Appendix B
Bibliography
-
[AG98]
-
Beatrice
Amrhein
and
Oliver
Gloor.
The
fractal
walk.
In
Bruno
Buchberger
and
Franz
Winkler,
editors,
Gröbner
Bases
and
Applications,
volume
251
of
London
Mathematical
Society
Lecture
Note
Series,
pages
305–322.
Cambridge
University
Press,
Feb
1998.
-
[AGK96a]
-
Beatrice
Amrhein,
Oliver
Gloor,
and
Wolfgang
Kuechlin.
How
fast
does
the
walk
run?
In
5th
Rhine
Workshop
on
Computer
Algebra,
volume
PR
801/96,
pages
8.1–8.9.
Institut
Franco–Allemand
de
Recherches
de
Saint–Louis,
Jan
1996.
-
[AGK96b]
-
Beatrice
Amrhein,
Oliver
Gloor,
and
Wolfgang
Kuechlin.
Walking
faster.
In
J. Calmet
and
C. Limongelli,
editors,
DISCO
1996:
Design
and
Implementation
of
Symbolic
Computation
Systems,
volume
1128
of
Lecture
Notes
in
Computer
Science,
pages
150–161.
Springer,
1996.
-
[All78]
-
J. R.
Allen.
The
Anatomy
of
LISP.
McGraw-Hill,
New
York,
1978.
-
[ALSU06]
-
Alfred V.
Aho,
Monica S.
Lam,
Ravi
Sethi,
and
Jeffrey D.
Ullman.
Compilers:
Principles,
Techniques,
and
Tools
(2nd
Edition).
Addison
Wesley,
August
2006.
-
[AM90]
-
V. S.
Adamchik
and
O. I.
Marichev.
The
algorithm
for
calculating
integrals
of
hypergeometric
type
functions
and
its
realization
in
REDUCE
system.
In
S. Watanabe
and
Morio
Nagata,
editors,
Proceedings
of
the
1990
International
Symposium
on
Symbolic
and
Algebraic
Computation,
pages
212–224.
ACM,
Addison-Wesley,
1990.
-
[Ape92]
-
Joachim
Apel.
A
relationship
between
Gröbner
bases
of
ideals
and
vector
modules
of
G-algebras,
volume
131.2
of
Contemporary
Mathematics,
pages
195–204.
AMS,
1992.
-
[AS72]
-
Milton
Abramowitz
and
Irene A.
Stegun,
editors.
Handbook
of
Mathematical
Functions.
Dover
Publications,
1972.
-
[ASW89]
-
Werner
Antweiler,
Andreas
Strotmann,
and
Volker
Winkelmann.
A
TEX-REDUCE-interface.
SIGSAM
Bulletin,
23(2):26–33,
February
1989.
-
[ASY74]
-
E A
Arais,
V P
Shapeev,
and
N N
Yanenko.
Computer
realization
of
cartan’s
exterior
calculus,.
Soviet
Math
Dokl,
15:203–205,
1974.
-
[Bar67a]
-
D. Barton.
A
scheme
for
manipulative
algebra
on
a
computer.
Computer
Journal,
9(4):340–344,
Feb
1967.
-
[Bar67b]
-
D. Barton.
A
scheme
for
manipulative
algebra
on
a
computer.
Astronomical
Journal,
72:1281–1287,
1967.
-
[BB89]
-
A. V.
Bocharov
and
M. L.
Bronstein.
Efficiently
Implementing
Two
Methods
of
the
Geometrical
Theory
of
Differential
Equations:
An
Experience
in
Algorithm
and
Software
Design,
pages
143–166.
Number 16
in
Acta
Applicandae
Mathematicae.
Kluwer,
1989.
-
[BC82]
-
Gregory
Butler
and
John J.
Cannon.
Computing
in
permutation
and
matrix
groups.
I.
Normal
closure,
commutator
subgroups,
series.
Math.
Comp.,
39:663–670,
1982.
-
[BC94]
-
A. Burnel
and
H. Caprasse.
Computing
the
BRST
operator
used
in
quantization
of
gauge
theories.
International
Journal
of
Modern
Physics
C,
5(6):1035–1047,
December
1994.
-
[BCD\(^{+}\)99]
-
A. V.
Bocharov,
V. N.
Chetverikov,
S. V.
Duzhin,
N. G.
Khor’kova,
A. V.
Samokhin,
Yu. N.
Torkhov,
and
A. M.
Verbovetsky:.
Symmetries
and
Conservation
Laws
for
Differential
Equations
of
Mathematical
Physics,
volume
182
of
Translations
of
Math.
Monographs.
AMS,
1999.
-
[BCG\(^{+}\)91]
-
Robert L
Byrant,
S S
Chern,
Robert B
Gardner,
Hubert L
Goldschmidt,
and
P A
Griffiths.
Computer
realization
of
Cartan’s
exterior
calculus,
volume 18
of
Mathematical
Sciences
Research
Institute
Publications.
Springer-Verlag
New
York,
1991.
-
[BCRT93]
-
Anna Maria
Bigatti,
Pasqualina
Conti,
Lorenzo
Robbiano,
and
Carlo
Traverso.
A
“Divide
and
conquer”
algorithm
for
Hilbert-Poincaré
series,
multiplicity
and
dimension
of
monomial
ideals.
In
G. Cohen,
T. Mora,
and
O. Moreno,
editors,
Applied
Algebra,
Algebraic
Algorithms
and
Error-Correcting
Codes.
AAECC
1993,
volume
673
of
Lecture
Notes
in
Computer
Science,
pages
76–88,
Berlin,
Heidelberg,
1993.
Springer-Verlag.
-
[BF72]
-
D. Barton
and
J. P.
Fitch.
The
application
of
symbolic
algebra
system
to
physics.
Reports
on
Progress
in
Physics,
35(1):235–314,
Jan
1972.
-
[BGDW95]
-
P. A.
Broadbery,
T. Gómez-Díaz,
and
S. M.
Watt.
On
the
Implementation
of
Dynamic
Evaluation.
In
A. Levelt,
editor,
Proceedings
of
the
1995
International
Symposium
on
Symbolic
and
Algebraic
Computation,
ISSAC
’95,
pages
77–84,
New
York,
NY,
USA,
1995.
ACM
Press.
-
[BGK86]
-
W. Boege,
R. Gebauer,
and
H. Kredel.
Some
examples
for
solving
systems
of
algebraic
equations
by
calculating
Groebner
bases.
Journal
of
Symbolic
Computation,
2(1):83–98,
March
1986.
-
[BGM96]
-
George A.
Baker
and
Peter
Graves-Morris.
Padé
Approximants,
volume 13
of
Encyclopedia
of
Mathematics
and
its
Applications.
Cambridge
University
Press,
2nd
edition,
1996.
-
[BH10]
-
D. E.
Baldwin
and
W. Hereman.
A
symbolic
algorithm
for
computing
recursion
operators
of
nonlinear
partial
differential
equations.
International
Journal
of
Computer
Mathematics,
87(5):1094–1119,
2010.
-
[BHPS86]
-
R. J.
Bradford,
A. C.
Hearn,
J. A.
Padget,
and
E. Schrüfer.
Enlarging
the
reduce
domain
of
computation.
In
SYMSAC
’86:
Proceedings
of
the
fifth
ACM
symposium
on
Symbolic
and
algebraic
computation,
pages
100–106,
New
York,
NY,
USA,
1986.
ACM.
-
[BL85]
-
Gregory
Butler
and
Clement
W. H.
Lam.
A
General
Backtrack
Algorithm
for
the
Isomorphism
Problem
of
Combinatorial
Objects.
Journal
of
Symbolic
Computation,
1(4):363–381,
1985.
-
[BO78]
-
Carl M.
Bender
and
Steven A.
Orszag.
Advanced
Mathematical
Methods
for
Scientists
and
Engineers.
McGraw-Hill,
1978.
-
[Bou72]
-
S. R.
Bourne.
Literal
expressions
for
the
co-ordinates
of
the
moon.
I.
The
First
Degree
Terms.
Celestial
Mechanics,
6(2):167–186,
Sep
1972.
-
[Bra93]
-
Russell
Bradford.
Algebraic
simplification
of
multiple-valued
functions.
In
John
Fitch,
editor,
DISCO
1992:
Design
and
Implementation
of
Symbolic
Computation
Systems,
volume
721
of
Lecture
Notes
in
Computer
Science,
pages
13–21,
Berlin,
Heidelberg,
1993.
Springer
Berlin
Heidelberg.
-
[Bro96]
-
E. W.
Brown.
An
Introductory
Treatise
on
the
Lunar
Theory.
Cambridge
University
Press,
1896.
-
[Bro97]
-
Manuel
Bronstein.
Symbolic
Integration
I:
Transcendental
Functions.
Springer-Verlag,
Heidelberg,
1997.
-
[BS81]
-
Ilja N.
Bronstein
and
K. A.
Semendjajew.
Taschenbuch
der
Mathematik.
Verlag
Harri
Deutsch,
Thun
und
Frankfurt
(Main),
1981.
-
[BS92]
-
Dave
Bayer
and
Mark
Stillman.
Computation
of
Hilbert
functions.
Journal
of
Symbolic
Computation,
14(1):31–50,
1992.
-
[Buc85]
-
Bruno
Buchberger.
Gröbner
Bases:
An
Algorithmic
Method
in
Polynomial
Ideal
Theory,
pages
184–232.
Reidel,
Dordrecht,
01
1985.
-
[Buc88]
-
Bruno
Buchberger.
Applications
of
Gröbner
bases
in
non-linear
computational
geometry.
In
R. Janssen,
editor,
Mathematical
Aspects
of
Scientific
Software,
volume
296
of
Lecture
Notes
in
Computer
Science,
pages
52–88.
Springer,
New
York,
1988.
-
[But82]
-
Gregory
Butler.
Computing
in
permutation
and
matrix
groups
ii:
Backtrack
algorithm.
Mathematics
of
Computation,
39(160):671–680,
1982.
-
[BW92]
-
A. Brand
and
T. Wolf.
The
computer
algebra
package
CRACK
for
investigating
PDEs.
In
Proc.
of
ERCIM,
Partial
Differential
Equations
and
Group
Theory,
Bonn,
page 24,
1992.
-
[BW95]
-
A. Brand
and
T. Wolf.
Investigating
DEs
with
CRACK
and
related
programs.
SIGSAM
Bulletin,
Special
Issue:1–8,
June
1995.
-
[BWK93]
-
Thomas
Becker,
Volker
Weispfenning,
and
Heinz
Kredel.
Gröbner
bases,
a
computational
approach
to
commutative
algebra.
Springer
-
Verlag
(Graduate
Texts
in
Mathematics
141),
1993.
-
[Cap97]
-
H. Caprasse.
Brst
charge
and
poisson
algebras.
Discrete
Mathematics
and
Theoretical
Computer
Science.
DMTCS
[electronic
only],
1(2):115–127,
1997.
-
[Car88]
-
B.C.
Carlson.
A
table
of
elliptic
integrals
of
the
third
kind.
51
(183),
pp.
267–280,
s1–s5.
Mathematics
of
Computation,
51
(183):267–280,
1988.
-
[Car99]
-
B.C.
Carlson.
Toward
symbolic
integration
of
elliptic
integrals.
J.
Symbolic
Computation,
28(6):739–753,
1999.
-
[CAT]
-
Cathode
(computer
algebra
tools
for
handling
ordinary
differential
equations).
OBSOLETE:
http://www-lmc.imag.fr/CATHODE/,
http://www-lmc.imag.fr/CATHODE2/.
-
[CF00]
-
B.C.
Carlson
and
J. FitzSimmons.
Reduction
theorems
for
elliptic
integrands
with
the
square
root
of
two
quadratic
factors.
J.
Computational
Applied
Mathematics,
118(1-2):71–85,
2000.
-
[CGG\(^{+}\)91]
-
Bruce W.
Char,
Keith O.
Geddes,
Gaston H.
Gonnet,
Benton
Leong,
Michael B.
Monagan,
and
Stephen M.
Watt.
Maple
V
Language
Reference
Manual.
Springer,
1991.
-
[Chi63]
-
J. S. R.
Chisholm.
Relativistic
scalar
products
of
\(\gamma \)
matrices.
Il
Nuovo
Cimento
(1955-1965),
30(1):426–428,
Oct
1963.
-
[CHW91]
-
B. Champagne,
W. Hereman,
and
P. Winternitz.
The
computer
calculation
of
Lie
point
symmetries
of
large
systems
of
differential
equations.
Computer
Physics
Communications,
66(2–3):319–340,
1991.
-
[CJ92]
-
Robert M.
Corless
and
David J.
Jeffrey.
Well
…
It
Isn’t
Quite
That
Simple.
SIGSAM
Bull.,
26(3):2–6,
Aug
1992.
-
[CKM97]
-
S. Collart,
M. Kalkbrenner,
and
D. Mall.
Converting
Bases
with
the
Gröbner
Walk.
Journal
of
Symbolic
Computation,
24(3-4):465–469,
Sep
1997.
-
[CLO92]
-
D. Cox,
J. Little,
and
D. O’Shea.
Ideals,
Varieties
and
Algorithms:
An
Introduction
of
Computational
Algebraic
Geometry
and
Commutative
Algebra.
Springer-Verlag,
1992.
-
[CLV20]
-
M. Casati,
P. Lorenzoni,
and
R. Vitolo.
Three
computational
approaches
to
weakly
nonlocal
poisson
brackets.
Studies
in
Applied
Mathematics,
2020.
-
[CN81]
-
B.C.
Carlson
and
E.M.
Notis.
Algorithm
577:
Algorithm
for
incomplete
elliptic
intergrals
[s21].
ACM
Transactions
Mathematical
Software,
7(3):398–403,
1981.
-
[Col75]
-
George E.
Collins.
Quantifier
elimination
for
real
closed
fields
by
cylindrical
algebraic
decompostion.
In
H. Brakhage,
editor,
Automata
Theory
and
Formal
Languages
2nd
GI
Conference
Kaiserslautern,
May
20–23,
1975,
volume 33
of
Lecture
Notes
in
Computer
Science,
pages
134–183,
Berlin,
Heidelberg,
1975.
Springer
Berlin
Heidelberg.
-
[Cvi76]
-
Predrag
Cvitanović.
Group
theory
for
Feynman
diagrams
in
non-Abelian
gauge
theories.
Phys.
Rev.
D,
14:1536,
Sep
1976.
-
[Dan63]
-
George Bernard
Dantzig.
Linear
programming
and
extensions.
Technical
report,
RAND
Corporation,
1963.
-
[Dav81]
-
James Harold
Davenport.
On
the
Integration
of
Algebraic
Functions,
volume
102
of
Lecture
Notes
in
Computer
Science.
Springer-Verlag,
1981.
-
[Del86]
-
C. Delaunay.
Théorie
du
Mouvement
de
la
Lune.
Extraits
des
Mém.
Acad.
Sci.
Mallet-Bachelier,
Paris,
186.
-
[DF94]
-
James
Davenport
and
Christèle
Faure.
The
“unknown”
in
computer
algebra.
Programming
and
Computer
Software,
20:1–5,
01
1994.
-
[DGV96]
-
D. Duval
and
Laureano
González-Vega.
Dynamic
evaluation
and
real
closure.
Mathematics
and
Computers
in
Simulation,
42:551–560,
11
1996.
-
[DLS90]
-
Ladislav
Drska,
Richard
Liska,
and
Milan
Sinor.
Two
practical
packages
for
computational
physics-GCPM,
RLFI.
Computer
Physics
Communications,
61(1-2):225–230,
November
1990.
-
[DN83]
-
B. A.
Dubrovin
and
S. P.
Novikov.
Hamiltonian
formalism
of
one-dimensional
systems
of
hydrodynamic
type
and
the
bogolyubov-whitham
averaging
method.
Dokl.
Akad.
Nauk
SSSR,
27(3):665–669,
1983.
Translated
by
J. R. Schulenberger.
-
[DN84]
-
B. A.
Dubrovin
and
S. P.
Novikov.
Poisson
brackets
of
hydrodynamic
type.
Dokl.
Akad.
Nauk
SSSR,
30(3):651–654,
Jan
1984.
-
[DP85]
-
James
Davenport
and
Julian
Padget.
HEUGCD:
how
elementary
upperbounds
generate
cheaper
data.
In
Proc.
EUROCAL
1985,
Lecture
Notes
in
Computer
Science,
volume
204,
pages
18–28.
Springer-Verlag,
1985.
-
[DR94a]
-
Dominique
Duval
and
Jean-Claude
Reynaud.
Sketches
and
computation
I:
basic
definitions
and
static
evaluation.
Mathematical
Structures
in
Computer
Science,
4(2):185–238,
1994.
-
[DR94b]
-
Dominique
Duval
and
Jean-Claude
Reynaud.
Sketches
and
computation
II:
dynamic
evaluation
and
applications.
Mathematical
Structures
in
Computer
Science,
4(2):239–271,
1994.
-
[DS96]
-
Andreas
Dolzmann
and
Thomas
Sturm.
Redlog
user
manual.
Technical
Report
MIP-9616,
FMI,
Universität
Passau,
D-94030
Passau,
Germany,
October
1996.
Edition
1.0
for
Version
1.0.
-
[DS97a]
-
Andreas
Dolzmann
and
Thomas
Sturm.
REDLOG:
Computer
algebra
meets
computer
logic.
ACM
SIGSAM
Bulletin,
31(2):2–9,
June
1997.
-
[DS97b]
-
Andreas
Dolzmann
and
Thomas
Sturm.
Simplification
of
quantifier-free
formulae
over
ordered
fields.
Journal
of
Symbolic
Computation,
24(2):209–231,
August
1997.
-
[DS99]
-
Andreas
Dolzmann
and
Thomas
Sturm.
Redlog
User
Manual.
FMI,
Universität
Passau,
D-94030
Passau,
Germany,
April
1999.
Edition
2.0
for
Version
2.0.
-
[DST93]
-
J. H.
Davenport,
Y. Siret,
and
E. Tournier.
Computer
Algebra,
Systems
and
Algorithms
for
Algebraic
Computation.
Academic
Press,
2nd
edition,
1993.
-
[Dub96]
-
Boris
Dubrovin.
Geometry
of
2D
topological
field
theories.
In
Mauro
Francaviglia
and
Silvio
Greco,
editors,
Integrable
Systems
and
Quantum
Groups:
Lectures
given
at
the
1st
Session
of
the
Centro
Internazionale
Matematico
Estivo
(C.I.M.E.)
held
in
Montecatini
Terme,
Italy,
June
14–22,
1993,
volume
1620
of
Lecture
Notes
in
Mathematics,
pages
120–348.
Springer,
Berlin
Heidelberg,
1996.
-
[Eas87]
-
James W.
Eastwood.
Orthovec:
A
REDUCE
program
for
3-D
vector
analysis
in
orthogonal
curvilinear
coordinates.
Computer
Physics
Communications,
47(1):139–147,
October
1987.
-
[Eas91]
-
James W.
Eastwood.
ORTHOVEC:
version
2
of
the
REDUCE
program
for
3-D
vector
analysis
in
orthogonal
curvilinear
coordinates.
Computer
Physics
Communications,
64(1):121–122,
April
1991.
-
[Edm57]
-
A. R.
Edmonds.
Angular
Momentum
in
Quantum
Mechanics.
Princeton
University
Press,
1957.
-
[Eis95]
-
David
Eisenbud.
Commutative
Algebra
with
a
View
Toward
Algebraic
Geometry,
volume
150
of
Graduate
Texts
in
Math.
Springer-Verlag,
Berlin
and
New
York,
1995.
-
[Eul48]
-
L. Euler.
Introductio
in
analysin
infinitorum,
Vol.
1,
chapter 18.
Lausanne:
Marcum-Michaelem
Bousquet,
1748.
-
[Fat74]
-
Richard J.
Fateman.
On
the
multiplication
of
poisson
series.
Celestial
Mechanics,
10(2):243–247,
Oct
1974.
-
[Fat87]
-
Richard J.
Fateman.
TEX
output
from
macsyma-like
systems.
ACM
SIGSAM
Bulletin,
21(4):1–5,
1987.
-
[FGLM93]
-
J. C.
Faugère,
P. Gianni,
D. Lazard,
and
T .Mora.
Efficient
Computation
of
Zero-dimensional
Gröbner
Bases
by
Change
of
Ordering.
Journal
of
Symbolic
Computation,
16(4):329–344,
Oct
1993.
-
[FGMN97]
-
E. V.
Ferapontov,
C .A. P.
Galvao,
O. Mokhov,
and
Y. Nutku.
Bi-hamiltonian
structure
in
2-d
field
theory.
Communications
in
Mathematical
Physics,
186(3):649–669,
Jan
1997.
-
[FH74]
-
John A.
Fox
and
Anthony C.
Hearn.
Analytic
computation
of
some
integrals
in
fourth
order
quantum
electrodynamics.
Journal
of
Computational
Physics,
14(3):301–317,
March
1974.
-
[Fil92]
-
Sandra
Fillebrown.
Faster
computation
of
bernoulli
numbers.
Journal
of
Algorithms,
13(3):431–445,
September
1992.
-
[Fit75]
-
J. P.
Fitch.
Syllabus
for
algebraic
manipulation
lectures
in
cambridge.
SIGSAM
Bulletin,
32:15,
1975.
-
[Fit83]
-
J. P.
Fitch.
CAMAL
User’s
Manual.
Technical
report,
University
of
Cambridge
Computer
Laboratory,
1983.
2nd
edition.
-
[FJ03]
-
J. C.
Faugère
and
A. Joux.
Algebraic
Cryptanalysis
of
Hidden
Field
Equation
(HFE)
Cryptosystems
Using
Gröbner
Bases.
In
Advances
in
Cryptology
-
CRYPTO
2003,
volume
2729
of
Lecture
Notes
in
Computer
Science,
pages
44–60.
Springer,
Berlin,
Heidelberg,
2003.
-
[FK89]
-
W. I.
Fushchich
and
V. V.
Kornyak.
Computer
algebra
application
for
determining
lie
and
lie–bäcklund
symmetries
of
differential
equations.
Journal
of
Symbolic
Computation,
7:611–619,
1989.
-
[FPV14]
-
E. V.
Ferapontov,
M .V.
Pavlov,
and
R.F.
Vitolo.
Projective-geometric
aspects
of
homogeneous
third-order
hamiltonian
operators.
Journal
of
Geometry
and
Physics,
85:16–28,
2014.
-
[FPV16]
-
E. V.
Ferapontov,
M. V.
Pavlov,
and
R. F.
Vitolo.
Towards
the
classification
of
homogeneous
third-order
hamiltonian
operators.
International
Mathematics
Research
Notices,
2016(22):6829–6855,
Jan
2016.
-
[Gas95]
-
G. Gasper.
Lecture
notes
for
an
introductory
minicourse
on
q-series.
Mini-course
lecture
notes,
1995.
-
[GB98]
-
V. P.
Gerdt
and
Yu. A.
Blinkov.
Involutive
bases
of
polynomial
ideals.
Math.
Comp.
Simul.,
45(5-6):519–541,
March
1998.
-
[GCL92]
-
K.O.
Geddes,
S.R.
Czapor,
and
G. Labahn.
Algorithms
for
Computer
Algebra.
Kluwer
Academic
Publishers,
1992.
-
[GD96]
-
Teresa
Gómez-Díaz.
Examples
of
using
dynamic
constructible
closure.
Mathematics
and
Computers
in
Simulation,
42(4–6):375–383,
1996.
-
[gde]
-
Geometry
of
differential
equations
web
site.
-
[Ger05]
-
Vladimir P.
Gerdt.
Involutive
algorithms
for
computing
groebner
bases.
In
Svetlana
Cojocaru,
Gerhard
Pfister,
and
Victor
Ufnarovski,
editors,
Computational
Commutative
and
Non-Commutative
Algebraic
Geometry,
volume
196
of
NATO
Science
Series,
III:
Computer
and
Systems
Sciences,
pages
199–225,
2005.
Proceedings
of
the
NATO
Advanced
Research
Workshop
"Computational
commutative
and
non-commutative
algebraic
geometry"
(Chishinau,
June
6-11,
2004).
-
[Get02]
-
Ezra
Getzler.
A
darboux
theorem
for
hamiltonian
operators
in
the
formal
calculus
of
variations.
Duke
Mathematical
Journal,
111,
Mar
2002.
-
[GH79]
-
Martin L.
Griss
and
Anthony C.
Hearn.
Portable
LISP
compiler.
Software
-
Practice
and
Experience,
11(6):541–605,
June
1979.
-
[GK82]
-
Daniel H.
Greene
and
Donald E.
Knuth.
Mathematics
for
the
Analysis
of
Algorithms,
volume 1
of
Progress
in
Computer
Science
and
Applied
Logic
(PCS).
Birkhäuser
Boston,
2nd
edition,
1982.
-
[GM88]
-
Rüdiger
Gebauer
and
H. Michael
Möller.
On
an
installation
of
Buchberger’s
algorithm.
Journal
of
Symbolic
Computation,
6(2
and
3):275–286,
1988.
-
[GMM\(^{+}\)81]
-
V G
Ganzha,
S V
Meleshko,
F A
Murzin,
V P
Shapeev,
and
N N
Yanenko.
Computer
realization
of
an
algorithm
for
investigating
the
compatibility
of
systems
of
partial
differential
equations,.
Soviet
Math
Dokl,
24:638–640,
1981.
-
[GMN\(^{+}\)91]
-
Alessandro
Giovini,
Teo
Mora,
Gianfranco
Niesi,
Lorenzo
Robbiano,
and
Carlo
Traverso.
“;one
sugar
cube,
please”
or
selection
strategies
in
the
buchberger
algorithm.
In
Watt
[Wat91],
pages
49–54.
-
[Gos78]
-
R. W.
Gosper,
Jr.
Decision
procedure
for
indefinite
hypergeometric
summation.
Proc.
Natl.
Acad.
Sci.
USA,
75(1):40–42,
January
1978.
-
[GR90]
-
G. Gasper
and
M. Rahman.
Basic
Hypergeometric
Series.
Number 35
in
Encyclopedia
of
Mathematics
and
its
Applications.
Cambridge
University
Press,
London
and
New
York,
1990.
-
[Grä93]
-
Hans-Gert
Gräbe.
Two
remarks
on
independent
sets.
Journal
of
Algebraic
Combinatorics,
2:137–145,
1993.
-
[Grä94a]
-
Hans-Gert
Gräbe.
On
factorized
gröbner
bases.
In
J. Fleischer,
J. Grabmeier,
F. W.
Hehl,
and
W. Küchlin,
editors,
Computer
Algebra
in
Science
and
Engineering.
World
Scientific,
1994.
Proceedings
of
the
Conference,
Bielefeld,
Germany,
28–31
August
1994.
-
[Grä94b]
-
Hans-Gert
Gräbe.
The
tangent
cone
algorithm
and
homogenization.
Journal
of
Pure
and
Applied
Algebra,
97(3):303–312,
Dec
1994.
-
[Grä95a]
-
Hans-Gert
Gräbe.
Algorithms
in
local
algebra.
Journal
of
Symbolic
Computation,
19(6):545–577,
Jun
1995.
-
[Grä95b]
-
Hans-Gert
Gräbe.
Triangular
systems
and
factorized
gröbner
bases.
In
Gérard
Cohen,
Marc
Giusti,
and
Teo
Mora,
editors,
Applied
Algebra,
Algebraic
Algorithms,
and
Error-Correcting
Codes,
Lecture
Notes
in
Computer
Science,
pages
248–261,
Berlin,
Heidelberg,
1995.
Springer.
-
[Grä97]
-
Hans-Gert
Gräbe.
Minimal
primary
decomposition
and
factorized
gröbner
bases.
Applicable
Algebra
in
Engineering,
Communication
and
Computing,
8(4):265–278,
1997.
-
[Gru96]
-
Dominik
Gruntz.
On
Computing
Limits
in
a
Symbolic
Manipulation
System.
PhD
thesis,
ETH
Zürich,
1996.
-
[GTZ88]
-
Patrizia
Gianni,
Barry
Trager,
and
Gail
Zacharias.
Gröbner
bases
and
primary
decomposition
of
polynomial
ideals.
Journal
of
Symbolic
Computation,
6(2-3):149–167,
Oct
1988.
-
[GZ08a]
-
V. P.
Gerdt
and
M. V.
Zinin.
Involutive
method
for
computing
Gröbner
bases
over
\(\mathbb {F}_{2}\).
Programming
and
Computer
Software,
34(4):191–203,
Jul
2008.
-
[GZ08b]
-
Vladimir P.
Gerdt
and
Mikhail V.
Zinin.
A
pommaret
division
algorithm
for
computing
grobner
bases
in
boolean
rings.
In
Proceedings
of
the
Twenty-first
International
Symposium
on
Symbolic
and
Algebraic
Computation,
ISSAC
’08,
pages
95–102,
New
York,
NY,
USA,
2008.
ACM.
-
[GZB10]
-
V. P.
Gerdt,
M. V.
Zinin,
and
Yu. A.
Blinkov.
Programming
and
computing
software.
Program.
Comput.
Softw.,
36(2):117–123,
mar
2010.
-
[Han75]
-
E. R.
Hansen.
A
table
of
series
and
products.
Prentice-Hall,
Englewood
Cliffs,
NJ,
1975.
-
[Har12]
-
G. H.
Hardy.
Properties
of
logarithmico-exponential
functions.
Proceedings
of
the
London
Mathematical
Society,
s2-10(1):54–90,
1912.
-
[Har79]
-
Steven J.
Harrington.
A
new
symbolic
integration
system
in
REDUCE.
The
Computer
Journal,
22(2):127–131,
1979.
-
[Har89]
-
David
Harper.
Vector33:
a
REDUCE
program
for
vector
algebra
and
calculus
in
orthogonal
curvilinear
coordinates.
Computer
Physics
Communications,
54(2
and
3):295–305,
June
and
July
1989.
-
[HCGJ92]
-
D. E. G.
Hare,
R. M.
Corless,
G. H.
Gonnet,
and
D. J.
Jeffrey.
On
Lambert’s
W
Function.
Preprint,
University
of
Waterloo,
1992.
-
[HCJE93]
-
H. Hong,
G. E.
Collins,
J. R.
Johnson,
and
M. J.
Encarnacion.
QEPCAD
interactive
version
12,
Sep
1993.
Kindly
communicated
to
us
by
Hoon
Hong.
-
[Hea68]
-
Anthony C.
Hearn.
REDUCE:
A
user-oriented
interactive
system
for
algebraic
simplification.
In
M. Klerer
and
J. Reinfelds,
editors,
Interactive
Systems
for
Experimental
Applied
Mathematics,
pages
79–90,
New
York,
1968.
Academic
Press.
-
[Hea69]
-
Anthony C.
Hearn.
The
problem
of
substitution.
In
R.G.
Tobey,
editor,
Proc.
of
the
1968
Summer
Institute
on
Symbolic
Mathematical
Computation,
pages
3–19,
Cambridge,
Mass,
1969.
IBM
Boston
Prog.
Center.
-
[Hea71]
-
Anthony C.
Hearn.
REDUCE
2:
A
system
and
language
for
algebraic
manipulation.
In
S.R.
Petrick,
editor,
SYMSAC
’71:
Proceedings
of
the
second
ACM
symposium
on
Symbolic
and
algebraic
manipulation,
pages
128–133.
ACM,
New
York,
1971.
-
[Hea79]
-
Anthony C.
Hearn.
Non-modular
computation
of
polynomial
GCDs
using
trial
division.
In
Symbolic
and
Algebraic
Computation
(EUROSAM
’79,
An
International
Symposium
on
Symbolic
and
Algebraic
Manipulation,
Marseille,
France,
June
1979),
volume 72
of
Lecture
Notes
in
Computer
Science,
pages
227–239.
Springer
Berlin
/
Heidelberg,
1979.
-
[Hea95]
-
Anthony C.
Hearn.
REDUCE
User’s
Manual,
Version
3.6.
Report
CP
78,
RAND,
July
1995.
-
[Her95]
-
W. Hereman.
Symbolic
Software
for
Lie
Symmetry
Analysis,
volume 3,
chapter 13.
CRC
Press,
Boca
Raton,
Florida,
1995.
Systems
described
in
this
paper
are
among
others:
DELiA
(Alexei
Bocharov
et.al.)
Pascal
DIFFGROB2
(Liz
Mansfield)
Maple
DIMSYM
(James
Sherring
and
Geoff
Prince)
REDUCE
HSYM
(Vladimir
Gerdt)
Reduce
LIE
(V.
Eliseev,
R.N.
Fedorova
and
V.V.
Kornyak)
Reduce
LIE
(Alan
Head)
muMath
Lie
(Gerd
Baumann)
Mathematica
LIEDF/INFSYM
(Peter
Gragert
and
Paul
Kersten)
Reduce
Liesymm
(John
Carminati,
John
Devitt
and
Greg
Fee)
Maple
MathSym
(Scott
Herod)
Mathematica
NUSY
(Clara
Nucci)
Reduce
PDELIE
(Peter
Vafeades)
Macsyma
SPDE
(Fritz
Schwarz)
Reduce
and
Axiom
SYM_DE
(Stanly
Steinberg)
Macsyma
Symmgroup.c
(Dominique
Berube
and
Marc
de
Montigny)
Mathematica
STANDARD
FORM
(Gregory
Reid
and
Alan
Wittkopf)
Maple
SYMCAL
(Gregory
Reid)
Macsyma
and
Maple
SYMMGRP.MAX
(Benoit
Champagne,
Willy
Hereman
and
Pavel
Winternitz)
Macsyma
LIE
package
(Khai
Vu)
Maple
Toolbox
for
symmetries
(Mark
Hickman)
Maple
Lie
symmetries
(Jeffrey
Ondich
and
Nick
Coult)
Mathematica.
-
[Hil99]
-
Dietmar
Hillebrand.
Triangulierung
nulldimensionaler
Ideale
-
Implementierung
und
Vergleich
zweier
Algorithmen
-
in
German
.
Diplomarbeit
im
Studiengang
Mathematik
der
Universität
Dortmund.
Betreuer:
Prof.
Dr.
H.
M.
Möller.
Technical
report,
Universität
Dortmund,
1999.
-
[HT91]
-
David
Hartley
and
Robin W.
Tucker.
A
constructive
implementation
of
the
Cartan-Kähler
theory
of
exterior
differential
systems.
Journal
of
Symbolic
Computation,
12(6):655–667,
December
1991.
-
[HT93]
-
David
Hartley
and
Philip
Tuckey.
A
direct
characterisation
of
gröbner
bases
in
clifford
and
grassmann
algebras.
Preprint
MPI-Ph/93–96,
Max-Planck-Institut
für
Physik,
1993.
-
[IK96]
-
V. A.
Ilyin
and
A. P.
Kryukov.
ATENSOR
-
REDUCE
program
for
tensor
simplification.
Computer
Physics
Communications,
96(1):36–52,
July
1996.
-
[IKRT89]
-
V. A.
Ilyin,
A. P.
Kryukov,
A. Ya.
Rodioniov,
and
A. Yu.
Taranov.
Fast
algorithm
for
calculation
of
Dirac’s
gamma-matrices
traces.
SIGSAM
Bulletin,
23(4):15–24,
Oct
1989.
-
[IVV04]
-
Sergei
Igonin,
A. Verboretsky,
and
Raffaele
Vitolo.
Variational
multivectors
and
brackets
in
the
geometry
of
jet
spaces.
In
R.O. Popovych
A.G. Nikitin,
V.M. Boyko
and
I.A.
Yehorchenko,
editors,
V
Int.
Conf.
on
on
Symmetry
in
Nonlinear
Mathematical
Physics,
Kyiv
2003,
volume 50,
pages
1335–1342,
Oct
2004.
-
[Jan29]
-
M. Janet.
Leçons
sur
les
systèmes
d’équations
aux
dérivées.
Gauthier-Villars,
Paris,
1929.
-
[JDB65]
-
Sidney
D. Drell
James
D. Bjorken.
Relativistic
Quantum
Mechanics.
International
Series
In
Pure
and
Applied
Physics.
McGraw-Hill,
New
York,
1965.
-
[Jef70]
-
W. H.
Jefferys.
A
fortran-based
list
processor
for
poisson
series.
Celestial
Mechanics,
2(4):474–480,
Dec
1970.
-
[JR94]
-
D. J.
Jeffrey
and
A. D.
Rich.
The
evaluation
of
trigonometric
integrals
avoiding
spurious
discontinuities.
ACM
Trans.
Math.
Softw.,
20(1):124–135,
March
1994.
-
[JT80]
-
W.B.
Jones
and
W.J.
Thron.
Continued
Fractions:
Analytic
Theory
and
Applications,
volume 11
of
Encyclopedia
of
mathematics
and
its
applications.
Addison-Wesley
Publishing
Company,
1980.
-
[Kah68]
-
Joseph
Kahane.
Algorithm
for
reducing
contracted
products
of
\(\gamma \)
matrices.
Journal
of
Mathematical
Physics,
9(10):1732–1738,
1968.
-
[Kah87]
-
W. Kahan.
Branch
cuts
for
complex
elementary
functions
or
much
ado
about
nothing’s
sign
bit.
In
A. Iserles
and
M.J.D.
Powell,
editors,
The
State
of
the
Art
in
Numerical
Analysis
:
Proceedings
of
the
Joint
IMA/SIAM
Conference
on
the
State
of
the
Art
in
Numerical
Analysis
held
at
the
University
of
Birmingham,
14-18
April
1986,
Oxford,
1987.
Clarendon
Press.
-
[Kam59]
-
E. Kamke.
Differentialgleichungen,
Lösungsmethoden
und
Lösungen,
Band
1,
Gewöhnliche
Differentialgleichungen,
volume 1.
Chelsea
Publishing
Company,
New
York,
1959.
-
[Kaz87]
-
C. Kazasov.
Laplace
transformations
in
REDUCE
3.
In
Proc.
EUROCAL
’87,
Lecture
Notes
in
Computer
Science,
volume
378,
pages
132–133.
Springer-Verlag,
1987.
-
[Ken82]
-
A. D.
Kennedy.
Diagrammatic
methods
for
spinors
in
feynman
diagrams.
Phys.
Rev.
D,
26:1936–1955,
Oct
1982.
-
[Khi64]
-
Aleksandr J.
Khinchin.
Continued
Fractions.
University
of
Chicago
Press,
1964.
-
[KKV04]
-
P. H. M.
Kersten,
I. S.
Krasil’shchik,
and
A. M.
Verbovetsky.
Hamiltonian
operators
and
\(\ell ^*\)-covering.
Journal
of
Geometry
and
Physics,
50(1–4):273–302,
2004.
-
[KKV06]
-
P. H. M.
Kersten,
I. S.
Krasil’shchik,
and
A. M.
Verbovetsky.
A
geometric
study
of
the
dispersionless
boussinesq
type
equation.
Acta
Applicandae
Mathematica,
90:143–178,
2006.
1–2.
-
[KKVV09]
-
P. Kersten,
I. S.
Krasil’shchik,
A. Verboretsky,
and
Vito.
Hamiltonian
Structures
for
General
PDEs,
pages
187–198.
Abel
Symposia.
Springer
Berlin
Heidelberg,
Berlin,
Heidelberg,
2009.
-
[Knu81]
-
Donald E.
Knuth.
The
Art
of
Computer
Programming,
Volume
2:
Seminumerical
Algorithms.
Addison-Wesley
Publishing
Company,
2nd
edition,
1981.
-
[Knu84]
-
Donald E.
Knuth.
The
TEXbook.
Computers
&
typesetting.
Addison-Wesley,
Reading,
1984.
-
[Koe92]
-
Wolfram
Koepf.
Power
series
in
computer
algebra.
Journal
of
Symbolic
Computation,
13(6):581–603,
June
1992.
-
[Koe93a]
-
Wolfram
Koepf.
Algorithmic
development
of
power
series.
In
J. Calmet
and
J. A.
Campbell,
editors,
Artificial
intelligence
and
symbolic
mathematical
computing,
volume
737
of
Lecture
Notes
in
Computer
Science,
pages
195–213,
Berlin,
Heidelberg,
1993.
Springer-Verlag.
International
Conference
AISMC-1,
Karlsruhe,
Germany,
August
1992,
Proceedings.
-
[Koe93b]
-
Wolfram
Koepf.
Examples
for
the
algorithmic
calculation
of
formal
puiseux,
laurent
and
power
series.
ACM
SIGSAM
Bulletin,
27(1):20–32,
Jan
1993.
-
[Koe94a]
-
Wolfram
Koepf.
Algorithmic
work
with
orthogonal
polynomials
and
special
functions.
Preprint
sc
94-5,
Konrad-Zuse-Zentrum
Berlin
(ZIB),
1994.
-
[Koe94b]
-
Wolfram
Koepf.
Algorithms
for
the
indefinite
and
definite
summation.
Preprint
SC
94-33,
Konrad-Zuse-Zentrum
für
Informationstechnik
Berlin,
December
1994.
-
[Koe95a]
-
Wolfram
Koepf.
Algorithms
for
\(m\)-fold
hypergeometric
summation.
Journal
of
Symbolic
Computation,
20(4):399–417,
Oct
1995.
-
[Koe95b]
-
Wolfram
Koepf.
REDUCE
package
for
the
indefinite
and
definite
summation.
SIGSAM
Bulletin,
29(1):14–30,
January
1995.
-
[Koo93]
-
T. H.
Koornwinder.
On
Zeilberger’s
algorithm
and
its
\(q\)-analogue:
a
rigorous
description.
J.
of
Comput.
and
Appl.
Math.,
48(1-2):91–111,
October
1993.
-
[KR88]
-
A. P.
Kryukov
and
A. Ya.
Rodionov.
Program
“COLOR”
for
computing
the
group-theoretic
weight
of
Feynman
diagrams
in
Non-Abelian
gauge
theories.
Computer
Physics
Communications,
48(2):327–334,
February
1988.
-
[Kre87]
-
Heinz
Kredel.
Primary
ideal
decomposition.
In
James
Davenport,
editor,
EUROCAL
’87,
European
Conference
on
Computer
Algebra,
Leipzig,
GDR,
June
2-5,
1987,
Proceedings,
1987.
-
[Kre88]
-
Heinz
Kredel.
Admissible
term
orderings
used
in
computer
algebra
systems.
SIGSAM
Bulletin,
22(1):28–31,
January
1988.
-
[KS94]
-
R. Koekoek
and
R. F.
Swarttouw.
The
askey-scheme
of
hypergeometric
orthogonal
polynomials
and
its
\(q\)-analogue.
Report
94-05,
Faculty
of
Technical
Mathematics
and
Informatics,
Technische
Universiteit
Delft,
Delft,
1994.
-
[Kub]
-
M. Kubitza.
Private
communication.
-
[Kup94]
-
B. A.
Kuperschmidt.
Geometric
Hamiltonian
forms
for
the
Kadomtsev-Petviashvili
and
Zabolotskaya-Khokhlov
equations,
pages
155–172.
World
Scientific,
1994.
-
[KV11]
-
Joseph
Krasil’shchik
and
Alexander
Verbovetsky.
Geometry
of
jet
spaces
and
integrable
systems.
Journal
of
Geometry
and
Physics,
61:1633–1674,
2011.
-
[KVV12]
-
Joseph
Krasil’shchik,
Alexander
Verbovetsky,
and
Raffaele
Vitolo.
A
unified
approach
to
computation
of
integrable
structures.
Acta
Applicandae
Mathematicae,
120(1):199–218,
Aug
2012.
-
[KVV18]
-
Joseph
Krasil’shchik,
Alexander
Verbovetsky,
and
Raffaele
Vitolo.
The
Symbolic
Computation
of
Integrability
Structures
for
Partial
Differential
Equations.
Texts
&
Monographs
in
Symbolic
Computation.
Springer
International
Publishing,
1
edition,
2018.
-
[KW88]
-
Heinz
Kredel
and
Volker
Weispfenning.
Computing
dimension
and
independent
sets
for
polynomial
ideals.
Journal
of
Symbolic
Computation,
6(2-3):231–247,
Oct
1988.
-
[Lam86]
-
Leslie
Lamport.
LATEX
-
A
Document
Preparation
System.
Addison-Wesley,
Reading,
1986.
-
[Law89]
-
Derek F.
Lawden.
Elliptic
Functions
and
Applications.
Springer-Verlag,
1989.
-
[Laz83]
-
D. Lazard.
Gröbner
bases,
gaussian
elimination
and
resolution
of
systems
of
algebraic
equations.
In
Proceedings
of
EUROCAL
’83,
volume
162
of
Lecture
Notes
in
Computer
Science,
pages
146–157.
Springer,
1983.
-
[LB68]
-
Landolt-Boernstein.
Zahlenwerte
und
Funktionen
aus
Naturwissenschaften
und
Technik.
Springer,
1968.
-
[LD90]
-
R. Liska
and
L. Drska.
FIDE:
A
REDUCE
package
for
automation
of
FInite
difference
method
for
solving
pDE.
In
S. Watanabe
and
Morio
Nagata,
editors,
Proceedings
of
the
1990
International
Symposium
on
Symbolic
and
Algebraic
Computation,
pages
169–176.
ACM,
Addison-Wesley,
1990.
-
[Leo80]
-
Jeffrey S.
Leon.
On
an
algorithm
for
finding
a
base
and
a
strong
generating
set
for
a
group
given
by
generating
permutations.
Mathematics
of
Computation,
35(151):941–974,
1980.
-
[Leo84]
-
Jeffrey S.
Leon.
Computing
Automorphism
Groups
of
Combinatorial
Objects.
In
Michael D.
Atkinson,
editor,
Computational
Group
Theory:
Proceedings
of
the
London
Mathematical
Society
Symposium
on
Computational
Group
Theory.
London
Mathematical
Society,
Academic
Press,
1984.
-
[Leo91]
-
Jeffrey S.
Leon.
Permutation
group
algorithms
based
on
partitions,
i:
Theory
and
algorithms.
Journal
of
S,
12:533–583,
1991.
-
[Lie67]
-
S. Lie.
Differentialgleichungen.
Chelsea
Publishing
Company,
New
York,
1967.
-
[Lie75]
-
S. Lie.
Sophus
lie’s
1880
transformation
group
paper.
Translated
by
M.
Ackerman,
comments
by
R.
Hermann,
Mathematical
Sciences
Press,
Brookline,
1975.
-
[Lin91]
-
Stephen A.
Linton.
Double
coset
enumeration.
Journal
of
Symbolic
Computation,
12(4):415–426,
1991.
-
[Lis91]
-
R. Liska.
Numerical
code
generation
for
finite
difference
schemes
solving.
In
R. Vichnevetsky
and
J.J.H.
Miller,
editors,
IMACS’91
13th
World
Congress
on
Computation
and
Applied
Mathematics,
July
22-26,
1991,
pages
92–93,
Dublin,
1991.
IMACS.
-
[LM94]
-
Eugenio Roanes
Lozano
and
Eugenio Roanes
Macias.
An
implementatio
of
“turtle
graphics”
in
maple
v.
In
Tony
Scott,
editor,
Maple
in
Mathematics
and
the
Sciences
-
Maple
Technical
Newsletter
Special
Issue,
pages
82–85.
Birkhäuser,
12
1994.
-
[Loo82]
-
R. Loos.
Computing
in
Algebraic
Extensions,
pages
173–187.
Springer
Vienna,
Vienna,
1982.
-
[Loo83]
-
Rüdiger
Loos.
Computing
rational
zeros
of
integral
polynomials
by
\(p\)-adic
expansion.
SIAM
J.
Computing,
12(2):286–293,
1983.
-
[Mac88]
-
M. A. H.
MacCallum.
An
ordinary
differential
equation
solver
for
REDUCE.
In
Proc.
of
ISSAC
’88,
volume
358,
pages
196–205.
Springer-Verlag,
1988.
-
[Mac89]
-
M. A. H.
MacCallum.
An
ordinary
differential
equation
solver
for
reduce.
In
P. Gianni,
editor,
Symbolic
and
Algebraic
Computation,
pages
196–205,
Berlin,
Heidelberg,
1989.
Springer
Berlin
Heidelberg.
-
[Mac99]
-
M. A. H.
MacCallum.
On
the
Classification
of
the
Real
Four-Dimensional
Lie
Algebras,
pages
299–317.
Springer
New
York,
New
York,
NY,
1999.
-
[MAE\(^{+}\)62]
-
John
McCarthy,
Paul W.
Abrahams,
Daniel J.
Edwards,
Timothy P.
Hart,
and
Michael I.
Levin.
LISP
1.5
Programmer’s
Manual.
M.I.T.
Press,
Aug
1962.
-
[Man94]
-
Y.-K.
Man.
Algorithmic
Solution
of
ODEs
and
Symbolic
Summation
Using
Computer
Algebra.
PhD
thesis,
School
of
Mathematical
Sciences,
Queen
Mary
and
Westfield
College,
University
of
London,
July
1994.
-
[Man96]
-
Elizabeth L.
Mansfield.
The
differential
algebra
package
diffgrob2.
Mapletech,
3(2):33–37,
1996.
-
[Mar93]
-
Jed
Marti.
RLISP
’88
An
Evolutionary
Approach
to
Program
Design
and
Reuse,
volume 42
of
World
Scientific
Series
in
Computer
Science.
World
Scientific,
Singapore,
1993.
-
[Mar09]
-
M. Marvan.
Sufficient
set
of
integrability
conditions
of
an
orthonomic
system.
Foundations
of
Computational
Mathematics,
9:652–674,
Jan
2009.
-
[McI85]
-
Kevin
McIsaac.
Pattern
matching
algebraic
identities.
SIGSAM
Bulletin,
19(2):4–13,
May
1985.
-
[McK78]
-
Brendan D.
McKay.
Computing
automorphisms
and
canonical
labellings
of
graphs.
In
D. A.
Holton
and
Jennifer
Seberry,
editors,
Combinatorial
Mathematics,
pages
223–232,
Berlin,
Heidelberg,
1978.
Springer
Berlin
Heidelberg.
-
[Mel95]
-
Herbert
Melenk.
Reduce
symbolic
mode
primer.
Technical
report,
Konrad-Zuse-Institut,
1995.
-
[MF93]
-
E Mansfield
and
E D
Fackerell.
Differential
gröbner
bases
and
involutivity
of
systems
of
non-linear
partial
differential
equations.
Preprint
92/108,
School
of
Mathematics,
Physics,
Computer
Science,
and
Electronics,
Macquarie
University,
Sydney,
Australia,
1993.
-
[MHGG80]
-
J. Marti,
A. C.
Hearn,
M. L.
Griss,
and
C. Griss.
Standard
Lisp
report.
SIGSAM
Bulletin,
14(1):23–41,
February
1980.
-
[Mis93]
-
Bhubaneswar
Mishra.
Algorithmic
Algebra.
Monographs
in
Computer
Science.
Springer,
New
York,
1993.
-
[MM86]
-
H. Michael
Möller
and
Ferdinando
Mora.
New
constructive
methods
in
classical
ideal
theory.
Journal
of
Algebra,
100(1):138–178,
1986.
-
[MM97]
-
Yiu-Kwong
Man
and
Malcolm
A. H.
MacCallum.
A
rational
approach
to
the
prelle-singer
algorithm.
Journal
of
Symbolic
Computation,
24(1):31–43,
Jul
1997.
-
[MMM91]
-
M. G.
Marinari,
H. M.
Möller,
and
T. Mora.
Gröbner
bases
of
ideals
given
by
dual
bases.
In
Watt
[Wat91].
-
[MMN88]
-
H. Melenk,
H. M.
Möller,
and
W. Neun.
On
Gröbner
bases
computation
on
a
supercomputer
using
REDUCE.
Preprint
SC
88-2,
Konrad-Zuse-Zentrum
für
Informationstechnik
Berlin,
January
1988.
-
[MN02]
-
H. Melenk
and
W. Neun.
Rr:
Parallel
symbolic
algorithm
support
for
psl
based
reduce.
Technical
report,
ZIB,
2002.
-
[Möl93]
-
H. Michael
Möller.
On
decomposing
systems
of
polynomial
equations
with
finitely
many
solutions.
Applicable
Algebra
in
Engineering,
Communication
and
Computing,
4(4):217–230,
1993.
-
[MPT89]
-
Teo
Mora,
Gerhard
Pfister,
and
Carlo
Traverso.
An
Introduction
to
the
Tangent
Cone
Algorithm.
Publications
de
l’Institut
de
recherche
mathématiques
de
Rennes,
(4):133–171,
1989.
-
[MR88]
-
Teo
Mora
and
Lorenzo
Robbiano.
The
gröbner
fan
of
an
ideal.
Journal
of
Symbolic
Computation,
6(2-3):183–208,
Oct
1988.
-
[MY73]
-
Joel
Moses
and
David
Y. Y.
Yun.
The
ez
gcd
algorithm.
In
ACM
Annual
Conference,
pages
159–166.
Association
for
Computing
Machinery,
08
1973.
-
[ND79]
-
A. C.
Norman
and
J. H.
Davenport.
Symbolic
integration
-
the
dust
settles?
In
Proc.
EUROSAM
1979,
volume 72
of
Lecture
Notes
in
Computer
Science,
pages
398–407.
Springer-Verlag,
1979.
-
[NM77]
-
A. C.
Norman
and
P. M. A.
Moore.
Implementing
the
new
Risch
integration
algorithm.
In
Proc.
of
the
Fourth
Colloquium
on
Advanced
Comp.
Methods
in
Theor.
Phys.,
St.
Maximin,
France,
March
1977.
-
[NM81]
-
Arthur C.
Norman
and
Mary Ann
Moore.
Implementing
a
polynomial
factorization
and
gcd
package.
In
SYMSAC
’81:
Proceedings
of
the
fourth
ACM
symposium
on
Symbolic
and
algebraic
computation,
pages
109–116.
Association
for
Computing
Machinery,
08
1981.
-
[NNS05]
-
F. Neyzi,
Y. Nutku,
and
M. B.
Sheftel.
Multi-hamiltonian
structure
of
plebanski’s
second
heavenly
equation.
Journal
of
Physics
A,
38(39):8473,
2005.
-
[Nuc92]
-
C. M.
Nucci.
Interactive
REDUCE
programs
for
calculating
classical,
non-classical,
and
approximate
symmetries
of
differential
equations,
pages
345–350.
Elsevier,
Amsterdam,
1992.
-
[Nuc96]
-
Maria Clara
Nucci.
Interactive
REDUCE
programs
for
calculating
Lie
point,
non-classical,
Lie-Bäcklund,
and
approximate
symmetries
of
differential
equations:
Manual
and
floppy
disk,
volume 3
of
CRC
Handbook
of
Lie
Group
Analysis
of
Differential
Equations,
pages
415–481.
CRC
Press,
Boca
Raton,
Jan
1996.
-
[NUS91]
-
Arnold F.
Nikiforov,
Vasilii B.
Uvarov,
and
Sergei K.
Suslov.
Classical
Orthogonal
Polynomials
of
a
Discrete
Variable.
Springer
Verlag,
Berlin–Heidelberg–New
York,
1991.
-
[NV]
-
A. C.
Norman
and
R. Vitolo.
Inside
reduce.
Part
of
the
official
REDUCE
documentation
included
in
the
source
code.
-
[OK94]
-
A. W.
Overhauser
and
Y. J.
Kim.
An
infinite
sum
from
a
diffusion
problem.
SIAM
Review,
36(1):107,
1994.
-
[Oli]
-
P. Oliveri.
ReLie,
Reduce
software
and
user
guide.
online.
-
[Olv86]
-
Peter J.
Olver.
Applications
of
Lie
Groups
to
Differential
Equations,
volume
107
of
Graduate
Texts
in
Mathematics.
Springer
Verlag,
New
York,
1986.
-
[Olv93]
-
Peter J.
Olver.
Applications
of
Lie
Groups
to
Differential
Equations,
volume
107
of
Graduate
Texts
in
Mathematics.
Springer-Verlag
New
York,
2nd
edition,
1993.
-
[PB90]
-
Julian
Padget
and
Alan
Barnes.
Univariate
power
series
expansions
in
REDUCE.
In
S. Watanabe
and
Morio
Nagata,
editors,
Proceedings
of
the
International
Symposium
on
Symbolic
and
Algebraic
Computation,
pages
82–87.
ACM,
Addison-Wesley,
1990.
-
[PBM89]
-
A. P.
Prudnikov,
Yu. A.
Brychkov,
and
O. I.
Marichev.
Integrals
and
Series,
Volume
3:
More
Special
Functions.
Gordon
and
Breach
Science
Publishers,
Oct
1989.
Transl.
from
the
Russian
by
G.
G.
Gould.
-
[Pos96]
-
G. F.
Post.
A
manual
for
the
package
TOOLS
2.1.
Memorandum
1331,
Dept.
Appl.
Math.,
University
of
Twente,
1996.
-
[PR95]
-
P. Paule
and
A. Riese.
A
mathematica
\(q\)-analogue
of
zeilberger’s
algorithm
based
on
an
algebraically
motivated
approach
to
\(q\)-hypergeometric
telescoping.
In
Fields
Proceedings
of
the
Workshop
’Special
Functions,
\(q\)-Series
and
Related
Topics,
pages
179–210,
Toronto,
Ontario,
12-23
June
1995.
Fields
Institute
for
Research
in
Mathematical
Sciences
at
University
College.
-
[PS83]
-
M. J.
Prelle
and
M. F.
Singer.
Elementary
first
integrals
of
differential
equations.
Transactions
of
the
American
Mathematical
Society,
279:215–229,
1983.
-
[PS95]
-
Peter
Paule
and
Markus
Schorn.
A
MATHEMATICA
version
of
zeilberger’s
algorithm
for
proving
binomial
coefficient
identities.
Journal
of
Symbolic
Computation,
20(5–6):673–698,
Nov
1995.
-
[PV15]
-
Maxim V.
Pavlov
and
Raffaele.
Vitolo.
On
the
bi-hamiltonian
geometry
of
wdvv
equations.
Letters
in
Mathematical
Physics,
105(8):1135–163,
Aug
2015.
-
[PZ96]
-
F. Postel
and
P. Zimmermann.
A
Review
of
the
ODE
Solvers
of
AXIOM,
DERIVE,
MAPLE
MATHEMATICA,
MACSYMA,
MUPAD
and
REDUCE.
In
Proceedings
of
the
5th
Rhine
Workshop
on
Computer
Algebra,
April
1-3,
1996,
Saint-Louis,
France,
1996.
Specific
references
are
to
the
version
dated
April
11,
1996.
-
[PZ97]
-
Zoran
Putnik
and
Budimac
Zoran.
Implementation
of
turtle
graphics
for
teaching
purposes.
In
EDUGRAPHICS
’97.
Third
International
Conference
on
Graphics
Education.
COMPUGRAPHICS
’97.
Sixth
International
Conference
on
Computational
Graphics
and
Visualization
Techniques.
Combined
Proceedings,
01
1997.
-
[RED]
-
Obtaining
reduce.
-
[Rei90]
-
G. J.
Reid.
A
triangularization
algorithm
which
determines
the
lie
symmetry
algebra
of
any
system
of
pdes.
Journal
of
Physics
A,
23(17):L853–L859,
1990.
-
[Rei91]
-
Gregory J.
Reid.
Algorithms
for
reducing
a
system
of
pdes
to
standard
form,
determining
the
dimension
of
its
solution
space
and
calculating
its
taylor
series
solution.
European
Journal
of
Applied
Mathematics,
2(4):293–318,
1991.
-
[Riq10]
-
C. Riquier.
Les
systèmes
d’équations
aux
dérivées
partielles.
Gauthier–Villars,
Paris,
1910.
-
[RLW93]
-
R. R. Loos
and
V. Weispfenning.
Applying
linear
quantifier
elimination.
The
Computer
Journal,
36:450–462,
Jan
1993.
-
[RLW01]
-
Gregory J.
Reid,
Ping
Lin,
and
Allan D.
Wittkopf.
Differential
elimination–completion
algorithms
for
dae
and
pdae.
Studies
in
Applied
Mathematics,
106(1):1–45,
2001.
-
[Roa]
-
Kelly
Roach.
Difficulties
with
trigonometrics.
Notes
of
a
talk.
-
[Rob89]
-
Lorenzo
Robbiano.
Computer
and
commutative
algebra.
In
Teo
Mora,
editor,
Applied
Algebra,
Algebraic
Algorithms
and
Error-Correcting
Codes,
pages
31–44,
Berlin,
Heidelberg,
1989.
Springer
Berlin
Heidelberg.
-
[Roe92a]
-
G. H. M.
Roelofs.
The
INTEGRATOR
package
for
REDUCE,
Version
1.0.
Memorandum
1100,
Dept.
Appl.
Math.,
University
of
Twente,
1992.
-
[Roe92b]
-
G. H. M.
Roelofs.
The
SUPER
VECTOR
FIELD
package
for
REDUCE,
Version
1.0.
Memorandum
1099,
Dept.
Appl.
Math.,
University
of
Twente,
1992.
-
[RT89]
-
A.Ya.
Rodionov
and
A.Yu.
Taranov.
Combinatorial
aspects
of
simplification
of
algebraic
expressions.
In
James H.
Davenport,
editor,
Eurocal
’87,
pages
192–201,
Berlin,
Heidelberg,
1989.
Springer
Berlin
Heidelberg.
-
[Rut92]
-
Elizabeth W.
Rutman.
Gröbner
bases
and
primary
decomposition
of
modules.
Journal
of
Symbolic
Computation,
14(5):483–503,
Nov
1992.
-
[RWB96]
-
Gregory J.
Reid,
Allan D.
Wittkopf,
and
Alan
Boulton.
Reduction
of
systems
of
nonlinear
partial
differential
equations
to
simplified
involutive
forms.
European
Journal
of
Applied
Mathematics,
7(6):635–666,
1996.
-
[Sch85a]
-
Eberhard
Schrüfer.
Excalc:
A
system
for
doing
calculations
in
modern
differential
geometry,
user’s
manual.
Technical
report,
Rand
Publication,
The
Rand
Corporation,
Santa
Monica,
1985.
-
[Sch85b]
-
F. Schwarz.
Automatically
determining
symmetries
of
partial
differential
equations.
Computing,
34(2):91–106,
Jun
1985.
-
[Sch85c]
-
F. Schwarz.
Automatically
determining
symmetries
of
partial
differential
equations.
Computing,
34(2):91–106,
November
1985.
-
[Sch87]
-
Fritz
Schwarz.
Computer
algebra
and
differential
equations
of
mathematical
physics.
Technical
report,
GMD
internal
report,
1987.
-
[Sch88]
-
F. Schwarz.
Symmetries
of
differential
equations:
From
Sophus
Lie
to
computer
algebra.
SIAM
Review,
30(3):450–481,
1988.
-
[Sch92]
-
Franziska
Schoebel.
The
symbolic
classification
of
real
four-dimensional
lie
algebras.
Technical
Report
Preprint
27/92,
Naturwissenschaftlich-Theoretisches
Zentrum,
Universitaet
Leipzig,
Germany,
1992.
-
[Sch93]
-
Carsten
Schöbel.
A
classification
of
real
finite-dimensional
lie
algebras
with
a
low-dimensional
derived
algebra.
Reports
on
Mathematical
Physics,
33(1–2):175..186,
Aug–Oct
1993.
-
[Sei95]
-
W M
Seiler.
Applying
axiom
to
partial
differential
equations.
Internal
Report
95-17,
Universität
Karlsruhe,
Fakultät
für
Informatik,
1995.
-
[Sei10]
-
Werner M.
Seiler.
Involution:
The
Formal
Theory
of
Differential
Equations
and
its
Applications
in
Computer
Algebra,
volume 24
of
Algorithms
and
Computation
in
Mathematics.
Springer
Verlag
Berlin
Heidelberg,
2010.
-
[Ser77]
-
Jean-Pierre
Serre.
Linear
Representations
of
Finite
Groups,
volume 42
of
Graduate
Texts
in
Mathematics.
Springer,
New
York,
NY,
1977.
-
[SF79]
-
Eduard
Stiefel
and
Albert
Fässler.
Gruppentheoretische
Methoden
und
ihre
Anwendung.
Teubner,
Stuttgart,
1979.
-
[Sim71a]
-
Charles C.
Sims.
Computation
with
permutation
groups.
In
S. R.
Petrick,
editor,
Proceedings
of
the
Second
ACM
Symposium
on
Symbolic
and
Algebraic
Manipulation,
SYMSAC
’71,
page
23–28,
New
York,
NY,
USA,
1971.
Association
for
Computing
Machinery.
-
[Sim71b]
-
Charles C.
Sims.
Determining
the
conjugation
classes
of
a
permutation
group.
In
G. Birckhoff
and
M. Hall
Jr.,
editors,
Computer
in
Algebra
and
Number
Theory,
volume 4
of
SIAM-AMS
Proceedings,
pages
191–195.
American
Mathematical
Society,
1971.
-
[SLT65]
-
Pascual
Sconzo,
A. R.
Leschack,
and
Robert G.
Tobey.
Symbolic
computation
of
f
and
g
series
by
computer.
The
Astronomical
Journal,
70:269,
May
1965.
-
[Spi59]
-
Murray R.
Spiegel.
Vector
Analysis.
Schaum’s
Outline.
McGraw-Hill
Education
(India)
Pvt
Limited,
1959.
-
[Spi79]
-
Michael
Spivak.
A
comprehensive
introduction
to
differential
geometry.
Publish
or
Perish,
Berkeley,
1979.
-
[Ste89]
-
Hans
Stephani.
Differential
equations,
Their
solution
using
symmetries.
Cambridge
University
Press,
1989.
-
[Sto90]
-
Timothy
Stokes.
Gröbner
bases
in
exterior
algebra.
Journal
of
Automated
Reasoning,
6(3):233–250,
Sep
1990.
-
[Str93]
-
Volker
Strehl.
Binomial
sums
and
identities.
Maple
Technical
Newsletter,
10:37–49,
1993.
-
[SV14]
-
Giuseppe
Saccomandi
and
Raffaele
Vitolo.
On
the
mathematical
and
geometrical
structure
of
the
determining
equations
for
shear
waves
in
nonlinear
isotropic
incompressible
elastodynamics.
Journal
of
Mathematical
Physics,
55(8):081502,
2014.
-
[SW06]
-
Vladimir V
Sokolov
and
Thomas
Wolf.
Integrable
quadratic
classical
hamiltonians
on
so(4)
and
so(3,
1).
Journal
of
Physics
A:
Mathematical
and
General,
39(8):1915,
feb
2006.
-
[Tar48]
-
A. Tarski.
A
decision
method
for
elementary
algebra
and
geometry.
Technical
report,
University
of
California,
1948.
Second
edn.,
rev.
1951.
-
[Tho37]
-
Joseph Miller
Thomas.
Differential
systems,
volume 21.
American
Mathematical
Soc.,
1937.
-
[Tou84]
-
David S.
Touretzky.
LISP:
A
Gentle
Introduction
to
Symbolic
Computation.
Harper
&
Row,
New
York,
1984.
-
[Tra76]
-
Barry M.
Trager.
Algebraic
factoring
and
rational
function
integration.
In
Richard D.
Jenks,
editor,
SYMSAC
’76:
Proceedings
of
the
Third
ACM
Symposium
on
Symbolic
and
Algebraic
Computation,
SYMSAC
’76,
pages
196–208,
New
York,
NY,
USA,
1976.
ACM.
-
[Vit]
-
R. Vitolo.
CDE:
a
reduce
package
for
computations
in
geometry
of
Differential
Equations.
Available
at
https://gdeq.org.
-
[Vit19]
-
R. Vitolo.
Computing
with
hamiltonian
operators.
Comput.
Phys.
Commun.,
244:228–245,
2019.
ArXiv:
https://arxiv.org/abs/1808.03902.
-
[Wah93]
-
Hugo D
Wahlquist.
Monte
carlo
calculation
of
cartan
characters:
using
the
maximal-slicing,
ricci-flat
ideal
as
an
example.
In
Nora
Bretón,
Riccardo
Capovilla,
and
Tonatiuh
Matos,
editors,
Proceedings,
12th
International
School-Seminar
on
The
Actual
Problems
of
Microworld
Physics.
Vol.1:
Gomel,
Belarus,
July
22
-
September
28,
2013,
pages
168–174,
1993.
-
[War91]
-
M. Warns.
Software
extensions
of
reduce
for
operator
calculus
in
quantum
theory.
In
Proceedings
of
the
IV
international
Conference
on
Computer
Algebra
in
Physical
Research,
Dubna
1990.
WORLD
SCIENTIFIC,
1991.
-
[Wat91]
-
Stephen M.
Watt,
editor.
ISSAC
’91.
Proceedings
of
the
1991
international
symposium
on
Symbolic
and
algebraic
computation.
Bonn,
Germany,
July
15–17,
1991,
New
York,
NY,
USA,
1991.
ACM.
-
[WB]
-
Thomas
Wolf
and
Andreas
Brand.
Demonstration
of
the
reduce
package
crack
for
investigating
partial
differential
equations.
online.
-
[Wei88]
-
Volker
Weispfenning.
The
complexity
of
linear
problems
in
fields.
Journal
of
Symb,
5(1–2):3–27,
Feb
1988.
-
[Wei92]
-
Volker
Weispfenning.
Comprehensive
Gröbner
bases.
Journal
of
Symbolic
Computation,
14(1):1–29,
July
1992.
-
[Wei94]
-
Volker
Weispfenning.
Quantifier
elimination
for
real
algebra
–
the
cubic
case.
In
Symbolic
and
Algebraic
Computation,
ISSAC,
pages
258–263.
SIGSAM,
ACM,
1994.
-
[Wei97]
-
V. Weispfenning.
Quantifier
elimination
for
real
algebra
—
the
quadratic
case
and
beyond.
Applicable
Algebra
in
Engineering,
Communication
and
Computing,
8(2):85–101,
Jan
1997.
-
[WH81]
-
Patrick
Winston
and
Berthold
Horn.
LISP.
Addison-Wesley,
1981.
-
[Wil90]
-
Herbert S.
Wilf.
Generatingfunctionology.
Academic
Press,
Boston,
1990.
-
[Wil93]
-
Herbert S.
Wilf.
Identities
and
their
computer
proofs.
SPICE
lecture
notes
31.
Available
by
anonymous
ftp
to
ftp.cis.upenn.edu
as
file
pub/wilf/lecnotes.ps.,
1993.
A
series
of
lectures
delivered
at
the
U.S.
National
Security
Agency.
-
[Wir80]
-
M. C.
Wirth.
On
the
Automation
of
Computational
Physics.
PhD
thesis,
Lawrence
Livermore
National
Lab.,
CA.,
jan
1980.
-
[Wol93]
-
T. Wolf.
An
efficiency
improved
program
liepde
for
determining
lie-symmetries
of
pdes.
In
Proc.
of
Modern
Group
Analysis:
advanced
analytical
and
computational
methods
in
mathematical
physics,
Catania,
Italy,
October
1992,
pages
377–385.
Kluwer
Academic
Publishers,
1993.
-
[Wol95]
-
Thomas
Wolf.
APPLYSYM
-
a
package
for
the
application
of
Lie-symmetries,
1995.
Software
distributed
together
with
the
computer
algebra
system
REDUCE.
-
[Wol00]
-
Thomas
Wolf.
The
symbolic
integration
of
exact
pdes.
Journal
of
Symbolic
Computation,
30(5):619–629,
2000.
-
[Wol02a]
-
T. Wolf.
A
comparison
of
four
approaches
to
the
calculation
of
conservation
laws.
Euro.
Jnl
of
Applied
Mathematics,
13(2):129–152,
2002.
-
[Wol02b]
-
Thomas
Wolf.
Size
reduction
and
partial
decoupling
of
systems
of
equations.
Journal
of
Symbolic
Computation,
33(3):367–383,
2002.
-
[Wri97]
-
F. J.
Wright.
An
Enhanced
ODE
Solver
for
REDUCE.
Programmirovanie,
3:5–22,
1997.
-
[Wri99]
-
F. J.
Wright.
Design
and
Implementation
of
ODESolve
1+
:
An
Enhanced
REDUCE
ODE
Solver.
Technical
report,
Queen
Mary,
University
of
London,
May
1999.
-
[Wt87]
-
Wu Wen-tsun.
A
zero
structure
theorem
for
polynomial-equations-solving
and
its
applications.
In
Proceedings
of
the
European
Conference
on
Computer
Algebra,
EUROCAL
’87,
page 44,
Berlin,
Heidelberg,
1987.
Springer-Verlag.
-
[WW69]
-
E. T.
Whittaker
and
G. N.
Watson.
A
Course
in
Modern
Analysis.
Cambridge
University
Press,
1969.
-
[ZB96]
-
A. Yu.
Zharkov
and
Yu. A.
Blinkov.
Involution
approach
to
investigating
polynomial
systems.
Mathematics
and
Computers
in
Simulation,
42:323–332,
11
1996.
-
[Zei90]
-
D. Zeilberger.
A
fast
algorithm
for
proving
terminating
hypergeometric
identities.
Discrete
Math.,
80(2):207–211,
March
1990.
-
[Zei91]
-
D. Zeilberger.
The
method
of
creative
telescoping.
Journal
of
Symbolic
Computation,
11(3):195–204,
1991.
-
[Zwi92]
-
D. Zwillinger.
Handbook
of
Differential
Equations.
Academic
Press,
second
edition,
1992.
Hosted by
