Documentation"
+\verbhttp://axiom.axiomdeveloper.org/axiomwebsite/documentation.html
+\bibitem[Du95]{Du95}
+Duval, D. ``Evaluation dynamique et cl\^oture alg\'ebrique en Axiom''.
+Journal of Pure and Applied Algebra, no99, 1995, pp. 267295.
+\bibitem[Fl01]{Fl01}
+Fletcher, John P. ``Symbolic processing of Clifford Numbers in C++'',
+Paper 25, AGACSE 2001.
+\bibitem[Fl09]{Fl09}
+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.
+\verbwww.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php
+\bibitem[Ga95]{Ga95}
+Garcia, A. and Stichtenoth, H.
+``A tower of ArtinSchreier extensions of function fields attaining the
+DrinfeldVladut bound'' Invent. Math., vol. 121, 1995, pp. 211222.
+\bibitem[Ha1896]{Ha1896}
+Hathway, Arthur S., "A Primer Of Quaternions" (1896)
+\bibitem[Ha95]{Ha95}
+Hach\'e, G. ``Computation in algebraic function fields for effective
+construction of algebraicgeometric codes''
+Lecture Notes in Computer Science, vol. 948, 1995, pp. 262278.
+\bibitem[Ha96]{Ha96}
+Hach\'e, G. ``Construction effective des codes g\'eom\'etriques''
+Th\'ese de doctorat de l'Universit\'e Pierre et Marie Curie (Paris 6),
+Septembre 1996.
+\bibitem[Her1872]{Her1872}
+E. Hermite. Sur l'int\'{e}gration des fractions
rationelles. {\sl Nouvelles Annales de Math\'{e}matiques}
($2^{eme}$ s\'{e}rie), 11:145148, 1872
\bibitem{8} Daniel Lazard and Renaud Rioboo. Integration of rational
+\bibitem[HI96]{HI96}
+Huang, M.D. and Ierardi, D.
+``Efficient algorithms for RiemannRoch problem and for addition in the
+jacobian of a curve''
+Proceedings 32nd Annual Symposium on Foundations of Computer Sciences.
+IEEE Comput. Soc. Press, pp. 678687.
+\bibitem[HL95]{HL95}
+Hach\'e, G. and Le Brigand, D.
+``Effective construction of algebraic geometry codes''
+IEEE Transaction on Information Theory, vol. 41, n27 6,
+November 1995, pp. 16151628.
+\bibitem[JS92]{JS92}
+Richard D. Jenks and Robert S. Sutor ``AXIOM: The Scientific Computation
+System''
+SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1992 ISBN 0387978550 (New York), 3540978550 (Berlin) 742pp
+LCCN QA76.95.J46 1992
+\bibitem[Knu84]{Knu84}
+Knuth, Donald, {\it The \TeX{}book} \\
+Reading, Massachusetts, AddisonWesley Publishing Company, Inc.,
+1984. ISBN 0201134489
+\bibitem[Kn92]{Kn92}
+Knuth, Donald E., ``Literate Programming''
+Center for the Study of Language and Information
+ISBN 0937073814 Stanford CA (1992)
+\bibitem[La86]{La86}
+Lamport, Leslie,
+{\it LaTeX: A Document Preparation System,} \\
+Reading, Massachusetts, AddisonWesley Publishing Company, Inc.,
+1986. ISBN 020115790X
+\bibitem[LR88]{LR88}
+Le Brigand, D. and Risler, J.J.
+``Algorithme de BrillNoether et codes de Goppa''
+Bull. Soc. Math. France, vol. 116, 1988, pp. 231253.
+\bibitem[LR90]{LR90}
+Daniel Lazard and Renaud Rioboo. Integration of rational
functions: Rational coputation of the logarithmic part {\sl Journal of
Symbolic Computation}, 9:113116:1990
\bibitem{9} Joseph Liouville. Premier m\'{e}moire sur la
+\bibitem[Lio1833a]{Lio1833a}
+Joseph Liouville. Premier m\'{e}moire sur la
d\'{e}termination des int\'{e}grales dont la valeur est
alg\'{e}brique. {\sl Journal de l'Ecole Polytechnique}, 14:124148,
1833
\bibitem{10} Joseph Liouville. Second m\'{e}moire sur la
+alg\'{e}brique. {\sl Journal de l'Ecole Polytechnique}, 14:124148, 1833
+\bibitem[Lio1833b]{Lio1833b}
+Joseph Liouville. Second m\'{e}moire sur la
d\'{e}termination des int\'{e}grales dont la valeur est
alg\'{e}brique. {\sl Journal de l'Ecole Polytechnique}, 14:149193,
1833
\bibitem{11} Thom Mulders. A note on subresultants and a correction to
+\bibitem[Mul97]{Mul97}
+Thom Mulders. A note on subresultants and a correction to
the lazard/rioboo/trager formula in rational function integration {\sl
Journal of Symbolic Computation}, 24(1):4550, 1997
\bibitem{12} M.W. Ostrogradsky. De l'int\'{e}gration des fractions
+\bibitem[Ost1845]{Ost1845}
+M.W. Ostrogradsky. De l'int\'{e}gration des fractions
rationelles. {\sl Bulletin de la Classe PhysicoMath\'{e}matiques de
l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,}
IV:145167,286300, 1845
\bibitem{13} Robert Risch. On the integration of elementary functions
+\bibitem[Pu09]{Pu09}
+Puffinware LLC ``Singular Value Decomposition (SVD) Tutorial''
+\verbwww.puffinwarellc.com/p3a.htm
+\bibitem[Ra03]{Ra03}
+Ramsey, Norman ``Noweb  A Simple, Extensible Tool for Literate Programming''
+\verbwww.eecs.harvard.edu/~nr/noweb
+\bibitem[Ris68]{Ris68}
+Robert Risch. On the integration of elementary functions
which are built up using algebraic operations. Research Report
SP2801/002/00, System Development Corporation, Santa Monica, CA, USA,
1968
\bibitem{14} Robert Risch. Further results on elementary
functions. Research Report RC2042, IBM Research, Yorktown Heights,
NY, USA, 1969
\bibitem{15} Robert Risch, The problem of integration in finite
terms. {\sl Transactions of the American Mathematical Society}
139:167189, 1969
\bibitem{16} Robert Risch. The solution of problem of integration in
finite terms. {\sl Transactions of the American Mathematical Society}
76:605608, 1970
\bibitem{17} Robert Risch. Algebraic properties of the elementary
functions of analysis. {\sl American Journal of Mathematics},
101:743759, 1979
\bibitem{18} Maxwell Rosenlicht. Integration in finite terms. {\sl
American Mathematical Monthly}, 79:963972, 1972
\bibitem{19} Michael Rothstein. A new algorithm for the integration of
exponential and logarithmic functions. In {\sl Proceedings of the 1977
+SP2801/002/00, System Development Corporation, Santa Monica, CA, USA, 1968
+\bibitem[Ris69a]{Ris69a}
+Robert Risch. ``Further results on elementary functions''
+Research Report RC2042, IBM Research, Yorktown Heights, NY, USA, 1969
+\bibitem[Ris69b]{Ris69b}
+Robert Risch, ``The problem of integration in finite terms''
+{\sl Transactions of the American Mathematical Society} 139:167189, 1969
+\bibitem[Ris70]{Ris70}
+Robert Risch. ``The solution of problem of integration in finite terms''
+{\sl Transactions of the American Mathematical Society} 76:605608, 1970
+\bibitem[Ris79]{Ris79}
+Robert Risch. ``Algebraic properties of the elementary functions of analysis''
+{\sl American Journal of Mathematics}, 101:743759, 1979
+\bibitem[Ro72]{Ro72}
+Maxwell Rosenlicht. Integration in finite terms.
+{\sl American Mathematical Monthly}, 79:963972, 1972
+\bibitem[Ro77]{Ro77}
+Michael Rothstein. ``A new algorithm for the integration of
+exponential and logarithmic functions'' In {\sl Proceedings of the 1977
MACSYMA Users Conference}, pages 263274. NASA Pub CP2012, 1977
\bibitem{20} Barry Trager. Algebraic factoring and rational function
integration. In {Proceedings of SYMSAC'76} pages 219226, 1976
\bibitem{21} Barry Trager {\sl On the integration of algebraic
functions}, PhD thesis, MIT, Computer Science, 1984
\bibitem{22} M. van Hoeij. An algorithm for computing an integral
basis in an algebraic function field. {\sl J. Symbolic Computation}
+\bibitem[St93]{St93}
+Stichtenoth, H. ``Algebraic function fields and codes''
+SpringerVerlag, 1993, University Text.
+\bibitem[Ta1890]{Ta1980}
+Tait, P.G.,{\it An Elementary Treatise on Quaternions}
+C.J. Clay and Sons, Cambridge University Press Warehouse, Ave Maria Lane 1890
+\bibitem[Tr76]{Tr76}
+Trager, Barry ``Algebraic factoring and rational function integration''
+In {Proceedings of SYMSAC'76} pages 219226, 1976
+\bibitem[Tr84]{Tr84}
+Trager Barry {\sl On the integration of algebraic functions},
+PhD thesis, MIT, Computer Science, 1984
+\bibitem[vH94]{vH94}
+M. van Hoeij. ``An algorithm for computing an integral
+basis in an algebraic function field'' {\sl J. Symbolic Computation}
18(4):353364, October 1994
\bibitem{23} Andr\'{e} Weil, {\sl Courbes alg\'{e}briques et
+\bibitem[Wa03]{Wa03}
+Watt, Stephen, ``Aldor'', \verbwww.aldor.org
+\bibitem[We71]{We71}
+Andr\'{e} Weil, {\sl Courbes alg\'{e}briques et
vari\'{e}t\'{e}s Abeliennes} Hermann, Paris, 1971
\bibitem{24} D.Y.Y. Yun. On squarefree decomposition algorithms. In
+\bibitem[Wo09]{Wo09}
+Wolfram Research, \verbmathworld.wolfram.com/Quaternion.html
+\bibitem[Yu76]{Yu76}
+D.Y.Y. Yun. ``On squarefree decomposition algorithms''
{\sl Proceedings of SYMSAC'76} pages 2635, 1976
\bibitem{25} Bronstein, Manuel "Symbolic Integration Tutorial"
INRIA Sophia Antipolis ISSAC 1998 Rostock
\bibitem{26} Jenks, R.J. and Sutor, R.S.
``Axiom  The Scientific Computation System''
SpringerVerlag New York (1992)
ISBN 0387978550
\bibitem{27} Knuth, Donald E., ``Literate Programming''
Center for the Study of Language and Information
ISBN 0937073814
Stanford CA (1992)
\bibitem{28} Daly, Timothy, ``The Axiom Wiki Website''\\
{\bf http://axiom.axiomdeveloper.org}
\bibitem{29} Watt, Stephen, ``Aldor'',\\
{\bf http://www.aldor.org}
\bibitem{30} Lamport, Leslie, ``Latex  A Document Preparation System'',
AddisonWesley, New York ISBN 0201529831
\bibitem{31} Ramsey, Norman ``Noweb  A Simple, Extensible Tool for
Literate Programming''\\
{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb}
\bibitem{32} Daly, Timothy, "The Axiom Literate Documentation"\\
{\bf http://axiom.axiomdeveloper.org/axiomwebsite/documentation.html}
\bibitem{33} {\bf http://www.puffinwarellc.com/p3a.htm}
\bibitem{34} Tait, P.G.,
{\it An Elementary Treatise on Quaternions} \\
C.J. Clay and Sons, Cambridge University Press Warehouse,
Ave Maria Lane 1890
\bibitem{35} Knuth, Donald, {\it The \TeX{}book} \\
Reading, Massachusetts,
AddisonWesley Publishing Company, Inc.,
1984. ISBN 0201134489
\bibitem{36} Hathway, Arthur S., "A Primer Of Quaternions" (1896)
\bibitem{37} Conway, John H. and Smith, Derek, A.,
"On Quaternions and Octonions", A.K Peters, Natick, MA. (2003)
ISBN 1568811349
\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.
\verbhttp://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. 23, December 11, 1998, pages 510535.

\end{thebibliography}
\end{document}

+\chapter{Index}
\printindex
\end{document}
diff git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet
index 0a3a5ca..99bcd11 100644
 a/books/bookvolbib.pamphlet
+++ b/books/bookvolbib.pamphlet
@@ 121,6 +121,9 @@ In Wang [Wan92] pp369375 ISBN 0897914899 (soft cover) 0897914902
T. Daly ``Axiom as open source'' SIGSAM Bulletin (ACM Special Interest Group
on Symbolic and Algebraic Manipulation) 36(1) pp28?? March 2002
CODEN SIGSBZ ISSN 01635824
+\bibitem[Dal03]{Dal03}
+Daly, Timothy, ``The Axiom Wiki Website''
+\verbhttp://axiom.axiomdeveloper.org
\bibitem[Dal09]{Dal09}
Daly, Timothy, "The Axiom Literate Documentation"
\verbhttp://axiom.axiomdeveloper.org/axiomwebsite/documentation.html
@@ 439,6 +442,9 @@ LCCN QA76.95.I59 1989
Online 72: conference proceedings ... international conference on online
interactive computing, Brunel University, Uxbridge, England, 47 September
1972 ISBN 0903796023 LCCN QA76.55.O54 1972 Two volumes.
+\bibitem[Pa07]{Pa07}
+Page, William S. ``Axiom  Open Source Computer Algebra System'' Poster
+ISSAC 2007 Proceedings Vol 41 No 3 Sept 2007 p114
\bibitem[Pet71]{Pet71}
S. R. Petric, editor. Proceedings of the second symposium on Symbolic and
Algebraic Manipulation, March 2325, 1971, Los Angeles, California, ACM Press,
@@ 575,17 +581,10 @@ Karlsruhe, Germany, 1992
\section{Axiom Citations of External Sources}
\begin{thebibliography}{999}
\bibitem[Lam]{Lam}
Lamport, Leslie,
{\it LaTeX: A Document Preparation System,} \\
Reading, Massachusetts,
AddisonWesley Publishing Company, Inc.,
1986. ISBN 020115790X
\bibitem[Knu84]{Knu84}
Knuth, Donald, {\it The \TeX{}book} \\
Reading, Massachusetts,
AddisonWesley Publishing Company, Inc.,
1984. ISBN 0201134489
+\bibitem[Ab98]{Ab98}
+Ablamowicz Rafal, ``Spinor Representations of Clifford
+Algebras: A Symbolic Approach'', Computer Physics Communications
+Vol. 115, No. 23, December 11, 1998, pages 510535.
\bibitem[Alt05]{Alt05}
Altmann, Simon L. Rotations, Quaternions, and Double Groups
Dover Publications, Inc. 2005 ISBN 0486445186
@@ 593,6 +592,9 @@ Dover Publications, Inc. 2005 ISBN 0486445186
Laurent Bertrand. Computing a hyperelliptic integral using
arithmetic in the jacobian of the curve. {\sl Applicable Algebra in
Engineering, Communication and Computing}, 6:275298, 1995
+\bibitem[Br98]{Br98}
+Bronstein, Manuel "Symbolic Integration Tutorial"
+INRIA Sophia Antipolis ISSAC 1998 Rostock
\bibitem[Bro90]{Bro90}
M. Bronstein. ``On the integration of elementary functions''
{\sl Journal of Symbolic Computation} 9(2):117173, February 1990
@@ 606,10 +608,24 @@ Springer, Heidelberg, 1997 ISBN 3540214933
\bibitem[Bro98]{Bro98}
M. Bronstein. ``The lazy hermite reduction'' Rapport de
Recherche RR3562, INRIA, 1998
+\bibitem[CS03]{CS03}
+Conway, John H. and Smith, Derek, A., ``On Quaternions and Octonions''
+A.K Peters, Natick, MA. (2003) ISBN 1568811349
+\bibitem[Fl01]{Fl01}
+Fletcher, John P. ``Symbolic processing of Clifford Numbers in C++'',
+Paper 25, AGACSE 2001.
+\bibitem[Fl09]{Fl09}
+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.
+\verbwww.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php
\bibitem[Ga95]{Ga95}
Garcia, A. and Stichtenoth, H.
``A tower of ArtinSchreier extensions of function fields attaining the
DrinfeldVladut bound'' Invent. Math., vol. 121, 1995, pp. 211222.
+\bibitem[Ha1896]{Ha1896}
+Hathway, Arthur S., "A Primer Of Quaternions" (1896)
\bibitem[Ha95]{Ha95}
Hach\'e, G. ``Computation in algebraic function fields for effective
construction of algebraicgeometric codes''
@@ 633,6 +649,19 @@ Hach\'e, G. and Le Brigand, D.
``Effective construction of algebraic geometry codes''
IEEE Transaction on Information Theory, vol. 41, n27 6,
November 1995, pp. 16151628.
+\bibitem[Knu84]{Knu84}
+Knuth, Donald, {\it The \TeX{}book} \\
+Reading, Massachusetts, AddisonWesley Publishing Company, Inc.,
+1984. ISBN 0201134489
+\bibitem[Kn92]{Kn92}
+Knuth, Donald E., ``Literate Programming''
+Center for the Study of Language and Information
+ISBN 0937073814 Stanford CA (1992)
+\bibitem[La86]{La86}
+Lamport, Leslie,
+{\it LaTeX: A Document Preparation System,} \\
+Reading, Massachusetts, AddisonWesley Publishing Company, Inc.,
+1986. ISBN 020115790X
\bibitem[LR88]{LR88}
Le Brigand, D. and Risler, J.J.
``Algorithme de BrillNoether et codes de Goppa''
@@ 658,6 +687,12 @@ M.W. Ostrogradsky. De l'int\'{e}gration des fractions
rationelles. {\sl Bulletin de la Classe PhysicoMath\'{e}matiques de
l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,}
IV:145167,286300, 1845
+\bibitem[Pu09]{Pu09}
+Puffinware LLC ``Singular Value Decomposition (SVD) Tutorial''
+\verbwww.puffinwarellc.com/p3a.htm
+\bibitem[Ra03]{Ra03}
+Ramsey, Norman ``Noweb  A Simple, Extensible Tool for Literate Programming''
+\verbwww.eecs.harvard.edu/~nr/noweb
\bibitem[Ris68]{Ris68}
Robert Risch. ``On the integration of elementary functions
which are built up using algebraic operations'' Research Report
@@ 674,8 +709,38 @@ Robert Risch. ``The solution of problem of integration in finite terms''
\bibitem[Ris79]{Ris79}
Robert Risch. ``Algebraic properties of the elementary functions of analysis''
{\sl American Journal of Mathematics}, 101:743759, 1979
+\bibitem[Ro72]{Ro72}
+Maxwell Rosenlicht. Integration in finite terms.
+{\sl American Mathematical Monthly}, 79:963972, 1972
+\bibitem[Ro77]{Ro77}
+Michael Rothstein. ``A new algorithm for the integration of
+exponential and logarithmic functions'' In {\sl Proceedings of the 1977
+MACSYMA Users Conference}, pages 263274. NASA Pub CP2012, 1977
\bibitem[St93]{St93}
Stichtenoth, H. ``Algebraic function fields and codes''
SpringerVerlag, 1993, University Text.
+\bibitem[Ta1890]{Ta1980}
+Tait, P.G.,{\it An Elementary Treatise on Quaternions}
+C.J. Clay and Sons, Cambridge University Press Warehouse, Ave Maria Lane 1890
+\bibitem[Tr76]{Tr76}
+Trager, Barry ``Algebraic factoring and rational function integration''
+In {Proceedings of SYMSAC'76} pages 219226, 1976
+\bibitem[Tr84]{Tr84}
+Trager Barry {\sl On the integration of algebraic functions},
+PhD thesis, MIT, Computer Science, 1984
+\bibitem[vH94]{vH94}
+M. van Hoeij. ``An algorithm for computing an integral
+basis in an algebraic function field'' {\sl J. Symbolic Computation}
+18(4):353364, October 1994
+\bibitem[Wa03]{Wa03}
+Watt, Stephen, ``Aldor'', \verbwww.aldor.org
+\bibitem[We71]{We71}
+Andr\'{e} Weil, {\sl Courbes alg\'{e}briques et
+vari\'{e}t\'{e}s Abeliennes} Hermann, Paris, 1971
+\bibitem[Wo09]{Wo09}
+Wolfram Research, \verbmathworld.wolfram.com/Quaternion.html
+\bibitem[Yu76]{Yu76}
+D.Y.Y. Yun. ``On squarefree decomposition algorithms''
+{\sl Proceedings of SYMSAC'76} pages 2635, 1976
\end{thebibliography}
\end{document}
diff git a/changelog b/changelog
index 3ff9317..01d63cd 100644
 a/changelog
+++ b/changelog
@@ 1,3 +1,6 @@
+20100419 tpd src/axiomwebsite/patches.html 20100419.02.tpd.patch
+20100419 tpd books/bookvol10.1 rename and align biblio with bookvolbib
+20100419 tpd books/bookvolbib rename and align biblio with bookvol10.1
20100419 tpd src/axiomwebsite/patches.html 20100419.01.tpd.patch
20100419 tpd books/bookvolbib Du95, Ga95, Ha95, Ha96, HI96, HL95, LR88, St93
20100418 tpd src/axiomwebsite/patches.html 20100418.05.tpd.patch
diff git a/src/axiomwebsite/patches.html b/src/axiomwebsite/patches.html
index 2132805..9611744 100644
 a/src/axiomwebsite/patches.html
+++ b/src/axiomwebsite/patches.html
@@ 2639,5 +2639,7 @@ books/bookvol10.3 add UnivariateTaylorSeriesCZero
books/bookvolbib add citation SDJ07
20100419.01.tpd.patch
books/bookvolbib Du95, Ga95, Ha95, Ha96, HI96, HL95, LR88, St93
+20100419.02.tpd.patch
+books/bookvolbib,bookvol1 rename and align biblio sections