diff git a/Makefile.pamphlet b/Makefile.pamphlet
index 1f08124..014a659 100644
 a/Makefile.pamphlet
+++ b/Makefile.pamphlet
@@ 221,6 +221,7 @@ clean:
@ rm f books/Makefile
@ rm f books/Makefile.dvi
@ rm f books/Makefile.pdf
+ @ rm f books/axiom.bib
@ rm f lsp/axiom.sty
@ rm f lsp/Makefile lsp/Makefile.dvi lsp/Makefile.pdf
@ rm rf lsp/gcl*
diff git a/books/Makefile.pamphlet b/books/Makefile.pamphlet
index 59dac4e..74df390 100644
 a/books/Makefile.pamphlet
+++ b/books/Makefile.pamphlet
@@ 23,6 +23,7 @@ PDF=${AXIOM}/doc
IN=${SPD}/books
LATEX=latex
MAKEINDEX=makeindex
+BIBTEX=bibtex
DVIPDFM=dvipdfm
DVIPS=dvips Ppdf
PS2PDF=ps2pdf
@@ 38,6 +39,7 @@ BOOKPDF=${PDF}/bookvol0.pdf ${PDF}/bookvol1.pdf ${PDF}/bookvol2.pdf \
${PDF}/bookvol11.pdf ${PDF}/bookvol12.pdf ${PDF}/bookvol13.pdf \
${PDF}/bookvolbib.pdf
+
OTHER= ${PDF}/refcard.pdf ${PDF}/endpaper.pdf ${PDF}/rosetta.pdf
all: announce ${BOOKPDF} ${PDF}/toc.pdf ${OTHER} spadedit
@@ 53,7 +55,14 @@ finish:
@ echo FINISHED BUILDING PDF FILES books/Makefile
@ echo ==========================================
${PDF}/%.pdf: ${IN}/%.pamphlet
+${PDF}/axiom.bib:
+ @ echo ===========================================
+ @ echo making ${PDF}/axiom.bib from ${IN}/bookvolbib.pamphlet
+ @ echo ===========================================
+ @${BOOKS}/tanglec ${BOOKS}/bookvolbib.pamphlet axiom.bib \
+ >${PDF}/axiom.bib
+
+${PDF}/%.pdf: ${IN}/%.pamphlet ${PDF}/axiom.bib
@ echo ===========================================
@ echo making ${PDF}/$*.pdf from ${IN}/$*.pamphlet
@ echo ===========================================
@@ 67,6 +76,8 @@ ${PDF}/%.pdf: ${IN}/%.pamphlet
${RM} $*.toc ; \
${LATEX} $*.pamphlet ; \
${MAKEINDEX} $*.idx 1>/dev/null 2>/dev/null ; \
+ ${BIBTEX} $*.aux ; \
+ ${LATEX} $*.pamphlet >/dev/null ; \
${LATEX} $*.pamphlet >/dev/null ; \
${DVIPDFM} $*.dvi 2>/dev/null ; \
${RM} $*.aux $*.dvi $*.log $*.ps $*.idx $*.tex $*.pamphlet ; \
@@ 76,6 +87,8 @@ ${PDF}/%.pdf: ${IN}/%.pamphlet
${RM} $*.toc ; \
${LATEX} $*.pamphlet >${TMP}/trace ; \
${MAKEINDEX} $*.idx 1>/dev/null 2>/dev/null ; \
+ ${BIBTEX} $*.aux 1>/dev/null 2>/dev/null ; \
+ ${LATEX} $*.pamphlet >${TMP}/trace ; \
${LATEX} $*.pamphlet >${TMP}/trace ; \
${DVIPDFM} $*.dvi 2>${TMP}/trace ; \
${RM} $*.aux $*.dvi $*.log $*.ps $*.idx $*.tex $*.pamphlet ; \
diff git a/books/bookvol0.pamphlet b/books/bookvol0.pamphlet
index 874493f..bcb7c0a 100644
 a/books/bookvol0.pamphlet
+++ b/books/bookvol0.pamphlet
@@ 9790,7 +9790,7 @@ running on an IBM workstation, for example, issue
Axiom can produce \TeX{} output for your \index{output formats!TeX
@{\TeX{}}} expressions. \index{TeX output format @{\TeX{}} output format}
The output is produced using macros from the \LaTeX{} document
preparation system by Leslie Lamport\cite{1}. The printed version
+preparation system by Leslie Lamport\cite{Lamp86}. The printed version
of this book was produced using this formatter.
To turn on \TeX{} output formatting, issue this.
@@ 88398,22 +88398,14 @@ SUCH DAMAGE.
\end{verbatim}
\eject
\eject
\begin{thebibliography}{99}
\bibitem{1} Lamport, Leslie,
{\it LaTeX: A Document Preparation System,} \\
Reading, Massachusetts,
AddisonWesley Publishing Company, Inc.,
1986. ISBN 020115790X
\bibitem{2} Knuth, Donald, {\it The \TeX{}book} \\
Reading, Massachusetts,
AddisonWesley Publishing Company, Inc.,
1984. ISBN 0201134489
\bibitem{3} Jenks, Richard D. and Sutor, Robert S.,\\
{\it Axiom, The Scientific Computation System} \\
SpringerVerlag, New York, NY 1992 ISBN 0387978550
\bibitem{4} Daly, Timothy, ``The Axiom Literate Documentation''\\
{\bf http://axiom.axiomdeveloper.org/axiomwebsite/documentation.html}
\end{thebibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Bibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\bibliographystyle{plain}
+\bibliography{axiom}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Index}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printindex
\end{document}
+
diff git a/books/bookvol1.pamphlet b/books/bookvol1.pamphlet
index ca534f1..ddc3fbb 100644
 a/books/bookvol1.pamphlet
+++ b/books/bookvol1.pamphlet
@@ 534,7 +534,8 @@ source code for the interpreter, compiler, graphics, browser, and
numerics is shipped with the system. There are several websites
that host Axiom source code.
Axiom is written using Literate Programming\cite{2} so each file is actually
+Axiom is written using Literate Programming\cite{Knut92}
+so each file is actually
a document rather than just machine source code. The goal is to make
the whole system completely literate so people can actually read the
system and understand it. This is the first volume in a series of books
@@ 924,8 +925,7 @@ interactive ``undo.''
\label{sec:Starting Up and Winding Down}
You need to know how to start the Axiom system and how to stop it.
We assume that Axiom has been correctly installed on your
machine. Information on how to install Axiom is available on
the wiki website\cite{3}.
+machine.
To begin using Axiom, issue the command {\bf axiom} to the
operating system shell.
@@ 6839,7 +6839,7 @@ plotting functions of one or more variables and plotting parametric
surfaces. Once the graphics figure appears in a window, move your
mouse to the window and click. A control panel appears immediately
and allows you to interactively transform the object. Refer to the
original Axiom book\cite{1} and the input files included with Axiom
+original Axiom book\cite{Jenk92} and the input files included with Axiom
for additional examples.
This is an example of Axiom's graphics. From the Control Panel you can
@@ 9497,7 +9497,7 @@ domains and their functions and how to write your own functions.
\index{Aldor!Spad}
\index{Spad}
\index{Spad!Aldor}
There is a second language, called {\bf Aldor}\cite{4} that is
+There is a second language, called {\bf Aldor}\cite{Watt03} that is
compatible with the {\bf Spad} language. They both can create
programs than can execute under Axiom. Aldor is a standalone
version of the {\bf Spad} language and contains some additional
@@ 11867,7 +11867,7 @@ running on an IBM workstation, for example, issue
Axiom can produce \TeX{} output for your \index{output formats!TeX
@{\TeX{}}} expressions. \index{TeX output format @{\TeX{}} output format}
The output is produced using macros from the \LaTeX{} document
preparation system by Leslie Lamport\cite{5}. The printed version
+preparation system by Leslie Lamport\cite{Lamp86}. The printed version
of this book was produced using this formatter.
To turn on \TeX{} output formatting, issue this.
@@ 14300,31 +14300,17 @@ The command synonym {\tt )apropos} is equivalent to
{\tt )show} \index{)show}.
\section{Makefile}
This book is actually a literate program\cite{2} and can contain
executable source code. In particular, the Makefile for this book
is part of the source of the book and is included below. Axiom
uses the ``noweb'' literate programming system by Norman Ramsey\cite{6}.
+This book is actually a literate program\cite{Knut92} and can contain
+executable source code.
\eject
\begin{thebibliography}{99}
\bibitem{1} Jenks, R.J. and Sutor, R.S.
``Axiom  The Scientific Computation System''
SpringerVerlag New York (1992)
ISBN 0387978550
\bibitem{2} Knuth, Donald E., ``Literate Programming''
Center for the Study of Language and Information
ISBN 0937073814
Stanford CA (1992)
\bibitem{3} Daly, Timothy, ``The Axiom Wiki Website''\\
{\bf http://axiom.axiomdeveloper.org}
\bibitem{4} Watt, Stephen, ``Aldor'',\\
{\bf http://www.aldor.org}
\bibitem{5} Lamport, Leslie, ``Latex  A Document Preparation System'',
AddisonWesley, New York ISBN 0201529831
\bibitem{6} Ramsey, Norman ``Noweb  A Simple, Extensible Tool for
Literate Programming''\\
{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb}
\bibitem{7} Daly, Timothy, "The Axiom Literate Documentation"\\
{\bf http://axiom.axiomdeveloper.org/axiomwebsite/documentation.html}
\end{thebibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Bibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\bibliographystyle{plain}
+\bibliography{axiom}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Index}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printindex
\end{document}
+
diff git a/books/bookvol10.1.pamphlet b/books/bookvol10.1.pamphlet
index 05c94b7..f8fee1e 100644
 a/books/bookvol10.1.pamphlet
+++ b/books/bookvol10.1.pamphlet
@@ 4,7 +4,7 @@
\mainmatter
\setcounter{chapter}{0} % Chapter 1
\chapter{Interval Arithmetic}
Lambov \cite{Lambov06} defines a set of useful formulas for
+Lambov \cite{Lamb06} defines a set of useful formulas for
computing intervals using the IEEE754 floatingpoint standard.
The first thing to note is that IEEE floating point defaults to
@@ 256,9 +256,9 @@ an exception. If it contains a negative part, the implementation will
crop it to only its nonnegative part to allow that computations
such as $\sqrt{0}$ ca be carried out in exact real arithmetic.
\chapter{Integration \cite{Bro98b}}
+\chapter{Integration}
An {\sl elementary function}
+An {\sl elementary function}\cite{Bro98b}
\index{elementary function}
of a variable $x$ is a function that can
be obtained from the rational functions in $x$ by repeatedly adjoining
@@ 289,7 +289,7 @@ 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{Ost1845,Ris68,Ris69a,Ris69b}.
+of reports \cite{Ostr1845,Risc68,Risc69a,Risc69b,Risc70}.
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
@@ 413,7 +413,7 @@ 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{Ris69a} that any field containing an
+example it is proven in \cite{Risc69a} 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}
@@ 423,8 +423,8 @@ 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{Her1872} and
Ostrogradsky \cite{Ost1845} invented methods for
+$19^{{\rm th}}$ century, when both Hermite \cite{Herm1872} and
+Ostrogradsky \cite{Ostr1845} invented methods for
computing the $v$ of \ref{Int4}
entirely within $K(x)$. We describe here only Hermite's method, since
it is the one that has been generalized to arbitrary elementary
@@ 445,7 +445,7 @@ finally that
D_1=\frac{D/R}{{\rm gcd}(R,D/R)}
\]
Computing recursively a squarefree factorization of $R$ completes the
one for $D$. Note that \cite{Yu76} presents a more efficient method for
+one for $D$. Note that \cite{Yun76} 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\\
${\rm deg}(A)<{\rm deg}(D)$.
@@ 487,7 +487,7 @@ follows from \ref{Int2} that
where the $\alpha_i$'s are the zeros of $D$ in $\overline{K}$, and the
$a_i$'s are the residues of $f$ at the $\alpha_i$'s. The problem
is then to compute those residues without splitting $D$. Rothstein
\cite{Ro77} and Trager \cite{Tr76} independently proved that the
+\cite{Roth77} and Trager \cite{Trag76} independently proved that the
$\alpha_i$'s are exactly the zeros of
\begin{equation}\label{Int5}
R={\rm resultant}_x(D,AtD^{\prime}) \in K[t]
@@ 502,7 +502,7 @@ where $R=\prod_{i=1}^m R_i^{e_i}$ is the irreducible factorization of
$R$ over $K$. Note that this algorithm requires factoring $R$ into
irreducibles over $K$, and computing greatest common divisors in
$(K[t]/(R_i))[x]$, hence computing with algebraic numbers. Trager and
Lazard \& Rioboo \cite{LR90} independently discovered that those
+Lazard \& Rioboo \cite{Laza90} independently discovered that those
computations can be avoided, if one uses the subresultant PRS
algorithm to compute the resultant of \ref{Int5}: let
$(R_0,R_1,\ldots R_k\ne 0,0,\ldots)$ be the subresultant PRS with
@@ 528,7 +528,7 @@ extension $K[t]/(Q_i)$. Even this step can be avoided: it is in fact
sufficient to ensure that $Q_i$ and the leading coefficient with
respect to $x$ of $R_{k_i}$ do not have a nontrivial common factor,
which implies then that the remainder by $Q_i$ is nonzero, see
\cite{Mul97} for details and other alternatives for computing
+\cite{Muld97} for details and other alternatives for computing
${\rm pp}_x(R_{k_i})(a,x)$
\section{Algebraic Functions}
@@ 718,7 +718,7 @@ and $F=27x^4+108x^3+418x^2+108x+27$. The system \ref{Int10} admits a unique
solution $f_1=f_2=0, f_3=2$ and $f_4=(x+1)/x$, whose denominator is
not coprime with $V$, so the Hermite reduction is not applicable.
The above problem was first solved by Trager \cite{Tr84}, who proved
+The above problem was first solved by Trager \cite{Trag84}, who proved
that if $w$ is an {\sl integral basis, i.e.} its elements generate
${\bf O}_{K[x]}$ over $K[x]$, then the system \ref{Int8} always has a
unique solution in $K(x)$ when $m > 1$, and that solution always has a
@@ 728,9 +728,9 @@ 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{Tr84} or the local alternative \cite{vH94},
+whether by the method of \cite{Trag84} or the local alternative \cite{Hoei94},
can be in general more expansive than the rest of the reduction
process. We describe here the lazy Hermite reduction \cite{REFBro98}, which
+process. We describe here the lazy Hermite reduction \cite{Bron98}, which
avoids the precomputation of an integral basis. It is based on the
observation that if $m > 1$ and \ref{Int8} does not have a solution allowing
us to perform the reduction, then either
@@ 745,7 +745,7 @@ 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{REFBro98})} {\sl Suppose that $m \ge 2$ and that
+{\bf Theorem 1 (\cite{Bron98})} {\sl Suppose that $m \ge 2$ and that
$\{S_1,\ldots,S_n\}$ as given by \ref{Int9} 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,
@@ 756,7 +756,7 @@ 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{REFBro98})} {\sl Suppose that $m \ge 2$ and that
+{\bf Theorem 2 (\cite{Bron98})} {\sl Suppose that $m \ge 2$ and that
$\{S_1,\ldots,S_n\}$ as given by \ref{Int9} are linearly independent over
$K(x)$, and let $Q,T_1,\ldots,T_n \in K[x]$ be such that
\[
@@ 771,7 +771,7 @@ 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{REFBro98})} {\sl Suppose that the denominator $F$ of
+{\bf Theorem 3 (\cite{Bron98})} {\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,}
\[
@@ 951,7 +951,7 @@ 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{Lio1833a,Lio1833b})} {\sl
+{\bf Theorem 4 (Liouville \cite{Liou1833a,Liou1833b})} {\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
@@ 960,9 +960,10 @@ $u_1,\ldots,u_k \in E(c_1,\ldots,c_k)^{*}$ such that}
f=v^{\prime}+c_1\frac{u_1^{\prime}}{u_1}+\cdots+c_k\frac{u_k^{\prime}}{u_k}
\end{equation}
The above is a restriction to algebraic functions of the strong
Liouville Theorem, whose proof can be found in \cite{Bro97,Ris69b}. An elegant
+Liouville Theorem, whose proof can be found in \cite{Bron97,Risc69b}.
+An elegant
and elementary algebraic proof of a slightly weaker version can be
found in \cite{Ro72}. As a consequence, we can look for an integral of
+found in \cite{Rose72}. As a consequence, we can look for an integral of
the form \ref{Int4}, 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
@@ 983,7 +984,7 @@ $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{Tr84} gave a change of variable that
+of polynomials, so Trager \cite{Trag84} gave a change of variable that
guarantees that either $v^{\prime}=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
@@ 1007,7 +1008,7 @@ and ${\rm deg}(C) \ge {\rm 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{Tr84} described an
+ramification index at infinity. Trager \cite{Trag84} 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:
@@ 1071,7 +1072,7 @@ $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{Tr84} describes an absolute factorization
+$K(x)$, and Trager \cite{Trag84} 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
@@ 1131,7 +1132,8 @@ 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 \ref{Int4}. Trager \cite{Tr84} describes a representation
+integral in the form \ref{Int4}. Trager \cite{Trag84}
+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$
@@ 1141,7 +1143,7 @@ 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{We71}, so we compute such an $M$ by testing $M=1,2,3,\ldots$ until
+\cite{Weil71}, 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
@@ 1149,7 +1151,7 @@ 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{Ber95} describes a
+simple radical extensions of degree 2, Bertrand \cite{Bert95} describes a
simpler representation of divisors for which the arithmetic and
principality tests are more efficient than the general methods.
@@ 1287,7 +1289,7 @@ 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{Ris79}, so we can assume that all the logarithms
+tower accordingly \cite{Risc79}, 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
@@ 1430,7 +1432,7 @@ $r_0,\ldots,r_{d1} \in K$. Again, it is easy to verify that for any
R=\frac{1}{{\rm 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
${\rm deg}_t(\overline{R}) < e1$. Furthermore, it can be proven \cite{Bro97}
+${\rm deg}_t(\overline{R}) < e1$. Furthermore, it can be proven \cite{Bron97}
that if $R+A/D$ has an elementary integral over $K(t)$, then
$r_{d1}/{c_d}$ is a constant, which implies that
\[
@@ 1480,7 +1482,7 @@ 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{Bro97} that
+case. It can then be proven \cite{Bron97} 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
@@ 1555,7 +1557,7 @@ $z$ be a new indeterminante and compute
\begin{equation}\label{Int16}
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{Bro90c} that if $f$ has an elementary integral
+It can then be proven \cite{Bron90c} that if $f$ has an elementary integral
over $E$, then $R\kappa(R)$ in $K[z]$.
{\bf Example 12} {\sl
@@ 1607,7 +1609,7 @@ to $f_d$, either proving that \ref{Int18} 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 \ref{Int18}, see \cite{Bro97} for details). Write then
+form \ref{Int18}, see \cite{Bron97} 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
@@ 1654,8 +1656,8 @@ The above problem is called a {\sl Risch differential equation over K}.
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{Bro90c,Bro91a,Bro97} and Risch
\cite{Ris68,Ris69a,Ris69b} describe algorithms for solving this type
+$K$. Bronstein \cite{Bron90c,Bron91a,Bron97} and Risch
+\cite{Risc68,Risc69a,Risc69b} describe algorithms for solving this type
of equation when $K$ is an elementary extension of the rational
function field.
@@ 1708,7 +1710,7 @@ 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{Bro97}. If it has no solution, then the
+differential equations \cite{Bron97}. 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
@@ 1898,7 +1900,7 @@ whose solution is $v_2=2$, implying that $h=2y'$, hence that
In the general case when $E$ is not a radical extension of $K(t)$,
\ref{Int21} is solved by bounding ${\rm deg}_t(v_i)$ and comparing the Puiseux
expansions at infinity of $\sum_{i=1}^n v_iw_i$ with those of the form
\ref{Int20} of $h$, see \cite{Bro90c,Ris68} for details.
+\ref{Int20} of $h$, see \cite{Bron90c,Risc68} for details.
\subsection{The algebraic exponential case}
The transcendental exponential case method also generalizes to the
@@ 2022,7 +2024,7 @@ $v=\sum_{i=1}^n v_iw_i/t^m$ where $v_1,\ldots,v_m \in K[t]$. We can
compute $v$ by bounding ${\rm deg}_t(v_i)$ 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 \ref{Int20} of the integrand,
see \cite{Bro90c,Ris68} for details.
+see \cite{Bron90c,Risc68} for details.
Once we are reduced to solving \ref{Int13} for $v \in K$, constants
$c_1,\ldots,c_k \in \overline{K}$ and
@@ 2032,13 +2034,14 @@ 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{Bro90c,Ris68,Ris69a}.
+\cite{Bron90c,Risc68,Risc69a}.
\chapter{Singular Value Decomposition \cite{Pu09}}
+\chapter{Singular Value Decomposition}
\section{Singular Value Decomposition Tutorial}
When you browse standard web sources like Wikipedia to learn about
Singular Value Decomposition or SVD you find many equations, but
+Singular Value Decomposition \cite{Puff09}
+or SVD you find many equations, but
not an intuitive explanation of what it is or how it works. SVD
is a way of factoring matrices into a series of linear approximations
that expose the underlying structure of the matrix. Two important
@@ 2445,7 +2448,7 @@ 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{Alt05}:
+from \cite{Altm05}:
\begin{quotation}
Quaternions are inextricably linked to rotations.
Rotations, however, are an accident of threedimensional space.
@@ 2467,8 +2470,8 @@ 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{Ta1890} is
the accepted textbook for advanced students.
+shortly after his death. The Elements of Quaternions by Tait \cite{Tait1890}
+is the accepted textbook for advanced students.
Large portions of this file are derived from a public domain version
of Tait's book combined with the algebra available in Axiom.
@@ 7651,13 +7654,13 @@ i =
\right]
$$
\chapter{Clifford Algebra \cite{Fl09}}
+\chapter{Clifford Algebra}
This is quoted from John Fletcher's web page \cite{Fl09} (with permission).
+This is quoted from John Fletcher's web page \cite{Flet09} (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{Ab98} gives examples of
+discuss this and in particular Ablamowicz \cite{Abla98} 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
@@ 7672,7 +7675,8 @@ 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{Ab98} shows that the matrices can be derived for each algebra
+Ablamowicz \cite{Abla98}
+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
@@ 7687,7 +7691,7 @@ 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{Fl01}.
+author \cite{Flet01}.
\section{Clifford Basis Matrix Theory}
@@ 8129,7 +8133,7 @@ simple cases of wide usefulness.
\subsection{Example 3: Clifford (2,2)}
The following basis matrices are given by Ablamowicz \cite{Ab98}
+The following basis matrices are given by Ablamowicz \cite{Abla98}
\[
\begin{array}{cc}
@@ 8379,7 +8383,7 @@ 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{Fl01}.
+system for Clifford algebra such as described in Fletcher \cite{Flet01}.
The result is
\[
@@ 8423,15 +8427,16 @@ 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 \cite{LR88}. In
\cite{Ha96}
+to any plane curve by D. LeBrigand and J.J. Risler \cite{LeBr88}. In
+\cite{Hach96}
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 \cite{Ha95}. This is the approach I used in my thesis
(\cite{Ha96}) and of course this is where you will find complete details about
+algebraic approach, see \cite{Hach95}. This is the approach I used in my thesis
+(\cite{Hach96})
+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
\cite{St93} a very good introduction to this theory and AGcodes.
+\cite{Stic93} 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
@@ 8444,7 +8449,7 @@ 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 \cite{Du95}.
+dynamic algebraic closure as proposed by D. Duval \cite{Duva95}.
Example 1
@@ 8484,7 +8489,7 @@ notation for the binomial coefficients
There are $n$ factors in the numerator and $n$ in the denominator.
Viewed as a function of $u$, $C(u+k,n)$ is a polynomial of degree $n$.
The figure above, Hamming \cite{Ham62}
+The figure above, Hamming \cite{Hamm62}
calls a lozenge diagram. A line starting at
a point on the left edge and following some path across the page
defines an interpolation formula if the following rules are used.
@@ 8560,182 +8565,13 @@ Gaussian Elimination
\chapter{Diophantine Equations}
Diophantine Equations
\begin{thebibliography}{99}

\bibitem[Ablamowicz 98]{Ab98} Ablamowicz, Rafal\\
``Spinor Representations of Clifford Algebras: A Symbolic Approach''\\
Computer Physics Communications
Vol. 115, No. 23, December 11, 1998, pages 510535.

\bibitem[Altmann 05]{Alt05} Altmann, Simon L.\\
``Rotations, Quaternions, and Double Groups''\\
Dover Publications, Inc. 2005 ISBN 0486445186

\bibitem[Bertrand 95]{Ber95} Bertrand, Laurent\\
``Computing a hyperelliptic integral using arithmetic in the jacobian
of the curve''\\
{\sl Applicable Algebra in Engineering, Communication and Computing},
6:275298, 1995

\bibitem[Bronstein 90c]{Bro90c} Bronstein, M.\\
``On the integration of elementary functions''\\
{\sl Journal of Symbolic Computation} 9(2):117173, February 1990

\bibitem[Bronstein 91a]{Bro91a} Bronstein, M.\\
``The Risch differential equation on an algebraic curve''\\
in Watt [Wat91], pp241246 ISBN 0897914376 LCCN QA76.95.I59 1991

\bibitem[Bronstein 97]{Bro97} Bronstein, M.\\
``Symbolic Integration ITranscendental Functions.''\\
Springer, Heidelberg, 1997 ISBN 3540214933
\verbevilwire.org/arrrXiv/Mathematics/Bronstein,_Symbolic_Integration_I,1997.pdf

\bibitem[Bronstein 98b]{Bro98b} Bronstein, Manuel\\
``Symbolic Integration Tutorial''\\
INRIA Sophia Antipolis ISSAC 1998 Rostock

\bibitem[Bronstein 98]{REFBro98} Bronstein, M.\\
``The lazy hermite reduction''\\
Rapport de Recherche RR3562, INRIA, 1998

\bibitem[Duval 95]{Du95} Duval, D.\\
``Evaluation dynamique et cl\^oture alg\'ebrique en Axiom''.\\
Journal of Pure and Applied Algebra, no99, 1995, pp. 267295.

\bibitem[Fletcher 01]{Fl01} Fletcher, John P.\\
``Symbolic processing of Clifford Numbers in C++''\\
Paper 25, AGACSE 2001.

\bibitem[Fletcher 09]{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[Hathway 1896]{Ha1896} Hathway, Arthur S.\\
``A Primer Of Quaternions''\\
(1896)

\bibitem[Hache 95a]{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[Hache 96]{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[Hamming 62]{Ham62} Hamming R W.\\
``Numerical Methods for Scientists and Engineers''\\
Dover (1973) ISBN 0486652416

\bibitem[Hermite 1872]{Her1872} Hermite, E.\\
``Sur l'int\'{e}gration des fractions rationelles.''\\
{\sl Nouvelles Annales de Math\'{e}matiques}
($2^{eme}$ s\'{e}rie), 11:145148, 1872

\bibitem[van Hoeij 94]{vH94} van Hoeij, M.\\
``An algorithm for computing an integral basis in an algebraic
function field''\\
Journal of Symbolic Computation, 18(4) pp353363 Oct. 1994
CODEN JSYCEH ISSN 07477171

\bibitem[Le Brigand 88]{LR88} Le Brigand, D.; Risler, J.J.\\
``Algorithme de BrillNoether et codes de Goppa''\\
Bull. Soc. Math. France, vol. 116, 1988, pp. 231253.

\bibitem[Lazard 90]{LR90} Lazard, Daniel; Rioboo, Renaud\\
``Integration of rational functions: Rational computation of the
logarithmic part''\\
{\sl Journal of Symbolic Computation}, 9:113116:1990

\bibitem[Liouville 1833a]{Lio1833a} Liouville, Joseph\\
``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[Liouville 1833b]{Lio1833b} Liouville, Joseph\\
``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[Mulders 97]{Mul97} Mulders. Thom\\
``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[Ostrogradsky 1845]{Ost1845} Ostrogradsky. M.W.\\
``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[Puffinware 09]{Pu09} Puffinware LLC.\\
``Singular Value Decomposition (SVD) Tutorial''\\
\verbwww.puffinwarellc.com/p3a.htm

\bibitem[Risch 68]{Ris68} Risch, Robert\\
``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[Risch 69a]{Ris69a} Risch, Robert\\
``Further results on elementary functions''\\
Research Report RC2042, IBM Research, Yorktown Heights, NY, USA, 1969

\bibitem[Risch 69b]{Ris69b} Risch, Robert\\
``The problem of integration in finite terms''\\
{\sl Transactions of the American Mathematical Society} 139:167189, 1969

\bibitem[Risch 79]{Ris79} Risch, Robert\\
``Algebraic properties of the elementary functions of analysis''\\
{\sl American Journal of Mathematics}, 101:743759, 1979

\bibitem[Rosenlicht 72]{Ro72} Rosenlicht, Maxwell\\
``Integration in finite terms''\\
{\sl American Mathematical Monthly}, 79:963972, 1972

\bibitem[Rothstein 77]{Ro77} Rothstein, Michael\\
``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[Stichtenoth 93]{St93} Stichtenoth, H.\\
``Algebraic function fields and codes''\\
SpringerVerlag, 1993, University Text.

\bibitem[Tait 1890]{Ta1890} Tait, P.G.\\
``An Elementary Treatise on Quaternions''\\
C.J. Clay and Sons, Cambridge University Press Warehouse, Ave Maria Lane 1890

\bibitem[Trager 76]{Tr76} Trager, Barry\\
``Algebraic factoring and rational function integration''\\
In {Proceedings of SYMSAC'76} pages 219226, 1976

\bibitem[Trager 84]{Tr84} Trager, Barry\\
``On the integration of algebraic functions''\\
PhD thesis, MIT, Computer Science, 1984

\bibitem[Lambov 06]{Lambov06} Lambov, Branimir\\
``Interval Arithmetic Using SSE2''\\
in Lecture Notes in Computer Science, Springer ISBN 9783540855200
(2006) pp102113

\bibitem[Weil 71]{We71} Weil, Andr\'{e}\\
``Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes''\\
Hermann, Paris, 1971

\bibitem[Yun 76]{Yu76} Yun, D.Y.Y.\\
``On squarefree decomposition algorithms''\\
{\sl Proceedings of SYMSAC'76} pages 2635, 1976

\end{thebibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Bibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\bibliographystyle{plain}
+\bibliography{axiom}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Index}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printindex
\end{document}
diff git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet
index 0015f59..4cad1e5 100644
 a/books/bookvol10.3.pamphlet
+++ b/books/bookvol10.3.pamphlet
@@ 18303,9 +18303,10 @@ CharacterClass: Join(SetCategory, ConvertibleTo String,
\end{chunk}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{domain CLIF CliffordAlgebra\cite{7,12}}
+\section{domain CLIF CliffordAlgebra}
\subsection{Vector (linear) spaces}
This information is originally from Paul Leopardi's presentation on
+This information is originally from Paul Leopardi's \cite{Leop03}
+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}.
@@ 18372,7 +18373,7 @@ 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}}
+\subsection{Quadratic Forms}
For vector space $\mathbb{V}$ over field $\mathbb{F}$, characteristic
$\ne 2$:
\begin{list}{}
@@ 18383,7 +18384,7 @@ $$b:\mathbb{V}{\rm\ x\ }\mathbb{V} \rightarrow \mathbb{F}{\rm\ ,given\ by\ }$$
$$b(x,y):=\frac{1}{2}(f(x+y)f(x)=f(y))$$
is a symmetric bilinear form
\end{list}
\subsection{Quadratic spaces, Clifford Maps\cite{1,2}}
+\subsection{Quadratic spaces, Clifford Maps}
\begin{list}{}
\item A quadratic space is the pair($\mathbb{V}$,$f$), where $f$ is a
quadratic form on $\mathbb{V}$
@@ 18392,7 +18393,7 @@ $$\rho : \mathbb{V} \rightarrow \mathbb{A}$$
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}}
+\subsection{Universal Clifford algebras}
\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
@@ 18402,7 +18403,7 @@ $\phi_{\mathbb{A}} \exists$ 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}}
+\subsection{Real Clifford algebras $\mathbb{R}_{p,q}$}
\begin{list}{}
\item The real quadratic space $\mathbb{R}^{p,q}$ is $\mathbb{R}^{p+q}$ with
$$\phi(x):=\sum_{k:=q}^{1}{x_k^2}+\sum_{k=1}^p{x_k^2}$$
@@ 18422,7 +18423,7 @@ $$\mathbb{P}(S):={\rm\ the\ }\ power\ set\ {\rm\ of\ }S$$
\item For $m \le n \in \mathbb{Z}$, define
$$\zeta(m,n):=\{m,m+1,\ldots,n1,n\}\backslash\{0\}$$
\end{list}
\subsection{Frames for Clifford algebras\cite{9,10,11}}
+\subsection{Frames for Clifford algebras}
\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
@@ 18433,7 +18434,7 @@ $$\gamma:\zeta(q,p) \rightarrow \mathbb{R}^{p,q}$$
$$\rho:\mathbb{R}^{p,q} \rightarrow \mathbb{R}_{p,q}$$
$$(\rho\gamma k)^2 = \phi\gamma k = {\rm\ sgn\ }k$$
\end{list}
\subsection{Real frame groups\cite{5,6}}
+\subsection{Real frame groups}
\begin{list}{}
\item For $p,q \in \mathbb{N}$, define the real {\sl frame group} $\mathbb{G}_{p,q}$
via the map
@@ 18449,7 +18450,7 @@ $$(g_k)^2 =
\right.$$
$$g_kg_m = \mu g_mg_k{\rm\ \ \ }\forall k \ne m\rangle$$
\end{list}
\subsection{Canonical products\cite{1,3,4}}
+\subsection{Canonical products}
\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
@@ 18457,7 +18458,7 @@ $$w=\mu^a\prod_{k \in T}{g_k}$$
$$\ =\mu^a\prod_{k=q,k\ne0}^p{g_k^{b_k}}$$
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}}
+\subsection{Clifford algebra of frame group}
\begin{list}{}
\item For $p,q \in \mathbb{N}$ embed $\mathbb{G}_{p,q}$ into
$\mathbb{R}_{p,q}$ via the map
@@ 18471,7 +18472,7 @@ $$e:\mathbb{P}\zeta(q,p) \rightarrow \mathbb{R}_{p,q},
\item Each $a \in \mathbb{R}_{p,q}$ can be expressed as
$$a = \sum_{T \subseteq \zeta(q,p)}{a_T e_T}$$
\end{list}
\subsection{Neutral matrix representations\cite{1,2,8}}
+\subsection{Neutral matrix representations}
The {\sl representation map} $P_m$ and {\sl representation matrix} $R_m$
make the following diagram commute:
\begin{tabular}{ccc}
@@ 28603,7 +28604,7 @@ from /home/greg/Axiom/DFLOAT.nrlib/code
So it is clear that he has added a new function called
{\tt doubleFloatFormat} which takes a string argument that
specifies the common lisp format control string (\"{}\~{},4,,F\"{}).
For reference we quote from the common lisp manual \cite{1}.
+For reference we quote from the common lisp manual.
On page 582 we find:
\begin{quote}
@@ 156871,6 +156872,11 @@ Note that this code is not included in the generated catdef.spad file.
\getchunk{domain XRPOLY XRecursivePolynomial}
\end{chunk}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Bibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\bibliographystyle{plain}
+\bibliography{axiom}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Index}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printindex
diff git a/books/bookvol10.4.pamphlet b/books/bookvol10.4.pamphlet
index 65a295f..0847029 100644
 a/books/bookvol10.4.pamphlet
+++ b/books/bookvol10.4.pamphlet
@@ 14767,7 +14767,7 @@ DistinctDegreeFactorize(F,FP): C == T
The special functions in this section are developed as special cases
but can all be expressed in terms of generalized hypergeomentric
functions ${}_pF_q$ or its generalization, the Meijer G function.
\cite{Luk169,Luk269}
+\cite{Luke69a,Luke69b}
The long term plan is to reimplement these functions using the
generalized version.
\begin{chunk}{DoubleFloatSpecialFunctions.input}
@@ 15716,7 +15716,7 @@ DoubleFloatSpecialFunctions(): Exports == Impl where
\end{chunk}
\subsection{The Exponential Integral}
\subsubsection{The E1 function}
(Quoted from Segletes\cite{2}):
+(Quoted from Segletes\cite{Segl98}):
A number of useful integrals exist for which no exact solutions have
been found. In other cases, an exact solution, if found, may be
@@ 15775,7 +15775,7 @@ exponential integral family may be analytically related. However, this
technique only allows for the transformation of one integral into
another. There remains the problem of evaluating $E_1(x)$. There is an
exact solution to the integral of $(e^{t}/t)$, appearing in a number
of mathematical references \cite{4,5} which is obtainable by
+of mathematical references which is obtainable by
expanding the exponential into a power series and integrating term by
term. That exact solution, which is convergent, may be used to specify
$E_1(x)$ as
@@ 15789,7 +15789,7 @@ E_1(x)=\gammaln(x)
Euler's constant, $\gamma$, equal to $0.57721\ldots$, arises when the
power series expansion for $(e^{t}/t)$ is integrated and evaluated at
its upper limit, as $x\rightarrow\infty$\cite{6}.
+its upper limit, as $x\rightarrow\infty$.
Employing eqn (5), however, to evaluate $E_1(x)$ is problematic for
finite $x$ significantly larger than unity. One may well ask of the
@@ 15857,8 +15857,8 @@ fit. While some steps are taken to make the fits intelligent ({\sl
e.g.}, transformation of variables), the fits are all piecewise over
the domain of the integral.
Cody and Thatcher \cite{7} performed what is perhaps the definitive
work, with the use of Chebyshev\cite{18,19} approximations to the exponential
+Cody and Thatcher performed what is perhaps the definitive
+work, with the use of Chebyshev approximations to the exponential
integral $E_1$. Like others, they fit the integral over a piecewise
series of subdomains (three in their case) and provide the fitting
parameters necessary to evaluate the function to various required
@@ 16041,7 +16041,7 @@ $$E_{n+1}(z)=\frac{1}{n}\left(e^{z}zE_n(z)\right)\ \ \ (n=1,2,3,\ldots)$$
The base case of the recursion depends on E1 above.
The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229\cite{4}.
+The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229
\begin{chunk}{package DFSFUN DoubleFloatSpecialFunctions}
En(n:PI,x:R):OPR ==
n=1 => E1(x)
@@ 16051,12 +16051,12 @@ The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229\cite{4}.
\end{chunk}
\subsection{The Ei Function}
This function is based on Kin L. Lee's work\cite{8}. See also \cite{21}.
+This function is based on Kin L. Lee's work.
\subsubsection{Abstract}
The exponential integral Ei(x) is evaluated via Chebyshev series
expansion of its associated functions to achieve high relative
accuracy throughout the entire real line. The Chebyshev coefficients
for these functions are given to 30 significant digits. Clenshaw's\cite{20}
+for these functions are given to 30 significant digits. Clenshaw's
method is modified to furnish an efficient procedure for the accurate
solution of linear systems having neartriangular coefficient
matrices.
@@ 16067,13 +16067,13 @@ Ei(x)=\int_{\infty}^{X}{\frac{e^u}{u}}\ du=E_1(x), x \ne 0
\end{equation}
is usually based on the value of its associated functions, for
example, $xe^{x}Ei(x)$. High accuracy tabulations of integral (1) by
means of Taylor series techniques are given by Harris \cite{9} and
Miller and Hurst \cite{10}. The evaluation of $Ei(x)$ for
+means of Taylor series techniques are given by Harris and
+Miller and Hurst. The evaluation of $Ei(x)$ for
$4 \le x \le \infty$ by means of Chebyshev series is provided by
Clenshaw \cite{11} to have the absolute accuracy of 20 decimal
+Clenshaw to have the absolute accuracy of 20 decimal
places. The evaluation of the same integral (1) by rational
approximation of its associated functions is furnished by Cody and
Thacher \cite{12,13} for $\infty < x < \infty$, and has the relative
+Thacher for $\infty < x < \infty$, and has the relative
accuracy of 17 significant figures.
The approximation of Cody and Thacher from the point of view of
@@ 16089,7 +16089,7 @@ functions that are accurate to 30 significant figures by a
modification of Clenshaw's procedure. To verify the accuracy of the
several Chebyshev series, values of the associated functions were
checked against those computed by Taylor series and those of Murnaghan
and Wrench \cite{14} (see Remarks on Convergence and Accuracy).
+and Wrench (see Remarks on Convergence and Accuracy).
Although for most purposes fewer than 30 figures of accuracy are
required, such high accuracy is desirable for the following
@@ 16106,7 +16106,7 @@ approximated. To take account of the errors commited by these
routines, the function values must have an accuracy higher than the
approximation to be determined. Consequently, highprecision results
are useful as a master function for finding approximations for (or
involving) $Ei(x)$ (e.g. \cite{12,13}) where prescribed accuracy is
+involving) $Ei(x)$ where prescribed accuracy is
less than 30 figures.
\subsubsection{Discussion}
@@ 16199,10 +16199,10 @@ coefficients $A_k^{(0)}$ and $A_k^{(1)}$ can be obtained analytically
(if possible) or by numerical quadrature. However, since each function
in table 1 satisfies a linear differential equation with polynomial
coefficients, the Chebyshev coefficients can be more readily evaluated
by the method of Clenshaw \cite{16}.
+by the method of Clenshaw.
There are several variations of Clenshaw's procedure (see,
e.g. \cite{17}), but for highprecision computation, where multiple
+There are several variations of Clenshaw's procedure,
+but for highprecision computation, where multiple
precision arithmetic is employed, we find his original procedure
easiest to implement. However, straightforward application of it may
result in a loss of accuracy if the trial solutions selected are not
@@ 16248,7 +16248,7 @@ p(pt+q)
\end{equation}
It can be demonstrated that if $B_k$ are the Chebyshev coefficients of
a function $\Psi(t)$, then $C_k$, the Chebyshev coefficients of
$t^r\Psi(t)$ for positive integers r, are given by \cite{16}
+$t^r\Psi(t)$ for positive integers r, are given by
\begin{equation}
C_k=2^{r}\sum_{i=0}^r\binom{r}{i}B_{\vert kr+2i\vert}
\end{equation}
@@ 16277,7 +16277,7 @@ p^2A_{k+1}^{(0)}\\
\end{array}
\right\}
\end{equation}
The relation \cite{16}
+The relation
\begin{equation}
2kA_k^{(0)}=A_{k1}^{(1)}A_{k+1}^{(1)}
\end{equation}
@@ 16304,7 +16304,7 @@ p^2A_{k1}+2p(2k+q2)A_k+8q(k+1)A_{k+1}+2p(2kq+6)A_{k+2}p^2A_{k+3}\\
\right\}
\end{equation}
The superscript of $A_k^{(0)}$ is dropped for simplicity. In order to
solve the infinite system 20, Clenshaw \cite{11} essentially
+solve the infinite system 20, Clenshaw essentially
considered the required solution as the limiting solution of the
sequence of truncated systems consisting of the first $M+1$ equations
of the same system, that is, the solution of the system
@@ 16380,7 +16380,7 @@ $S(\alpha)$ are equal, respectively, to the left members of equations
designation holds for $R(\beta)$ and $S(\beta)$.)
The quantities $\alpha_k$ and $\beta_k$ are known as trial solutions
in reference \cite{12}. Clenshaw has pointed out that if $\alpha_k$
+in reference. Clenshaw has pointed out that if $\alpha_k$
and $\beta_k$ are not sufficiently independent, loss of significance
will occur in the formation of the linear combination 24, with
consequent loss of accuracy. Clenshaw suggested the GaussSeidel
@@ 16728,7 +16728,7 @@ evaluation also checks with that of the function values of table 4
(computed with 30digit floatingpoint arithmetic using the
coefficients of table 3) for at least 281/2 significant
digits. Evaluation of Ei(x) using the coefficients of table 3 also
checked with Murnaghan and Wrench \cite{14} for 281/2 significant
+checked with Murnaghan and Wrench for 281/2 significant
figures.
{\vbox{\vskip 1cm}}
@@ 17555,7 +17555,7 @@ $\infty$ & 1.000 & 0.100000000 0000000000 00000000001 E 01\\
32 & 1.000 & 0.103341356 4216241049 43493552567 E 01\\
\end{tabular}
\subsection{The Fresnel Integral\cite{PEA56,LOS60}}
+\subsection{The Fresnel Integral\cite{Pear56,Losc60}}
The Fresnel function is
\[C(x)  iS(x) = \int_0^x{i^{t^2}}~dt = \int_0^x{\exp(i\pi{}t^2/2)}~dt\]
@@ 17587,7 +17587,7 @@ $\rm{arc\ }z \le \pi\epsilon$, ($\epsilon > 0$), for $z \gg 1$ is given by
\left(1\frac{1\cdot{}3}{(2z)^2}+\frac{1\cdot{}3\cdot{}5\cdot{}7}{(2z)^4}
\cdots\right)\frac{\cos z}{\sqrt{2\pi{}z}}\left(\frac{1}{(2z)}
\frac{1\cdot{}3\cdot{}5}{(2z)^3}+\cdots\right)\]
(Note: Pearcey has a sign error for the second term (\cite{PEA56},p7)
+(Note: Pearcey has a sign error for the second term (\cite{Pear56},p7)
The first approximation is
\[C(z) \approx \frac{1}{2} + \frac{\sin z}{\sqrt{2\pi{}z}}\]
@@ 52980,7 +52980,7 @@ IntegerFactorizationPackage(I): Exports == Implementation where
\end{chunk}
\subsection{PollardSmallFactor}
This is Brent's\cite{1} optimization of Pollard's\cite{2} rho factoring.
+This is Brent's optimization of Pollard's rho factoring.
Brent's algorithm is about 24 percent faster than Pollard's. Pollard;s
algorithm has complexity $O(p^{1/2})$ where $p$ is the smallest prime
factor of the composite number $N$.
@@ 140129,7 +140129,7 @@ PolynomialGcdPackage(E,OV,R,P):C == T where
if degree gcd(uf,differentiate uf)=0 then return [uf,ltry]
\end{chunk}
In Gathen \cite{GG99} we find a discussion of applying the Euclidean
+In Gathen \cite{Gath99} we find a discussion of applying the Euclidean
algorithm to elements of a field. In a field every nonzero rational
number is a unit. If we want to define a single element such that
\[gcd(f,g) \in {\bf Q}[x]\] we choose a monic polynomial, that is, the
@@ 175476,6 +175476,11 @@ ZeroDimensionalSolvePackage(R,ls,ls2): Exports == Implementation where
\getchunk{package ZDSOLVE ZeroDimensionalSolvePackage}
\end{chunk}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Bibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\bibliographystyle{plain}
+\bibliography{axiom}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Index}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printindex
diff git a/books/bookvol10.5.pamphlet b/books/bookvol10.5.pamphlet
index 09485ba..5a36774 100644
 a/books/bookvol10.5.pamphlet
+++ b/books/bookvol10.5.pamphlet
@@ 4,10 +4,10 @@
\mainmatter
\setcounter{secnumdepth}{0} % override the one in bookheader.tex
\setcounter{chapter}{0} % Chapter 1
\chapter{Numerical Analysis \cite{4}}
+\chapter{Numerical Analysis}
We can describe each number as $x^{*}$ which has a machinerepresentable
form which differs from the number $x$ it is intended to represent.
Quoting Householder we get:
+Quoting Householder \cite{Hous81} we get:
\[x^{*}=\pm(x_1\beta^{1} + x_2\beta^{2}+\cdots+x_\lambda\beta^\lambda)
\beta^\sigma\]
where $\beta$ is the base, usually 2 or 10, $\lambda$ is a positive
@@ 128,7 +128,7 @@ For real matrices, TRANSx=T and TRANSx=C have the same meaning.
For Hermitian matrices, TRANSx=T is not allowed.
For complex symmetric matrices, TRANSx=H is not allowed.
There were 38 BLAS Level 1 routines defined in \cite{REFLAW79}. They are
+There were 38 BLAS Level 1 routines defined in \cite{Laws79}. They are
\begin{itemize}
\item Dot product SDSDOT, DSDOT, DQIDOT DQADOT CUDOT CCDOT DDOT SDOT
\item Constant times a vector plus a vector CAXPY DAXPY SAXPY
diff git a/books/bookvol10.pamphlet b/books/bookvol10.pamphlet
index c8f1684..6f77d64 100644
 a/books/bookvol10.pamphlet
+++ b/books/bookvol10.pamphlet
@@ 19266,7 +19266,7 @@ clean:
\chapter{Implementation}
\section{Elementary Functions\cite{4}}
+\section{Elementary Functions}
\subsection{Rationale for Branch Cuts and Identities}
Perhaps one of the most vexing problems to be addressed when
@@ 19279,7 +19279,7 @@ issue facing the mathematical library developer is the plethora of
possibilities, and while some choices are demonstrably inferior, there
is rarely a choice which is clearly best.
Following Kahan [1], we will refer to the mathematical formula we use
+Following Kahan\cite{Kaha86}, we will refer to the mathematical formula we use
to define the principal branch of each such function as its principal
expression. For the inverse trigonometric and inverse hyperbolic
functions, this principal expression is given in terms of the
@@ 19469,18 +19469,13 @@ $\begin{array}{l}
\end{tabular}
\eject
\begin{thebibliography}{99}
\bibitem{1} Kahan, W., “Branch cuts for complex elementary functions, or,
Much ado about nothing's sign bit”, Proceedings of the joint IMA/SIAM
conference on The State of the Art in Numerical Analysis, University of
Birmingham, A. Iserles and M.J.D. Powell, eds, Clarendon Press,
Oxford,1987, 165210.
\bibitem{2} IEEE standard 7541985 for binary floatingpoint arithmetic,
reprinted in ACM SIGPLAN Notices 22 \#2 (1987), 925.
\bibitem{3} IEEE standard 7542008
\bibitem{4} Numerical Mathematics Consortium
Technical Specification 1.0 (Draft)
\verbhttp://www.nmconstorium.org
\end{thebibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Bibliography}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\bibliographystyle{plain}
+\bibliography{axiom}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Index}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printindex
\end{document}
diff git a/books/bookvol11.pamphlet b/books/bookvol11.pamphlet
index 01dbce1..5cfa64e 100644
 a/books/bookvol11.pamphlet
+++ b/books/bookvol11.pamphlet
@@ 1,6 +1,9 @@
\documentclass[dvipdfm]{book}
\newcommand{\VolumeName}{Volume 11: Axiom Browser}
\input{bookheader.tex}
+\mainmatter
+\setcounter{chapter}{0} % Chapter 1
+\setcounter{secnumdepth}{0} % override the one in bookheader.tex
\chapter{Overview}
This book contains the Firefox browser AJAX routines.
@@ 840,8 +843,6 @@ result sent from the server.
This is the standard CSS style section that gets included with every
page. We do this here but it could be a separate style sheet. It
hardly matters either way as the style sheet is trivial.
\begin{verbatim}
\end{verbatim}
\begin{chunk}{style}