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 71-73"
+}
+
+\end{chunk}
+
\index{Mahboubi, Assia}
\begin{chunk}{axiom.bib}
@article{Mahb06,
@@ -6250,24 +6264,73 @@ Proc ISSAC 97 pp172-175 (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 = "303-315",
- 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 = "324--330",
+ 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 = "40--42",
+ 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
+ \[a_n=S(n)-S(n-1)\]
+ An algorithm is exhibited which, given $a_n$, finds those $S(n)$
+ with the property
+ \[\displaystyle\frac{S(n)}{S(n-1)}=\textrm{a rational function of n}\]
+ With this algorithm, we can determine, for example, the three
+ identities
+ \[\displaystyle\sum_{n=1}^m{
+ \frac{\displaystyle\prod_{j=1}^{n-1}{bj^2+cj+d}}
+ {\displaystyle\prod_{j=1}^n{bj^2+cj+e}}=
+ \frac{1-{\displaystyle\prod_{j=1}^m{\frac{bj^2+cj+d}{bj^2+cj+e}}}}{e-d}}\]
+ \[\displaystyle\sum_{n=1}^m{
+ \frac{\displaystyle\prod_{j=1}^{n-1}{aj^3+bj^2+cj+d}}
+ {\displaystyle\prod_{j=1}^n{aj^3+bj^2+cj+e}}=
+ \frac{1-{\displaystyle\prod_{j=1}^m{
+ \frac{aj^3+bj^2+cj+d}{aj^3+bj^2+cj+e}}}}{e-d}}\]
+ \[\displaystyle\sum_{n=1}^m{
+ \displaystyle\frac{\displaystyle\prod_{j=1}^{n-1}{bj^2+cj+d}}
+ {\displaystyle\prod_{j=1}^{n+1}{bj^2+cj+e}}=
+ \displaystyle\frac{
+ \displaystyle\frac{2b}{e-d}-
+ \displaystyle\frac{3b+c+d-e}{b+c+e}-
+ \left(
+ \displaystyle\frac{2b}{e-d}-\frac{b(2m+3)+c+d-e}{b(m+1)^2+c(m+1)+e}
+ \right)
+ \displaystyle\prod_{j=1}^m{\frac{bj^2+cj+d}{bj^2+cj+e}}}
+ {b^2-c^2+d^2+e^2+2bd-2de+2eb}}\]"
}
\end{chunk}
@@ -6302,54 +6365,150 @@ Proc ISSAC 97 pp172-175 (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 = "455--472",
- 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 = "99--102",
+ 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 right-hand 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 = "322--325",
- 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 = "303-315",
+ 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 = "978-1-4471-6464-7",
+ 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 q-analogues 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 q-hypergeometric) 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 = "1-10",
+ 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 pp172-175 (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 = "99--102",
- paper = "Abra85.pdf",
+@article{Schn05,
+ author = "Schneider, Carsten",
+ title = "A new Sigma approach to multi-summation",
+ year = "2005",
+ journal = "Advances in Applied Mathematics",
+ volume = "34",
+ number = "4",
+ pages = "740--767",
+ 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 right-hand 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 multi-summation problems efficiently.
+ For instacne, the extended Sigma package has been applied successively
+ to provide a computer-assisted 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 = "324--330",
- 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 = "2073--2083",
+ 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 = "948--970",
+ 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 multi-sum 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 = "611--644",
+ 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. AAECC-2010",
+ year = "2010",
+ volume = "21",
+ pages = "1--32",
+ 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 right-hand 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 multi-summation",
- year = "2005",
- journal = "Advances in Applied Mathematics",
- volume = "34",
- number = "4",
- pages = "740--767",
- 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 = "322--325",
+ 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 multi-summation problems efficiently.
- For instacne, the extended Sigma package has been applied successively
- to provide a computer-assisted 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 = "948--970",
- 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 = "2073--2083",
- 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 = "455--472",
+ 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 = "611--644",
- 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 so-called
+ $R\prod\sum*$-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 first-order 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{Vazquez-Trejo, Javier}
\begin{chunk}{axiom.bib}
-@article{Schn09,
- author = "Schneider, Carsten",
- title = "Structural theorems for symbolic summation",
- journal = "Proc. AAECC-2010",
- year = "2010",
- volume = "21",
- pages = "1--32",
- 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 = "Vazquez-Trejo, Javier",
+ title = "Symbolic Summation in Difference Fields",
+ year = "2014",
+ school = "Carnegie-Mellon 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 = {2006-05-26}
+}
- 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 = {2003-03-28}
}
\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 multi-sum 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 = {2012-05-01}
}
\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 = "1-10",
- 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 = {2013-10-04},
+ 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 = {1993-08-26}
+ 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 Yang-Baxter Equation and
+ Quantum Groups: An Algebraic Approach},
+ year = "1997",
+ location = {Dordrecht ; Boston},
+ edition = {Auflage: 1997},
+ isbn = {9780792347217},
+ shorttitle = {Introduction to the Quantum Yang-Baxter 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 Yang-Baxter 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 Yang-Baxter equation and objects quantum groups.},
+ pagetotal = {300},
+ publisher = {Springer},
+ date = {1997-10-31}
+}
+
+\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 91--96. (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/axiom-website/patches.html 20141017.01.tpd.patch
+20141017 tpd books/bookvolbib add a section on Differential Forms
20141010 kxp src/axiom-website/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/axiom-website/patches.html b/src/axiom-website/patches.html
index 7a66eeb..f354e37 100644
--- a/src/axiom-website/patches.html
+++ b/src/axiom-website/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