diff git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet
index 2d07bec..edb7152 100644
 a/books/bookvolbib.pamphlet
+++ b/books/bookvolbib.pamphlet
@@ 2240,6 +2240,20 @@ Kelsey, Tom; Martin, Ursula; Owre, Sam
\end{chunk}
+\index{Bressoud, David}
+\begin{chunk}{axiom.bib}
+@article{Bres93,
+ author = "Bressoud, David",
+ title = "Review of ``The problems of mathematics'',
+ journal = "Math. Intell.",
+ volume = "15",
+ number = "4",
+ year = "1993",
+ pages 7173"
+}
+
+\end{chunk}
+
\index{Mahboubi, Assia}
\begin{chunk}{axiom.bib}
@article{Mahb06,
@@ 6250,24 +6264,73 @@ Proc ISSAC 97 pp172175 (1997)
\section{Symbolic Summation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\index{Karr, Michael}
+\index{Abramov, S.A.}
\begin{chunk}{axiom.bib}
@Article{Karr85,
 author = "Karr, Michael",
 title = "Theory of Summation in Finite Terms",
 year = "1985",
 journal = "Journal of Symbolic Computation",
 volume = "1",
 number = "3",
 month = "September",
 pages = "303315",
 paper = "Karr85.pdf",
+@article{Abra71,
+ author = "Abramov, S.A.",
+ title = "On the summation of rational functions",
+ year = "1971",
+ journal = "USSR Computational Mathematics and Mathematical Physics",
+ volume = "11",
+ number = "4",
+ pages = "324330",
+ paper = "Abra71.pdf",
abstract = "
 This paper discusses some of the mathematical aspects of an algorithm
 for finding formulas for finite sums. The results presented here
 concern a property of difference fields which show that the algorithm
 does not divide by zero, and an analogue to Liouville's theorem on
 elementary integrals."
+ An algorithm is given for solving the following problem: let
+ $F(x_1,\ldots,x_n)$ be a rational function of the variables
+ $x_i$ with rational (read or complex) coefficients; to see if
+ there exists a rational function $G(v,w,x_2,\ldots,x_n)$ with
+ coefficients from the same field, such that
+ \[\sum_{x_1=v}^w{F(x_1,\ldots,x_n)} = G(v,w,x_2,\ldots,x_n)\]
+ for all integral values of $v \le w$. If $G$ exists, to obtain it.
+ Realization of the algorithm in the LISP language is discussed."
+}
+
+\end{chunk}
+
+\index{Gosper, R. William}
+\begin{chunk}{axiom.bib}
+@article{Gosp78,
+ author = "Gosper, R. William",
+ title = "Decision procedure for indefinite hypergeometric summation",
+ year = "1978",
+ journal = "Proc. Natl. Acad. Sci. USA",
+ volume = "75",
+ number = "1",
+ pages = "4042",
+ month = "January",
+ paper = "Gosp78.pdf",
+ abstract = "
+ Given a summand $a_n$, we seek the ``indefinite sum'' $S(n)$
+ determined (within an additive constant) by
+ \[\sum_{n=1}^m{a_n} = S(m)=S(0)\]
+ or, equivalently, by
+ \[a_n=S(n)S(n1)\]
+ An algorithm is exhibited which, given $a_n$, finds those $S(n)$
+ with the property
+ \[\displaystyle\frac{S(n)}{S(n1)}=\textrm{a rational function of n}\]
+ With this algorithm, we can determine, for example, the three
+ identities
+ \[\displaystyle\sum_{n=1}^m{
+ \frac{\displaystyle\prod_{j=1}^{n1}{bj^2+cj+d}}
+ {\displaystyle\prod_{j=1}^n{bj^2+cj+e}}=
+ \frac{1{\displaystyle\prod_{j=1}^m{\frac{bj^2+cj+d}{bj^2+cj+e}}}}{ed}}\]
+ \[\displaystyle\sum_{n=1}^m{
+ \frac{\displaystyle\prod_{j=1}^{n1}{aj^3+bj^2+cj+d}}
+ {\displaystyle\prod_{j=1}^n{aj^3+bj^2+cj+e}}=
+ \frac{1{\displaystyle\prod_{j=1}^m{
+ \frac{aj^3+bj^2+cj+d}{aj^3+bj^2+cj+e}}}}{ed}}\]
+ \[\displaystyle\sum_{n=1}^m{
+ \displaystyle\frac{\displaystyle\prod_{j=1}^{n1}{bj^2+cj+d}}
+ {\displaystyle\prod_{j=1}^{n+1}{bj^2+cj+e}}=
+ \displaystyle\frac{
+ \displaystyle\frac{2b}{ed}
+ \displaystyle\frac{3b+c+de}{b+c+e}
+ \left(
+ \displaystyle\frac{2b}{ed}\frac{b(2m+3)+c+de}{b(m+1)^2+c(m+1)+e}
+ \right)
+ \displaystyle\prod_{j=1}^m{\frac{bj^2+cj+d}{bj^2+cj+e}}}
+ {b^2c^2+d^2+e^2+2bd2de+2eb}}\]"
}
\end{chunk}
@@ 6302,54 +6365,150 @@ Proc ISSAC 97 pp172175 (1997)
\end{chunk}
\index{Zima, Eugene V.}
+\index{Abramov, S.A.}
\begin{chunk}{axiom.bib}
@article{Zima13,
 author = "Zima, Eugene V.",
 title = "Accelerating Indefinite Summation: Simple Classes of Summands",
 journal = "Mathematics in Computer Science",
 year = "2013",
 month = "December",
 volume = "7",
 number = "4",
 pages = "455472",
 paper = "Zima13.pdf",
+@article{Abra85,
+ author = "Abramov, S.A.",
+ title = "Separation of variables in rational functions",
+ year = "1985",
+ journal = "USSR Computational Mathematics and Mathematical Physics",
+ volume = "25",
+ number = "5",
+ pages = "99102",
+ paper = "Abra85.pdf",
abstract = "
 We present the history of indefinite summation starting with classics
 (Newton, Montmort, Taylor, Stirling, Euler, Boole, Jordan) followed by
 modern classics (Abramov, Gosper, Karr) to the current implementation
 in computer algebra system Maple. Along with historical presentation
 we describe several ``acceleration techniques'' of algorithms for
 indefinite summation which offer not only theoretical but also
 practical improvements in running time. Implementations of these
 algorithms in Maple are compared to standard Maple summation tools"
+The problem of expanding a rational function of several variables into
+terms with separable variables is formulated. An algorithm for solving
+this problem is given. Programs which implement this algorithm can
+occur in sets of algebraic alphabetical transformations on a computer
+and can be used to reduce the multiplicity of sums and integrals of
+rational functions for investigating differential equations with
+rational righthand sides etc."
}
\end{chunk}
\index{Polyakov, S.P.}
+\index{Karr, Michael}
\begin{chunk}{axiom.bib}
@article{Poly11,
 author = "Polyadov, S.P.",
 title = "Indefinite summation of rational functions with factorization
 of denominators",
 year = "2011",
 month = "November",
 journal = "Programming and Computer Software",
 volume = "37",
 number = "6",
 pages = "322325",
 paper = "Poly11.pdf",
+@Article{Karr85,
+ author = "Karr, Michael",
+ title = "Theory of Summation in Finite Terms",
+ year = "1985",
+ journal = "Journal of Symbolic Computation",
+ volume = "1",
+ number = "3",
+ month = "September",
+ pages = "303315",
+ paper = "Karr85.pdf",
abstract = "
 A computer algebra algorithm for indefinite summation of rational
 functions based on complete factorization of denominators is
 proposed. For a given $f$, the algorithm finds two rational functions
 $g$, $r$ such that $f=g(x+1)g(x)+r$ and the degree of the denominator
 of $r$ is minimal. A modification of the algorithm is also proposed
 that additionally minimizes the degree of the denominator of
 $g$. Computational complexity of the algorithms without regard to
 denominator factorization is shown to be $O(m^2)$, where $m$ is the
 degree of the denominator of $f$."
+ This paper discusses some of the mathematical aspects of an algorithm
+ for finding formulas for finite sums. The results presented here
+ concern a property of difference fields which show that the algorithm
+ does not divide by zero, and an analogue to Liouville's theorem on
+ elementary integrals."
+}
+
+\end{chunk}
+
+\index{Koepf, Wolfram}
+\begin{chunk}{axiom.bib}
+@book{Koep98,
+ author = "Koepf, Wolfram",
+ title = "Hypergeometric Summation",
+ publisher = "Springer",
+ year = "1998",
+ isbn = "9781447164647",
+ paper = "Koep98.pdf",
+ abstract = "
+ Modern algorithmic techniques for summation, most of which were
+ introduced in the 1990s, are developed here and carefully implemented
+ in the computer algebra system Maple.
+
+ The algorithms of Fasenmyer, Gosper, Zeilberger, Petkovsek and van
+ Hoeij for hypergeometric summation and recurrence equations, efficient
+ multivariate summation as well as qanalogues of the above algorithms
+ are covered. Similar algorithms concerning differential equations are
+ considered. An equivalent theory of hyperexponential integration due
+ to Almkvist and Zeilberger completes the book.
+
+ The combination of these results gives orthogonal polynomials and
+ (hypergeometric and qhypergeometric) special functions a solid
+ algorithmic foundation. Hence, many examples from this very active
+ field are given.
+
+ The materials covered are sutiable for an introductory course on
+ algorithmic summation and will appeal to students and researchers
+ alike."
+}
+
+\end{chunk}
+
+\index{Schneider, Carsten}
+\begin{chunk}{axiom.bib}
+@InProceedings{Schn00,
+ author = "Schneider, Carsten",
+ title = "An implementation of Karr's summation algorithm in Mathematica",
+ year = "2000",
+ booktitle = "S\'eminaire Lotharingien de Combinatoire",
+ volume = "S43b",
+ pages = "110",
+ url = "",
+ paper = "Schn00.pdf",
+ abstract = "
+ Implementations of the celebrated Gosper algorithm (1978) for
+ indefinite summation are available on almost any computer algebra
+ platform. We report here about an implementation of an algorithm by
+ Karr, the most general indefinite summation algorithm known. Karr's
+ algorithm is, in a sense, the summation counterpart of Risch's
+ algorithm for indefinite integration. This is the first implementation
+ of this algorithm in a major computer algebra system. Our version
+ contains new extensions to handle also definite summation problems. In
+ addition we provide a feature to find automatically appropriate
+ difference field extensions in which a closed form for the summation
+ problem exists. These new aspects are illustrated by a variety of
+ examples."
+
+}
+
+\end{chunk}
+
+\index{Schneider, Carsten}
+\begin{chunk}{axiom.bib}
+@phdthesis{Schn01,
+ author = "Schneider, Carsten",
+ title = "Symbolic Summation in Difference Fields",
+ school = "RISC Research Institute for Symbolic Computation",
+ year = "2001",
+ url =
+ "http://www.risc.jku.at/publications/download/risc_3017/SymbSumTHESIS.pdf",
+ paper = "Schn01.pdf",
+ abstract = "
+
+ There are implementations of the celebrated Gosper algorithm (1978) on
+ almost any computer algebra platform. Within my PhD thesis work I
+ implemented Karr's Summation Algorithm (1981) based on difference
+ field theory in the Mathematica system. Karr's algorithm is, in a
+ sense, the summation counterpart of Risch's algorithm for indefinite
+ integration. Besides Karr's algorithm which allows us to find closed
+ forms for a big clas of multisums, we developed new extensions to
+ handle also definite summation problems. More precisely we are able to
+ apply creative telescoping in a very general difference field setting
+ and are capable of solving linear recurrences in its context.
+
+ Besides this we find significant new insights in symbolic summation by
+ rephrasing the summation problems in the general difference field
+ setting. In particular, we designed algorithms for finding appropriate
+ difference field extensions to solve problems in symbolic summation.
+ For instance we deal with the problem to find all nested sum
+ extensions which provide us with additional solutions for a given
+ linear recurrence of any order. Furthermore we find appropriate sum
+ extensions, if they exist, to simplify nested sums to simpler nested
+ sum expressions. Moreover we are able to interpret creative
+ telescoping as a special case of sum extensions in an indefinite
+ summation problem. In particular we are able to determine sum
+ extensions, in case of existence, to reduce the order of a recurrence
+ for a definite summation problem."
+
}
\end{chunk}
@@ 6376,49 +6535,136 @@ Proc ISSAC 97 pp172175 (1997)
\end{chunk}
\index{Abramov, S.A.}
+\index{Schneider, Carsten}
\begin{chunk}{axiom.bib}
@article{Abra85,
 author = "Abramov, S.A.",
 title = "Separation of variables in rational functions",
 year = "1985",
 journal = "USSR Computational Mathematics and Mathematical Physics",
 volume = "25",
 number = "5",
 pages = "99102",
 paper = "Abra85.pdf",
+@article{Schn05,
+ author = "Schneider, Carsten",
+ title = "A new Sigma approach to multisummation",
+ year = "2005",
+ journal = "Advances in Applied Mathematics",
+ volume = "34",
+ number = "4",
+ pages = "740767",
+ paper = "Schn05.pdf",
abstract = "
The problem of expanding a rational function of several variables into
terms with separable variables is formulated. An algorithm for solving
this problem is given. Programs which implement this algorithm can
occur in sets of algebraic alphabetical transformations on a computer
and can be used to reduce the multiplicity of sums and integrals of
rational functions for investigating differential equations with
rational righthand sides etc."
+ We present a general algorithmic framework that allows not only to
+ deal with summation problems over summands being rational expressions
+ in indefinite nested syms and products (Karr, 1981), but also over
+ $\delta$finite and holonomic summand expressions that are given by a
+ linear recurrence. This approach implies new computer algebra tools
+ implemented in Sigma to solve multisummation problems efficiently.
+ For instacne, the extended Sigma package has been applied successively
+ to provide a computerassisted proof of Stembridge's TSPP Theorem."
}
\end{chunk}
\index{Abramov, S.A.}
+\index{Schneider, Carsten}
+\index{Kauers, Manuel}
\begin{chunk}{axiom.bib}
@article{Abra71,
 author = "Abramov, S.A.",
 title = "On the summation of rational functions",
 year = "1971",
 journal = "USSR Computational Mathematics and Mathematical Physics",
 volume = "11",
 number = "4",
 pages = "324330",
 paper = "Abra71.pdf",
+@article{Kaue08,
+ author = "Kauers, Manuel and Schneider, Carsten",
+ title = "Indefinite summation with unspecified summands",
+ year = "2006",
+ journal = "Discrete Mathematics",
+ volume = "306",
+ number = "17",
+ pages = "20732083",
+ paper = "Kaue80.pdf",
+ abstract = "
+ We provide a new algorithm for indefinite nested summation which is
+ applicable to summands involving unspecified sequences $x(n)$. More
+ than that, we show how to extend Karr's algorithm to a general
+ summation framework by which additional types of summand expressions
+ can be handled. Our treatment of unspecified sequences can be seen as
+ a first illustrative application of this approach."
+}
+
+\end{chunk}
+
+\index{Kauers, Manuel}
+\begin{chunk}{axiom.bib}
+@article{Kaue07,
+ author = "Kauers, Manuel",
+ title = "Summation algorithms for Stirling number identities",
+ year = "2007",
+ journal = "Journal of Symbolic Computation",
+ volume = "42",
+ number = "10",
+ month = "October",
+ pages = "948970",
+ paper = "Kaue07.pdf",
+ abstract = "
+ We consider a class of sequences defined by triangular recurrence
+ equations. This class contains Stirling numbers and Eulerian numbers
+ of both kinds, and hypergeometric multiples of those. We give a
+ sufficient criterion for sums over such sequences to obey a recurrence
+ equation, and present algorithms for computing such recurrence
+ equations efficiently. Our algorithms can be used for verifying many
+ known summation identities on Stirling numbers instantly, and also for
+ discovering new identities."
+}
+
+\end{chunk}
+
+\index{Schneider, Carsten}
+\begin{chunk}{axiom.bib}
+@InProceedings{Schn07,
+ author = "Schneider, Carsten",
+ title = "Symbolic Summation Assists Combinatorics",
+ year = "2007",
+ booktitle = "S\'eminaire Lotharingien de Combinatoire",
+ volume = "56",
+ article = "B56b",
+ url = "",
+ paper = "Schn07.pdf",
+ abstract = "
+ We present symbolic summation tools in the context of difference
+ fields that help scientists in practical problem solving. Throughout
+ this article we present multisum examples which are related to
+ combinatorial problems."
+}
+
+\end{chunk}
+
+\index{Schneider, Carsten}
+\begin{chunk}{axiom.bib}
+@article{Schn08,
+ author = "Schneider, Carsten",
+ title = "A refined difference field theory for symbolic summation",
+ year = "2008",
+ journal = "Journal of Symbolic Computation",
+ volume = "43",
+ number = "9",
+ pages = "611644",
+ paper = "Schn08.pdf",
+ abstract = "
+ In this article we present a refined summation theory based on Karr's
+ difference field approach. The resulting algorithms find sum
+ representations with optimal nested depth. For instance, the
+ algorithms have been applied successively to evaluate Feynman
+ integrals from Perturbative Quantum Field Theory"
+}
+
+\end{chunk}
+
+\index{Schneider, Carsten}
+\begin{chunk}{axiom.bib}
+@article{Schn09,
+ author = "Schneider, Carsten",
+ title = "Structural theorems for symbolic summation",
+ journal = "Proc. AAECC2010",
+ year = "2010",
+ volume = "21",
+ pages = "132",
+ paper = "Schn09.pdf",
abstract = "
 An algorithm is given for solving the following problem: let
 $F(x_1,\ldots,x_n)$ be a rational function of the variables
 $x_i$ with rational (read or complex) coefficients; to see if
 there exists a rational function $G(v,w,x_2,\ldots,x_n)$ with
 coefficients from the same field, such that
 \[\sum_{x_1=v}^w{F(x_1,\ldots,x_n)} = G(v,w,x_2,\ldots,x_n)\]
 for all integral values of $v \le w$. If $G$ exists, to obtain it.
 Realization of the algorithm in the LISP language is discussed."
+ Starting with Karr's structural theorem for summation  the discrete
+ version of Liouville's structural theorem for integration  we work
+ out crucial properties of the underlying difference fields. This leads
+ to new and constructive structural theorems for symbolic summation.
+ E.g., these results can be applied for harmonic sums which arise
+ frequently in particle physics."
}
\end{chunk}
@@ 6512,206 +6758,303 @@ rational righthand sides etc."
\end{chunk}
\index{Schneider, Carsten}
+\index{Polyakov, S.P.}
\begin{chunk}{axiom.bib}
@article{Schn05,
 author = "Schneider, Carsten",
 title = "A new Sigma approach to multisummation",
 year = "2005",
 journal = "Advances in Applied Mathematics",
 volume = "34",
 number = "4",
 pages = "740767",
 paper = "Schn05.pdf",
+@article{Poly11,
+ author = "Polyadov, S.P.",
+ title = "Indefinite summation of rational functions with factorization
+ of denominators",
+ year = "2011",
+ month = "November",
+ journal = "Programming and Computer Software",
+ volume = "37",
+ number = "6",
+ pages = "322325",
+ paper = "Poly11.pdf",
abstract = "
 We present a general algorithmic framework that allows not only to
 deal with summation problems over summands being rational expressions
 in indefinite nested syms and products (Karr, 1981), but also over
 $\delta$finite and holonomic summand expressions that are given by a
 linear recurrence. This approach implies new computer algebra tools
 implemented in Sigma to solve multisummation problems efficiently.
 For instacne, the extended Sigma package has been applied successively
 to provide a computerassisted proof of Stembridge's TSPP Theorem."
+ A computer algebra algorithm for indefinite summation of rational
+ functions based on complete factorization of denominators is
+ proposed. For a given $f$, the algorithm finds two rational functions
+ $g$, $r$ such that $f=g(x+1)g(x)+r$ and the degree of the denominator
+ of $r$ is minimal. A modification of the algorithm is also proposed
+ that additionally minimizes the degree of the denominator of
+ $g$. Computational complexity of the algorithms without regard to
+ denominator factorization is shown to be $O(m^2)$, where $m$ is the
+ degree of the denominator of $f$."
}
\end{chunk}
\index{Kauers, Manuel}
+\index{Schneider, Carsten}
\begin{chunk}{axiom.bib}
@article{Kaue07,
 author = "Kauers, Manuel",
 title = "Summation algorithms for Stirling number identities",
 year = "2007",
 journal = "Journal of Symbolic Computation",
 volume = "42",
 number = "10",
 month = "October",
 pages = "948970",
 paper = "Kaue07.pdf",
+@article{Schn13,
+ author = "Schneider, Carsten",
+ title =
+ "Fast Algorithms for Refined Parameterized Telescoping in Difference Fields",
+ journal = "CoRR",
+ year = "2013",
+ volume = "abs/1307.7887",
+ paper = "Schn13.pdf",
+ keywords = "survey",
abstract = "
 We consider a class of sequences defined by triangular recurrence
 equations. This class contains Stirling numbers and Eulerian numbers
 of both kinds, and hypergeometric multiples of those. We give a
 sufficient criterion for sums over such sequences to obey a recurrence
 equation, and present algorithms for computing such recurrence
 equations efficiently. Our algorithms can be used for verifying many
 known summation identities on Stirling numbers instantly, and also for
 discovering new identities."
+ Parameterized telescoping (including telescoping and creative
+ telescoping) and refined versions of it play a central role in the
+ research area of symbolic summation. In 1981 Karr introduced
+ $\prod\sum$fields, a general class of difference fields, that enables
+ one to consider this problem for indefinite nested sums and products
+ covering as special cases, e.g., the (q)hypergeometric case and their
+ mixed versions. This survey article presents the available algorithms
+ in the framework of $\prod\sum$extensions and elaborates new results
+ concerning efficiency."
}
\end{chunk}
\index{Schneider, Carsten}
\index{Kauers, Manuel}
+\index{Zima, Eugene V.}
\begin{chunk}{axiom.bib}
@article{Kaue08,
 author = "Kauers, Manuel and Schneider, Carsten",
 title = "Indefinite summation with unspecified summands",
 year = "2006",
 journal = "Discrete Mathematics",
 volume = "306",
 number = "17",
 pages = "20732083",
 paper = "Kaue80.pdf",
+@article{Zima13,
+ author = "Zima, Eugene V.",
+ title = "Accelerating Indefinite Summation: Simple Classes of Summands",
+ journal = "Mathematics in Computer Science",
+ year = "2013",
+ month = "December",
+ volume = "7",
+ number = "4",
+ pages = "455472",
+ paper = "Zima13.pdf",
abstract = "
 We provide a new algorithm for indefinite nested summation which is
 applicable to summands involving unspecified sequences $x(n)$. More
 than that, we show how to extend Karr's algorithm to a general
 summation framework by which additional types of summand expressions
 can be handled. Our treatment of unspecified sequences can be seen as
 a first illustrative application of this approach."
+ We present the history of indefinite summation starting with classics
+ (Newton, Montmort, Taylor, Stirling, Euler, Boole, Jordan) followed by
+ modern classics (Abramov, Gosper, Karr) to the current implementation
+ in computer algebra system Maple. Along with historical presentation
+ we describe several ``acceleration techniques'' of algorithms for
+ indefinite summation which offer not only theoretical but also
+ practical improvements in running time. Implementations of these
+ algorithms in Maple are compared to standard Maple summation tools"
}
\end{chunk}
\index{Schneider, Carsten}
\begin{chunk}{axiom.bib}
@article{Schn08,
+@misc{Schn14,
author = "Schneider, Carsten",
 title = "A refined difference field theory for symbolic summation",
 year = "2008",
 journal = "Journal of Symbolic Computation",
 volume = "43",
 number = "9",
 pages = "611644",
 paper = "Schn08.pdf",
+ title = "A Difference Ring Theory for Symbolic Summation",
+ year = "2014",
+ paper = "Schn14.pdf",
abstract = "
 In this article we present a refined summation theory based on Karr's
 difference field approach. The resulting algorithms find sum
 representations with optimal nested depth. For instance, the
 algorithms have been applied successively to evaluate Feynman
 integrals from Perturbative Quantum Field Theory"
+ A summation framework is developed that enhances Karr's difference
+ field approach. It covers not only indefinite nested sums and products
+ in terms of transcendental extensions, but it can treat, e.g., nested
+ products defined over roots of unity. The theory of the socalled
+ $R\prod\sum*$extensions is supplemented by algorithms that support the
+ construction of such difference rings automatically and that assist in
+ the task to tackle symbolic summation problems. Algorithms are
+ presented that solve parameterized telescoping equations, and more
+ generally parameterized firstorder difference equations, in the given
+ difference ring. As a consequence, one obtains algorithms for the
+ summation paradigms of telescoping and Zeilberger's creative
+ telescoping. With this difference ring theory one obtains a rigorous
+ summation machinery that has been applied to numerous challenging
+ problems coming, e.g., from combinatorics and particle physics."
}
\end{chunk}
\index{Schneider, Carsten}
+\index{VazquezTrejo, Javier}
\begin{chunk}{axiom.bib}
@article{Schn09,
 author = "Schneider, Carsten",
 title = "Structural theorems for symbolic summation",
 journal = "Proc. AAECC2010",
 year = "2010",
 volume = "21",
 pages = "132",
 paper = "Schn09.pdf",
 abstract = "
 Starting with Karr's structural theorem for summation  the discrete
 version of Liouville's structural theorem for integration  we work
 out crucial properties of the underlying difference fields. This leads
 to new and constructive structural theorems for symbolic summation.
 E.g., these results can be applied for harmonic sums which arise
 frequently in particle physics."
+@phdthesis{Vazq14,
+ author = "VazquezTrejo, Javier",
+ title = "Symbolic Summation in Difference Fields",
+ year = "2014",
+ school = "CarnegieMellon University",
+ paper = "Vazq14.pdf",
+ abstract = "
+ We seek to understand a general method for finding a closed form for a
+ given sum that acts as its antidifference in the same way that an
+ integral has an antiderivative. Once an antidifference is found, then
+ given the limits of the sum, it suffices to evaluate the
+ antidifference at the given limits. Several algorithms (by Karr and
+ Schneider) exist to find antidifferences, but the apers describing
+ these algorithms leave out several of the key proofs needed to
+ implement the algorithms. We attempt to fill in these gaps and find
+ that many of the steps to solve difference equations rely on being
+ able to solve two problems: the equivalence problem and the homogenous
+ group membership problem. Solving these two problems is essential to
+ finding the polynomial degree bounds and denominator bounds for
+ solutions of difference equations. We study Karr and Schneider's
+ treatment of these problems and elaborate on the unproven parts of
+ their work. Section 1 provides background material; section 2 provides
+ motivation and previous work; Section 3 provides an outline of Karr's
+ Algorithm; section 4 examines the Equivalance Problem, and section 5
+ examines the Homogeneous Group Membership Problem. Section 6 presents
+ some proofs for the denominator and polynomial bounds used in solving
+ difference equations, and Section 7 gives some directions for future
+ work."
+}
+
+\end{chunk}
+
+\index{Petkov\overline{s}ek, Marko}
+\index{Wilf, Herbert S.}
+\index{Zeilberger, Doran}
+\begin{chunk}{axiom.bib}
+@book{Petk97,
+ author = "Petkov\overline{s}ek, Marko and Wilf, Herbert S. and
+ Zeilberger, Doran",
+ title = "A=B",
+ publisher = "A.K. Peters, Ltd",
+ year = "1997",
+ paper = "Petk97.pdf"
}
\end{chunk}
\index{Schneider, Carsten}
\begin{chunk}{axiom.bib}
@phdthesis{Schn01,
 author = "Schneider, Carsten",
 title = "Symbolic Summation in Difference Fields",
 school = "RISC Research Institute for Symbolic Computation",
 year = "2001",
 url =
 "http://www.risc.jku.at/publications/download/risc_3017/SymbSumTHESIS.pdf",
 paper = "Schn01.pdf",
 abstract = "
+\section{Differential Forms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 There are implementations of the celebrated Gosper algorithm (1978) on
 almost any computer algebra platform. Within my PhD thesis work I
 implemented Karr's Summation Algorithm (1981) based on difference
 field theory in the Mathematica system. Karr's algorithm is, in a
 sense, the summation counterpart of Risch's algorithm for indefinite
 integration. Besides Karr's algorithm which allows us to find closed
 forms for a big clas of multisums, we developed new extensions to
 handle also definite summation problems. More precisely we are able to
 apply creative telescoping in a very general difference field setting
 and are capable of solving linear recurrences in its context.
+\index{Cartan, Henri}
+\begin{chunk}{axiom.bib}
+@book{Cart06,
+ author = {Cartan, Henri},
+ title = {Differential Forms},
+ year = "2006",
+ location = {Mineola, N.Y},
+ edition = {Auflage: Tra},
+ isbn = {9780486450100},
+ pagetotal = {166},
+ publisher = {Dover Pubn Inc},
+ date = {20060526}
+}
 Besides this we find significant new insights in symbolic summation by
 rephrasing the summation problems in the general difference field
 setting. In particular, we designed algorithms for finding appropriate
 difference field extensions to solve problems in symbolic summation.
 For instance we deal with the problem to find all nested sum
 extensions which provide us with additional solutions for a given
 linear recurrence of any order. Furthermore we find appropriate sum
 extensions, if they exist, to simplify nested sums to simpler nested
 sum expressions. Moreover we are able to interpret creative
 telescoping as a special case of sum extensions in an indefinite
 summation problem. In particular we are able to determine sum
 extensions, in case of existence, to reduce the order of a recurrence
 for a definite summation problem."
+\end{chunk}
+\index{Flanders, Harley}
+\begin{chunk}{axiom.bib}
+ @book{Flan03,
+ author = {Flanders, Harley and Mathematics},
+ title = {Differential Forms with Applications to the Physical Sciences},
+ year = "2003",
+ location = {Mineola, N.Y},
+ isbn = {9780486661698}
+ pagetotal = {240},
+ publisher = {Dover Pubn Inc},
+ date = {20030328}
}
\end{chunk}
\index{Schneider, Carsten}
+\index{Whitney, Hassler}
\begin{chunk}{axiom.bib}
@InProceedings{Schn07,
 author = "Schneider, Carsten",
 title = "Symbolic Summation Assists Combinatorics",
 year = "2007",
 booktitle = "S\'eminaire Lotharingien de Combinatoire",
 volume = "56",
 article = "B56b",
 url = "",
 paper = "Schn07.pdf",
 abstract = "
 We present symbolic summation tools in the context of difference
 fields that help scientists in practical problem solving. Throughout
 this article we present multisum examples which are related to
 combinatorial problems."
+@book{Whit12,
+ author = {Whitney, Hassler},
+ title =
+ {Geometric Integration Theory: Princeton Mathematical Series, No. 21},
+ year = "2012",
+ isbn = {9781258346386},
+ shorttitle = {Geometric Integration Theory},
+ pagetotal = {402},
+ publisher = {Literary Licensing, {LLC}},
+ date = {20120501}
}
\end{chunk}
\index{Schneider, Carsten}
+\index{Federer, Herbert}
\begin{chunk}{axiom.bib}
@InProceedings{Schn00,
 author = "Schneider, Carsten",
 title = "An implementation of Karr's summation algorithm in Mathematica",
 year = "2000",
 booktitle = "S\'eminaire Lotharingien de Combinatoire",
 volume = "S43b",
 pages = "110",
 url = "",
 paper = "Schn00.pdf",
 abstract = "
 Implementations of the celebrated Gosper algorithm (1978) for
 indefinite summation are available on almost any computer algebra
 platform. We report here about an implementation of an algorithm by
 Karr, the most general indefinite summation algorithm known. Karr's
 algorithm is, in a sense, the summation counterpart of Risch's
 algorithm for indefinite integration. This is the first implementation
 of this algorithm in a major computer algebra system. Our version
 contains new extensions to handle also definite summation problems. In
 addition we provide a feature to find automatically appropriate
 difference field extensions in which a closed form for the summation
 problem exists. These new aspects are illustrated by a variety of
 examples."
+@book{Fede13,
+ author = {Federer, Herbert},
+ title = {Geometric Measure Theory},
+ year = "2013",
+ location = {Berlin ; New York},
+ edition = {Reprint of the 1st ed. Berlin, Heidelberg, New York 1969},
+ isbn = {9783540606567},
+ pagetotal = {700},
+ publisher = {Springer},
+ date = {20131004},
+ abstract = {
+ "This book is a major treatise in mathematics and is essential in the
+ working library of the modern analyst." (Bulletin of the London
+ Mathematical Society)}
+}
+
+\end{chunk}
+\index{Abraham, Ralph}
+\index{Marsden, Jerrold E.}
+\index{Ratiu, Tudor}
+\begin{chunk}{axiom.bib}
+@book{Abra93,
+ author = {Abraham, Ralph and Marsden, Jerrold E. and Ratiu, Tudor},
+ title = {Manifolds, Tensor Analysis, and Applications},
+ year = "1993",
+ location = {New York},
+ edition = {2nd Corrected ed. 1988. Corr. 2nd printing 1993},
+ isbn = {9780387967905},
+ pagetotal = {656},
+ publisher = {Springer},
+ date = {19930826}
+ abstract = {
+ The purpose of this book is to provide core material in nonlinear
+ analysis for mathematicians, physicists, engineers, and mathematical
+ biologists. The main goal is to provide a working knowledge of
+ manifolds, dynamical systems, tensors, and differential forms. Some
+ applications to Hamiltonian mechanics, fluid mechanics,
+ electromagnetism, plasma dynamics and control theory are given using
+ both invariant and index notation. The prerequisites required are
+ solid undergraduate courses in linear algebra and advanced calculus.}
+}
+
+\end{chunk}
+
+\index{Lambe, L. A.}
+\index{Radford, D. E.}
+\begin{chunk}{axiom.bib}
+@book{Lamb97,
+ author = {Lambe, L. A. and Radford, D. E.},
+ title = {Introduction to the Quantum YangBaxter Equation and
+ Quantum Groups: An Algebraic Approach},
+ year = "1997",
+ location = {Dordrecht ; Boston},
+ edition = {Auflage: 1997},
+ isbn = {9780792347217},
+ shorttitle = {Introduction to the Quantum YangBaxter Equation and
+ Quantum Groups},
+ abstract = {
+ Chapter 1 The algebraic prerequisites for the book are covered here
+ and in the appendix. This chapter should be used as reference material
+ and should be consulted as needed. A systematic treatment of algebras,
+ coalgebras, bialgebras, Hopf algebras, and represen tations of these
+ objects to the extent needed for the book is given. The material here
+ not specifically cited can be found for the most part in [Sweedler,
+ 1969] in one form or another, with a few exceptions. A great deal of
+ emphasis is placed on the coalgebra which is the dual of n x n
+ matrices over a field. This is the most basic example of a coalgebra
+ for our purposes and is at the heart of most algebraic constructions
+ described in this book. We have found pointed bialgebras useful in
+ connection with solving the quantum YangBaxter equation. For this
+ reason we develop their theory in some detail. The class of examples
+ described in Chapter 6 in connection with the quantum double consists
+ of pointed Hopf algebras. We note the quantized enveloping algebras
+ described Hopf algebras. Thus for many reasons pointed bialgebras are
+ elsewhere are pointed of fundamental interest in the study of the
+ quantum YangBaxter equation and objects quantum groups.},
+ pagetotal = {300},
+ publisher = {Springer},
+ date = {19971031}
+}
+
+\end{chunk}
+
+\index{Wheeler, James T.}
+\begin{chunk}{axiom.bib}
+@misc{Whee12,
+ author = "Wheeler, James T.",
+ title = "Differential Forms",
+ year = "2012",
+ month = "September",
+ url =
+"http://www.physics.usu.edu/Wheeler/ClassicalMechanics/CMDifferentialForms.pdf",
+ paper = "Whee12.pdf"
}
\end{chunk}
@@ 15239,19 +15582,6 @@ Math. Tables Aids Comput. 10 9196. (1956)
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Whee12,
 author = "Wheeler, James T.",
 title = "Differential Forms",
 year = "2012",
 month = "September",
 url =
"http://www.physics.usu.edu/Wheeler/ClassicalMechanics/CMDifferentialForms.pdf",
 paper = "Whee12.pdf"
}

\end{chunk}

\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Bibliography}
diff git a/changelog b/changelog
index cd8b464..08e6617 100644
 a/changelog
+++ b/changelog
@@ 1,3 +1,5 @@
+20141017 tpd src/axiomwebsite/patches.html 20141017.01.tpd.patch
+20141017 tpd books/bookvolbib add a section on Differential Forms
20141010 kxp src/axiomwebsite/patches.html 20141010.01.kxp.patch
20141010 kxp books/bookvolbib add references
20141010 kxp src/input/derham3.input test Pagani's functions
diff git a/patch b/patch
index 1583b4b..65b03ee 100644
 a/patch
+++ b/patch
@@ 1,3 +1,4 @@
books/bookvol10.3 add Pagani's functions to DERHAM
+books/bookvolbib add a section on Differential Forms
+
+Kurt has written new documentation. Add the references.
Additional functions in DERHAM
diff git a/src/axiomwebsite/patches.html b/src/axiomwebsite/patches.html
index 7a66eeb..f354e37 100644
 a/src/axiomwebsite/patches.html
+++ b/src/axiomwebsite/patches.html
@@ 4680,6 +4680,8 @@ books/bookvol10.1 add chapter on differential forms
books/bookvolbib add a section on Symbolic Summation
20141010.01.kxp.patch
books/bookvol10.3 add Pagani's functions to DERHAM
+20141017.01.tpd.patch
+books/bookvolbib add a section on Differential Forms