(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.0'

                    Mathematica-Compatible Notebook

This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.

To get the notebook into a Mathematica-compatible application, do
one of the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the
  application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info@wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from 
Wolfram Research.
*******************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[     45009,       1415]*)
(*NotebookOutlinePosition[     54767,       1706]*)
(*  CellTagsIndexPosition[     54417,       1691]*)
(*WindowFrame->Normal*)



Notebook[{
Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(First, \ 
    set\ the\ directory\ so\ that\ Mathematica\ knows\ where\ to\ get\ \ the\ \
code . \ For\ \(\(example\)\(:\)\)\)\)\)], "Input"],

Cell[BoxData[
    \(SetDirectory["\<d:\\invarsym\\testing\>"]\)], "Input",
  CellLabel->"In[1]:="],

Cell[BoxData[
    \(Second, \ 
    load\ the\ package . \ 
        The\ command\ below\ will\ generate\ two\ messages\ \ \((one\ will\ \
say\ that\ the\ package\ was\ successfully\ loaded)\)\)], "Input"],

Cell[BoxData[
    \(Get["\<InvariantsSymmetries.m\>"]\)], "Input",
  CellLabel->"In[2]:="],

Cell[BoxData[
    \(\(\(\ \ \ \ \ \ \ \)\(The\ code\ has\ been\ updated\ on\ October\ 23, 
    2005. \ \[IndentingNewLine]
    \ \ \ \ \ \ \ Reason : \ 
      make\ the\ code\ compatible\ with\ Mathematica\ v . \ 
          5.0\ and\ 5.1 . \ \[IndentingNewLine]This\ notebook\ has\ been\ \
updated\ on\ June\ 7, 2006. \)\)\)], "Input"],

Cell[CellGroupData[{

Cell["\[Copyright] Copyright Notice", "Section",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  CellTags->"Copyright"],

Cell[TextData[{
  "Copyright \[Copyright] by Unal Goktas and Willy Hereman (Department of \
Mathematical and Computer Sciences, Colorado School of Mines, Golden, \
Colorado, USA). No part of the ",
  StyleBox["Integrability ",
    FontSlant->"Italic"],
  "package can be sold or reproduced without written consent of the authors. \
",
  ButtonBox["Wolfram Research",
    ButtonData:>{
      URL[ "http://www.wolfram.com"], None},
    ButtonStyle->"Hyperlink"],
  ", Inc. (Champaign, Illinois, USA) is the holder of the copyright to the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " software system."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Copyright"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "About the Package: ",
  StyleBox["Integrability",
    FontFamily->"Helvetica",
    FontSlant->"Italic"]
}], "Section",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->0,
  AspectRatioFixed->True,
  CellTags->"Introduction"],

Cell[TextData[{
  "Nonlinear partial differential equations (PDEs) and differential \
difference equations (DDEs) possess a number of remarkable properties \
reflecting their rich mathematical structure. Such properties include the \
Painleve property, nontrivial prolongation and bi-Hamiltonian structures, \
soliton solutions, Lax pairs, Backlund transformations, symmetries and \
conservation laws (which we call invariants). They reveal the adequacy of \
PDEs and DDEs as models for physical relevant phenomena. The search for these \
intrinsic properties becomes more attractive due to the availability of CAS \
such as ",
  StyleBox["Mathematica.",
    FontSlant->"Italic"],
  " Indeed, ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is well suited and often necessary tool to perform the computations \
inherent in the investigation of integrability."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->0,
  AspectRatioFixed->True,
  CellTags->"Introduction"],

Cell[TextData[{
  StyleBox["Integrability ",
    FontSlant->"Italic"],
  "is a collection of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " functions addressing computations of invariants and symmetries in \
analyzing nonlinear PDEs and DDEs. The main topics covered in the ",
  StyleBox["Integrability",
    FontSlant->"Italic"],
  " package are as follows:\n\[FilledVerySmallSquare] Computations of \
invariants (that is conservation laws) for systems of nonlinear PDEs and \
DDEs,\n\[FilledVerySmallSquare] Computations of symmetries for systems of \
nonlinear PDEs and DDEs."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->0,
  AspectRatioFixed->True,
  CellTags->"Introduction"],

Cell[TextData[{
  "A complete set of mathematical definitions and the concepts used in the \
package and this manual are beyond the scope of this document. For the formal \
definitions of the concepts and the theory behind the functions in the ",
  StyleBox["Integrability",
    FontSlant->"Italic"],
  " package, consult the references given in the References section. In this \
manual, a basic knowledge of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is assumed. For further information on ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", consult the standard ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " reference: ",
  StyleBox["The Mathematica Book",
    FontSlant->"Italic"],
  ",  Stephen Wolfram, Third Edition, Wolfram Media and Cambridge University \
Press, 1996."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->0,
  AspectRatioFixed->True,
  CellTags->"Introduction"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Organization of the Package and Functionality", "Section",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  CellTags->"Organization"],

Cell[TextData[{
  "The package ",
  StyleBox["Integrability",
    FontSlant->"Italic"],
  " consists of the following  M",
  StyleBox["athematica",
    FontSlant->"Italic"],
  " packages.  "
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Organization"],

Cell[TextData[{
  StyleBox["InvariantsSymmetries.m", "cb",
    FontWeight->"Bold"],
  StyleBox[" ",
    FontWeight->"Bold"],
  "\t       Computations of invariants and symmetries of PDEs and DDEs\n",
  StyleBox["init.m\t                                           ",
    FontWeight->"Bold"],
  "Function declarations"
}], "Text",
  CellFrame->True,
  CellMargins->{{10.75, 24.125}, {0, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False},
  CellTags->"Organization"],

Cell[TextData[{
  StyleBox["Packages and file names in ",
    FontFamily->"Helvetica",
    FontSize->9],
  StyleBox["Integrability",
    FontFamily->"Helvetica",
    FontSize->9,
    FontSlant->"Italic"]
}], "Text",
  CellMargins->{{10.75, 24.125}, {10, 0}},
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  FrameBoxOptions->{BoxMargins->{{0.4, 0.4}, {0, 0}}},
  CellTags->"Organization"],

Cell[TextData[{
  StyleBox["PDEInvariants[",
    FontWeight->"Bold"],
  StyleBox["eqn, u, {x, t}, R, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the invariant with rank R of a partial differential equation for \
the function u.\n",
  StyleBox["PDEInvariants[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {x, t}, R, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["] ",
    FontWeight->"Bold"],
  "finds the invariant of a system of partial differential equations.\n",
  StyleBox["PDEInvariants[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {x, t}, {Rmax}, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["] ",
    FontWeight->"Bold"],
  "finds the invariants with rank 0 through Rmax.\n",
  StyleBox["PDEInvariants[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {x, t}, {Rmin, Rmax}, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the invariants with rank Rmin through Rmax.\n",
  StyleBox["PDESymmetries[",
    FontWeight->"Bold"],
  StyleBox["eqn, u, {x, t}, R, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the symmetry with rank R of a partial differential equation for the \
function u.\n",
  StyleBox["PDESymmetries[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {x, t}, R, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the symmetry of a system of partial differential equations, where R \
is the rank of the first equation in the desired symmetry.\n",
  StyleBox["PDESymmetries[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {x, t}, {Rmax}, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the symmetries with rank 0 through Rmax.\n",
  StyleBox["PDESymmetries[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {x, t}, {Rmin, Rmax}, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the symmetries with rank Rmin through Rmax.\nx is understood as the \
space variable and t as the time variable."
}], "Text",
  CellFrame->True,
  CellMargins->{{10.75, 24.125}, {0, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False},
  CellTags->"Organization"],

Cell[TextData[{
  StyleBox["Functions for the analysis of PDEs in ",
    FontFamily->"Helvetica",
    FontSize->9],
  StyleBox["Integrability",
    FontFamily->"Helvetica",
    FontSize->9,
    FontSlant->"Italic"]
}], "Text",
  CellMargins->{{10.75, 24.125}, {10, 0}},
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  FrameBoxOptions->{BoxMargins->{{0.4, 0.4}, {0, 0}}},
  CellTags->"Organization"],

Cell[TextData[{
  StyleBox["DDEInvariants[",
    FontWeight->"Bold"],
  StyleBox["eqn, u, {n, t}, R, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the invariant with rank R of a differential difference equation for \
the function u.\n",
  StyleBox["DDEInvariants[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {n, t}, R, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the invariant of a system of differential difference equations. \n",
  
  StyleBox["DDEInvariants[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {n, t}, {Rmax}, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the invariants with rank 0 through Rmax. \n",
  StyleBox["DDEInvariants[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {n, t}, {Rmin, Rmax}, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the invariants with rank Rmin through Rmax. \n",
  StyleBox["DDESymmetries[",
    FontWeight->"Bold"],
  StyleBox["eqn, u, {n, t}, R, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the symmetry with rank R of a differential difference equation for \
the function u.\n",
  StyleBox["DDESymmetries[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {n, t}, R, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the symmetry of a system of differential difference equations, \
where R is the rank of the first equation in the desired symmetry.\n",
  StyleBox["DDESymmetries[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {n, t}, {Rmax}, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the symmetries with rank 0 through Rmax.\n",
  StyleBox["DDESymmetries[",
    FontWeight->"Bold"],
  StyleBox["{eqn1, eqn2, ...}, {u1, u2, ...}, {n, t}, {Rmin, Rmax}, opts",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["]",
    FontWeight->"Bold"],
  " finds the symmetries with rank Rmin through Rmax.\nn is understood as the \
discrete space variable and t as the time variable."
}], "Text",
  CellFrame->True,
  CellMargins->{{10.75, 24.125}, {0, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False},
  CellTags->"Organization"],

Cell[TextData[{
  StyleBox["Functions for the analysis of DDEs in ",
    FontFamily->"Helvetica",
    FontSize->9],
  StyleBox["Integrability",
    FontFamily->"Helvetica",
    FontSize->9,
    FontSlant->"Italic"]
}], "Text",
  CellMargins->{{10.75, 24.125}, {10, 0}},
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  FrameBoxOptions->{BoxMargins->{{0.4, 0.4}, {0, 0}}},
  CellTags->"Organization"],

Cell[TextData[{
  StyleBox["Options[PDEInvariants]",
    FontWeight->"Bold"],
  " = ",
  StyleBox["Options[DDEInvariants]",
    FontWeight->"Bold"],
  " = ",
  StyleBox["Options[PDESymmetries]",
    FontWeight->"Bold"],
  " = ",
  StyleBox["Options[DDESymmetries]",
    FontWeight->"Bold"],
  " =                              {WeightRules->{},WeightedParameters -> {}, \
MaxExplicitDependency -> 0, UndeterminedCoefficients -> C}\t\n",
  StyleBox["WeightRules",
    FontWeight->"Bold"],
  " is an option that determines the rules for weights of the variables. If \
WeightRules ->{Weight[u] -> val, ...}, then scaling properties are determined \
under these rules. There is a built in checking mechanism to see if the given \
rules cause inconsistency.\:f3b5\n",
  StyleBox["WeightedParameters",
    FontWeight->"Bold"],
  " is an option that determines the parameters with weight. If \
WeightedParameters -> {p1, p2, ...}, then p1, p2,.... are considered as \
constant parameters with weight. The default is WeightedParameters -> {}.\n",
  StyleBox["MaxExplicitDependency",
    FontWeight->"Bold"],
  " is an option in finding the invariant and generalized symmetries of PDEs \
and DDEs. If MaxExplicitDependency-> Max_Integer, then program allows \
explicit dependency of independent variables of maximum degree of Max. The \
default is MaxExplicitDependency -> 0.\n",
  StyleBox["UndeterminedCoefficients",
    FontWeight->"Bold"],
  " is an option that determines the name of the constant coefficients to \
use. The default is UndeterminedCoefficients ->C.\:f3b5"
}], "Text",
  CellFrame->True,
  CellMargins->{{10.75, 24.125}, {0, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False},
  CellTags->"Organization"],

Cell[TextData[{
  StyleBox["Options of functions in ",
    FontFamily->"Helvetica",
    FontSize->9],
  StyleBox["Integrability",
    FontFamily->"Helvetica",
    FontSize->9,
    FontSlant->"Italic"]
}], "Text",
  CellMargins->{{10.75, 24.125}, {10, 0}},
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  FrameBoxOptions->{BoxMargins->{{0.4, 0.4}, {0, 0}}},
  CellTags->"Organization"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(IMPORTANT\ \(\(NOTE\)\(:\)\(\ \)\)\)\)\)], \
"Input"],

Cell[BoxData[{
    \(The\ user\ does\ not\ have\ to\ install\ the\ package\ Integrability\ \
to\ be\ able\ to\ use\ it . \ The\ core\ code\ is\), "\[IndentingNewLine]", 
    \(in\ the\ file\ InvariantsSymmetries . m . \ Therefore, \ 
    the\ \ Sections\ \((below)\)\ on\ Loading\ the\ Package\ and\ Palettes\ \
can\ be\ skipped, \ 
    provided\ \ the\ two\ Mathematica\ commands\ for\ setting\ the\ Directory\
\ and\ loading\ the\ file\ InvariantsSymmetries . 
        m\ via\ the\ Set\ command\ \((at\ the\ top\ of\ this\ notebook)\)\ \
have\ been\ executed . \ \[IndentingNewLine]So, \ 
    go\ to\ the\ section\ with\ Examples\ and\ execute\ the\ commands\ one\ \
by\ \(\(one\)\(.\)\(\ \)\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Loading the Package", "Section",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  CellTags->"Loading"],

Cell[TextData[{
  "To access all of the functions in ",
  StyleBox["Integrability",
    FontSlant->"Italic"],
  ", all you need to do is to load the file ",
  StyleBox["Integrability`init.m`",
    FontWeight->"Bold"],
  " by typing the following line in a ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " 3.0 notebook:"
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Loading"],

Cell["<<Integrability`", "Input",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  CellTags->"Loading"],

Cell["\<\
or simply, click at the following button \"Load Integrability\" in the help \
browser entry for the package. Note that this button can be copy-and-pasted \
in a notebook without losing its function.\
\>", "Text",
  CellTags->"Loading"],

Cell[BoxData[
    RowBox[{"       ", 
      TagBox[
        ButtonBox[\(Load\ Integrability\),
          ButtonFunction:>CompoundExpression[ 
              Get[ "Integrability`"]],
          ButtonEvaluator->Automatic,
          Active->True],
        DisplayForm]}]], "Input",
  CellTags->"Loading"],

Cell[TextData[{
  "Once ",
  StyleBox["Integrability`init.m`",
    FontWeight->"Bold"],
  " is loaded, the package will be automatically loaded when a function in \
the package is called from the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " environment. So, in addition to the execution time of a function, some \
delay will occur when a package is first loaded."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Loading"],

Cell[TextData[{
  "If the directory in which the package is located is not on the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " ",
  StyleBox["$Path",
    FontWeight->"Bold"],
  ", a warning message will be generated. To solve this problem, the \
directory containing the package should be included on the path ",
  StyleBox["$Path",
    FontWeight->"Bold"],
  ". The following ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " command puts the directory ",
  StyleBox["dir",
    FontSlant->"Italic"],
  " on your path."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Loading"],

Cell["AppendTo[$Path,\"dir\"]", "Input",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  CellTags->"Loading"],

Cell[TextData[{
  "It is customary to place the packages in a subdirectory called \
Applications (APPS under MS-DOS or Microsoft Windows), found under the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " root directory. In ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " 3.0, the canonical location for applications packages is the directory \
AddOns/Applications. Since the path to this directory is included in ",
  StyleBox["$Path",
    FontWeight->"Bold"],
  ", the user need not to set a path for the packages placed in \
AddOns/Applications."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Loading"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[ButtonBox["Palettes",
  ButtonData:>{"IntegrabilityPalettes.nb", None},
  ButtonStyle->"Hyperlink",
  ButtonNote->None]], "Section",
  CellTags->"Palettes"],

Cell[TextData[{
  "Use the above hyperlink to get the designed palettes for the ",
  StyleBox["Integrability",
    FontSlant->"Italic"],
  " package."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Palettes"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Examples", "Section",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Examples"],

Cell["\<\
If you haven't allready loaded the package, load it now by clicking \
at the following  button.\
\>", "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Examples"],

Cell[BoxData[
    RowBox[{"       ", 
      TagBox[
        ButtonBox[\(Load\ Integrability\),
          ButtonFunction:>CompoundExpression[ 
              Get[ "Integrability`"]],
          ButtonEvaluator->Automatic,
          Active->True],
        DisplayForm]}]], "Input",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[CellGroupData[{

Cell["Korteweg-de Vries (KdV) Equation", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(\(kdv := \[PartialD]\_t u[x, t] + 
            u[x, t]\ \[PartialD]\_x u[x, t] + \[PartialD]\_{x, 3}u[x, t] == 
          0;\)\)], "Input",
  CellLabel->"In[3]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(TableForm[PDEInvariants[kdv, u, {x, t}, {1, 8}], 
      TableHeadings -> {None, {"\<Rank\>", "\<Invariant\>"}}]\)], "Input",
  CellLabel->"In[4]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["\<\
PDEInvariants[kdv,u,{x,t},1,MaxExplicitDependency->1] //TableForm\
\>", "Input",
  CellLabel->"In[5]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["\<\
TableForm[PDESymmetries[kdv,u,{x,t},{9}],TableHeadings->{None,{\"Rank\",\"\
Symmetry\"}}]\
\>", "Input",
  CellLabel->"In[6]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDESymmetries[kdv,u,{x, t},2,MaxExplicitDependency->1]", "Input",
  CellLabel->"In[7]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDESymmetries[kdv,u,{x, t},0,MaxExplicitDependency->1]", "Input",
  CellLabel->"In[8]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Boussinesq Equation", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[{
    \(\(pde1 := \[PartialD]\_t u[x, t] + \[PartialD]\_x v[x, t] == 
          0;\)\), "\n", 
    \(\(pde2 := \[PartialD]\_t v[x, t] + \[PartialD]\_x u[x, t] - 
            3\ u[x, t]\ \[PartialD]\_x u[x, 
                  t] - \[Alpha]\ \[PartialD]\_{x, 3}u[x, t] == 
          0;\)\)}], "Input",
  CellLabel->"In[9]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(PDEInvariants[{pde1, pde2}, {u, v}, {x, t}, {2, 6}]\)], "Input",
  CellLabel->"In[11]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Nonlinear Schrodinger (NLS) Equation", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[{
    \(\(pde1 := \[PartialD]\_t q[x, t] - \[PartialD]\_{x, 2}q[x, t] + 
            2\ q[x, t]\^2\ r[x, t] == 0;\)\), "\n", 
    \(\(pde2 := \[PartialD]\_t r[x, t] + \[PartialD]\_{x, 2}r[x, t] - 
            2\ r[x, t]\^2\ q[x, t] == 0;\)\)}], "Input",
  CellLabel->"In[12]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDEInvariants[{pde1,pde2},{q,r},{x,t},{2,5}]", "Input",
  CellLabel->"In[14]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["\<\
PDEInvariants[{pde1,pde2},{q,r},{x,t},{2,5},WeightRules->{Weight[q]->Weight[r]\
}]\
\>", "Input",
  CellLabel->"In[15]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Non-dispersive Long Wave System", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[{
    \(\(pde1 := \[PartialD]\_t u[x, t] + u[x, t]\ \[PartialD]\_x v[x, t] + 
            v[x, t]\ \[PartialD]\_x u[x, t] == 0;\)\), "\n", 
    \(\(pde2 := \[PartialD]\_t v[x, t] + \[PartialD]\_x u[x, t] + 
            v[x, t]\ \[PartialD]\_x v[x, t] == 0;\)\)}], "Input",
  CellLabel->"In[16]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDEInvariants[{pde1,pde2},{u,v},{x,t},{1,5}]", "Input",
  CellLabel->"In[18]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["\<\
PDEInvariants[{pde1,pde2},{u,v},{x,t},{1,5},WeightRules->{Weight[u]->2}]\
\>", "Input",
  CellLabel->"In[19]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Ibragimov-Shabat System", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[{
    \(\(pde1 := \[PartialD]\_t u[x, t] - \[PartialD]\_{x, 2}u[x, t] - 
            1\/2\ v[x, t]\^2 == 0;\)\), "\n", 
    \(\(pde2 := \[PartialD]\_t v[x, t] - 2\ \[PartialD]\_{x, 2}v[x, t] == 
          0;\)\)}], "Input",
  CellLabel->"In[20]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDEInvariants[{pde1,pde2},{u,v},{x,t},{6}]", "Input",
  CellLabel->"In[22]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["\<\
PDEInvariants[{pde1,pde2},{u,v},{x,t},{6},WeightRules->{Weight[v]->2}]\
\>", "Input",
  CellLabel->"In[23]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["\<\
PDESymmetries[{pde1,pde2},{u,v},{x,t},{5,7},WeightRules->{Weight[v]->2}]//\
TableForm\
\>", "Input",
  CellLabel->"In[24]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Burgers Equation", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(\(burgers := \[PartialD]\_t u[x, t] - 
            2\ u[x, t]\ \[PartialD]\_x u[x, t] - \[PartialD]\_{x, 2}u[x, t] == 
          0;\)\)], "Input",
  CellLabel->"In[25]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDEInvariants[burgers,u,{x,t},{1,4}]", "Input",
  CellLabel->"In[26]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDESymmetries[burgers,u,{x,t},{4,6}]//TableForm", "Input",
  CellLabel->"In[27]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Family of Fifth-order KdV Equations", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(\(pde := \[PartialD]\_t u[x, 
                t] + \[Alpha]\ u[x, t]\^2\ \[PartialD]\_x u[x, 
                  t] + \[Beta]\ \[PartialD]\_x u[x, t]\ \[PartialD]\_{x, 2}u[
                  x, t] + \[Gamma]\ u[x, 
                t]\ \[PartialD]\_{x, 3}u[x, t] + \[PartialD]\_{x, 5}u[x, t] == 
          0;\)\)], "Input",
  CellLabel->"In[28]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDEInvariants[pde,u,{x,t},{2,10}];", "Input",
  CellLabel->"In[29]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["\<\
TableForm[%,TableHeadings->{None,{\"Rank\",\"Condition - Density\"}}]\
\>", "Input",
  CellLabel->"In[30]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDESymmetries[pde,u,{x,t},9];", "Input",
  CellLabel->"In[31]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["\<\
TableForm[%,TableHeadings->{None,{\"Rank\",\"Condition-Symmetry\"}}]\
\>", "Input",
  CellLabel->"In[32]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Hirota-Satsuma System", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[{
    \(\(pde1 := \[PartialD]\_t u[x, t] - 
            6\ \[Alpha]\ u[x, t]\ \[PartialD]\_x u[x, t] + 
            6\ v[x, t]\ \[PartialD]\_x v[x, 
                  t] - \[Alpha]\ \[PartialD]\_{x, 3}u[x, t] == 0;\)\), "\n", 
    \(\(pde2 := \[PartialD]\_t v[x, t] + 
            3\ u[x, t]\ \[PartialD]\_x v[x, t] + \[PartialD]\_{x, 3}v[x, t] == 
          0;\)\)}], "Input",
  CellLabel->"In[33]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["PDEInvariants[{pde1,pde2},{u,v},{x,t},{2,8}]//TableForm", "Input",
  CellLabel->"In[35]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(PDESymmetries[{pde1, pde2}, {u, v}, {x, t}, 7]\  // 
      TableForm\)], "Input",
  CellLabel->"In[36]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Toda lattice", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[{
    \(\(dde1 := \[PartialD]\_t u[n, t] - v[n - 1, t] + v[n, t] == 
          0;\)\), "\n", 
    \(\(dde2 := \[PartialD]\_t v[n, t] - 
            v[n, t]\ \((u[n, t] - u[n + 1, t])\) == 0;\)\)}], "Input",
  CellLabel->"In[6]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(DDEInvariants[{dde1, dde2}, {u, v}, {n, t}, {1, 5}] // 
      TableForm\)], "Input",
  CellLabel->"In[8]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["DDESymmetries[{dde1,dde2},{u,v},{n, t},{0,3}]//TableForm", "Input",
  CellLabel->"In[10]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Volterra equation", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(\(dde := 
      \[PartialD]\_t u[n, t] - u[n, t]\ \((u[n + 1, t] - u[n - 1, t])\) == 0; 
    \)\)], "Input",
  CellLabel->"In[39]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(DDEInvariants[dde, u, {n, t}, {1, 5}] // TableForm\)], "Input",
  CellLabel->"In[40]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["DDESymmetries[dde,u,{n,t},3]//TableForm", "Input",
  CellLabel->"In[41]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["DDESymmetries[dde,u,{n,t},1,MaxExplicitDependency->1]", "Input",
  CellLabel->"In[42]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Ablowitz-Ladik discretization of the NLS equation", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[{
    \(\(dde1 := 
      \[PartialD]\_t u[n, t] - \((u[n + 1, t] - 2\ u[n, t] + u[n - 1, t])\) - 
          u[n, t]\ v[n, t]\ \((u[n + 1, t] + u[n - 1, t])\) == 0; \)\), 
    \(\(dde2 := 
      \[PartialD]\_t v[n, t] + \((v[n + 1, t] - 2\ v[n, t] + v[n - 1, t])\) + 
          u[n, t]\ v[n, t]\ \((v[n + 1, t] + v[n - 1, t])\) == 0; \)\)}], 
  "Input",
  CellLabel->"In[43]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["DDEInvariants[{dde1,dde2},{u,v},{n,t},2]", "Input",
  CellLabel->"In[45]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(DDEInvariants[{dde1, dde2}, {u, v}, {n, t}, 2, 
      WeightRules -> {Weight[u] -> Weight[v]}]\)], "Input",
  CellLabel->"In[46]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Parameterized Toda lattice", "Subsection",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[{
    \(\(dde1 := 
      \[PartialD]\_t u[n, t] - \[Alpha]\ v[n - 1, t] + v[n, t] == 0; \)\), 
    \(\(dde2 := 
      \[PartialD]\_t v[n, t] - 
          v[n, t]\ \((\[Beta]\ u[n, t] - u[n + 1, t])\) == 0; \)\)}], "Input",\

  CellLabel->"In[47]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell[BoxData[
    \(DDEInvariants[{dde1, dde2}, {u, v}, {n, t}, {1, 3}] // TableForm\)], 
  "Input",
  CellLabel->"In[49]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"],

Cell["DDESymmetries[{dde1,dde2},{u,v},{n,t},3] //TableForm", "Input",
  CellLabel->"In[50]:=",
  TextAlignment->Left,
  TextJustification->1,
  CellTags->"Examples"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Scope and Limitations", "Section",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  CellTags->"Scope"],

Cell[TextData[{
  StyleBox["Integrability ",
    FontSlant->"Italic"],
  "can handle systems of evolution equations and first order DDEs that are \
polynomial in the dependent variables. One space variable and one time \
variable are allowed."
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Scope"],

Cell["\<\
Theoretically, there is no limit on the number of equations. In \
practice, for large systems, the computations may take a long time or require \
a lot of memory. Computational speed depends primarily on the amount of \
memory.\
\>", "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Scope"],

Cell[TextData[{
  "Only polynomial invariants and symmetries are computed. By design, ",
  StyleBox["Integrability",
    FontSlant->"Italic"],
  " constructs only invariants and symmetries that are uniform in rank. The \
uniform rank assumption allows one to compute independent invariants and \
symmetries piece by piece, without having to split linear combinations. Due \
to the superposition principle, a linear combination of invariants of unequal \
rank is still an invariant. This situation arises frequently when parameters \
with weights are introduced. "
}], "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Scope"],

Cell["\<\
The systems may have parameters, which are assumed to be nonzero. \
If a system has parameters, the program will compute the conditions for these \
parameters so that invariants or symmetries of a given rank might exist. The \
assumption that parameters are nonzero is necessary. As a result of setting \
parameters to zero in an equation, the weights and therefore the ranks might \
change.\
\>", "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Scope"],

Cell["\<\
The assumption that the systems are uniform in rank is not very \
restrictive. If the uniform rank condition is violated, users can introduce \
parameters with weights. This also allows some flexibility in the form of the \
invariants and symmetries such that invariants and symmetries may not be \
uniform in rank. In cases where it is not clear whether or not parameters \
with weights should be introduced, one should start without introducing any. \
If this causes incompatibilities in the assignment of weights, program tries \
to cure the incompatibility by introducing parameters with weights. In cases \
of  persistent incompatibility, corresponding diagnostic messages are \
printed.\
\>", "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Scope"],

Cell["\<\
Negative weights on the dependent variables are not permitted. Zero \
weights are allowed, but at least one of the dependent variables must have \
positive weight. Fractional weights and ranks are permitted. In the case of \
existence of free weights function calls are returned unevaluated.\
\>", 
  "Text",
  CellMargins->{{10.75, 24.125}, {Inherited, Inherited}},
  Evaluatable->False,
  CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}},
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Scope"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Acknowledgements", "Section",
  CellMargins->{{Inherited, 31.5}, {Inherited, Inherited}},
  Evaluatable->False,
  AspectRatioFixed->True,
  CellTags->"Acknowledgements"],

Cell[TextData[{
  "Thanks to R. Setinkaya, M. Ito, D. Lichtblau, J. Sanders, M. Trott, F. \
Verheest, J.P. Wang, P. Winternitz, S. Wolfram for many valuable comments and \
suggestions. ",
  ButtonBox["National Science Foundation",
    ButtonData:>{
      URL[ "http://www.nsf.gov"], None},
    ButtonStyle->"Hyperlink"],
  " (USA) and ",
  ButtonBox["Wolfram Research",
    ButtonData:>{
      URL[ "http://www.wolfram.com"], None},
    ButtonStyle->"Hyperlink"],
  ", Inc. are thanked for the support in development of this package."
}], "Text",
  CellMargins->{{Inherited, 31.5}, {Inherited, Inherited}},
  Evaluatable->False,
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Acknowledgements"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["References",
  FontWeight->"Bold"]], "Section",
  CellMargins->{{Inherited, 27.125}, {Inherited, Inherited}},
  Evaluatable->False,
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  FontWeight->"Plain",
  CellTags->"References"],

Cell[TextData[{
  "M.J. Ablowitz and P.A. Clarkson, ",
  StyleBox["Nonlinear Evolution Equations and Inverse Scattering",
    FontSlant->"Italic"],
  ", Cambridge University Press, Cambridge, 1991.\nU. Goktas and  W. Hereman, \
",
  StyleBox["Symbolic Computation of Conserved Densities for Systems of \
Nonlinear Evolution Equations",
    FontSlant->"Italic"],
  ",  J. Symbolic Computation 24(5), 591-621, 1997.\nU. Goktas and  W. \
Hereman, ",
  StyleBox["Computation of Conservation Laws for Nonlinear Lattices",
    FontSlant->"Italic"],
  ",  Physica D 123(1-4) , 425-436 , 1998.\nU. Goktas and  W. Hereman, ",
  StyleBox["Computation of Higher-order Symmetries for Nonlinear Evolution \
and Lattice Equations",
    FontSlant->"Italic"],
  ",  Adv. Comput. Math. 11 , 55-80 , 1999.\nU. Goktas, W. Hereman and G. \
Erdmann, ",
  StyleBox["Symbolic Computation of Conserved Densities for Systems of \
Nonlinear Differential-Difference Equations",
    FontSlant->"Italic"],
  ", Phys. Lett. A 236(1-2),  30-38, 1997.\nW. Hereman, U. Goktas, M. \
Colagrosso and A. Miller, ",
  StyleBox["Algorithmic Integrability Tests for", "SmallTextSans"],
  StyleBox[" ",
    FontSlant->"Italic"],
  StyleBox["Nonlinear Differential and Lattice Equations", "SmallTextSans",
    FontSlant->"Italic"],
  ", Comput. Phys. Commun. 115(2-3),  428-446, 1998.\nM. Hickman and W. \
Hereman, ",
  StyleBox["Computation of Densities and Fluxes of Nonlinear \
Differential-difference Equations",
    FontSlant->"Italic"],
  ",  Proc. Roy. Soc. London A 459 , 2705-2729, 2003.\nM. Ito, Comput. Phys. \
Commun. 79 , 547-554, 1994.\nA.V. Mikhailov, A.B. Shabat and V.V. Sokolov, \
Russian Math. Surveys 42, 1-63, 1987.\nP.J. Olver, ",
  StyleBox["Applications of  Lie Groups to Differential Equations",
    FontSlant->"Italic"],
  ", Springer-Verlag, Berlin, 1986.\nS. Wolfram, ",
  StyleBox["The Mathematica Book",
    FontSlant->"Italic"],
  ", Fifth Edition, Wolfram Media-Cambridge University Press, 2003."
}], "Text",
  CellMargins->{{Inherited, 27.125}, {Inherited, Inherited}},
  Evaluatable->False,
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"References"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Contact Information", "Section",
  CellMargins->{{Inherited, 31.5}, {Inherited, Inherited}},
  Evaluatable->False,
  AspectRatioFixed->True,
  CellTags->"Contact"],

Cell[TextData[{
  "For information about the features of ",
  StyleBox["Integrability, ",
    FontSlant->"Italic"],
  "or in case of trouble, contact Willy Hereman by email: ",
  ButtonBox["whereman@mines.edu",
    ButtonData:>{
      URL[ "mailto:whereman@mines.edu"], None},
    ButtonStyle->"Hyperlink"],
  " or mail to: Colorado School of Mines, Department of Mathematical and \
Computer Sciences, Golden, Colorado 80401-1887, U.S.A. You can also reach me \
by fax @ + 1-303-273-3875 (mention for Prof. Hereman) or by phone @ + \
1-303-273-3881. Authors will greatly appreciate hearing your comments about \
any of the material in the ",
  StyleBox["Integrability",
    FontSlant->"Italic"],
  " package."
}], "Text",
  CellMargins->{{Inherited, 31.5}, {Inherited, Inherited}},
  Evaluatable->False,
  TextAlignment->Left,
  TextJustification->1,
  AspectRatioFixed->True,
  CellTags->"Contact"]
}, Open  ]]
},
FrontEndVersion->"5.0 for Microsoft Windows",
ScreenRectangle->{{0, 1280}, {0, 941}},
ScreenStyleEnvironment->"AllBrackets",
PrintingStyleEnvironment->"EnhancedPrintout",
WindowToolbars->{"RulerBar", "EditBar"},
CellGrouping->Automatic,
WindowSize->{1272, 907},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
PrintingCopies->1,
PrintingStartingPageNumber->3,
PrintingPageRange->{Automatic, Automatic},
PageFooterLines->{True, True},
PrintingOptions->{"PrintingMargins"->{{72, 54}, {72, 72}},
"PrintCellBrackets"->False,
"PrintRegistrationMarks"->False,
"PrintMultipleHorizontalPages"->False},
PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}},
ShowCellLabel->True,
ShowCellTags->True,
RenderingOptions->{"ObjectDithering"->True,
"RasterDithering"->False},
CharacterEncoding->"ISO8859-1",
Magnification->1.5,
StyleDefinitions -> "HelpBrowser.nb"
]

(*******************************************************************
Cached data follows.  If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of  the file.  The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)

(*CellTagsOutline
CellTagsIndex->{
  "Copyright"->{
    Cell[2705, 80, 247, 5, 114, "Section",
      Evaluatable->False,
      CellTags->"Copyright"],
    Cell[2955, 87, 870, 23, 97, "Text",
      Evaluatable->False,
      CellTags->"Copyright"]},
  "Introduction"->{
    Cell[3862, 115, 386, 12, 114, "Section",
      Evaluatable->False,
      CellTags->"Introduction"],
    Cell[4251, 129, 1131, 24, 141, "Text",
      Evaluatable->False,
      CellTags->"Introduction"],
    Cell[5385, 155, 852, 21, 135, "Text",
      Evaluatable->False,
      CellTags->"Introduction"],
    Cell[6240, 178, 1075, 28, 119, "Text",
      Evaluatable->False,
      CellTags->"Introduction"]},
  "Organization"->{
    Cell[7352, 211, 266, 5, 114, "Section",
      Evaluatable->False,
      CellTags->"Organization"],
    Cell[7621, 218, 451, 15, 53, "Text",
      Evaluatable->False,
      CellTags->"Organization"],
    Cell[8075, 235, 714, 22, 107, "Text",
      Evaluatable->False,
      CellTags->"Organization"],
    Cell[8792, 259, 409, 12, 52, "Text",
      CellTags->"Organization"],
    Cell[9204, 273, 2768, 82, 347, "Text",
      Evaluatable->False,
      CellTags->"Organization"],
    Cell[11975, 357, 420, 12, 52, "Text",
      CellTags->"Organization"],
    Cell[12398, 371, 2795, 83, 347, "Text",
      Evaluatable->False,
      CellTags->"Organization"],
    Cell[15196, 456, 420, 12, 52, "Text",
      CellTags->"Organization"],
    Cell[15619, 470, 1964, 48, 285, "Text",
      Evaluatable->False,
      CellTags->"Organization"],
    Cell[17586, 520, 406, 12, 52, "Text",
      CellTags->"Organization"]},
  "Loading"->{
    Cell[18844, 554, 235, 5, 114, "Section",
      Evaluatable->False,
      CellTags->"Loading"],
    Cell[19082, 561, 585, 18, 53, "Text",
      Evaluatable->False,
      CellTags->"Loading"],
    Cell[19670, 581, 208, 4, 62, "Input",
      CellTags->"Loading"],
    Cell[19881, 587, 245, 5, 75, "Text",
      CellTags->"Loading"],
    Cell[20129, 594, 300, 9, 61, "Input",
      CellTags->"Loading"],
    Cell[20432, 605, 633, 17, 75, "Text",
      Evaluatable->False,
      CellTags->"Loading"],
    Cell[21068, 624, 798, 25, 75, "Text",
      Evaluatable->False,
      CellTags->"Loading"],
    Cell[21869, 651, 215, 4, 62, "Input",
      CellTags->"Loading"],
    Cell[22087, 657, 825, 21, 97, "Text",
      Evaluatable->False,
      CellTags->"Loading"]},
  "Palettes"->{
    Cell[22949, 683, 170, 4, 114, "Section",
      CellTags->"Palettes"],
    Cell[23122, 689, 407, 12, 53, "Text",
      Evaluatable->False,
      CellTags->"Palettes"]},
  "Examples"->{
    Cell[23566, 706, 272, 7, 114, "Section",
      Evaluatable->False,
      CellTags->"Examples"],
    Cell[23841, 715, 364, 10, 53, "Text",
      Evaluatable->False,
      CellTags->"Examples"],
    Cell[24208, 727, 348, 11, 61, "Input",
      CellTags->"Examples"],
    Cell[24581, 742, 125, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[24709, 747, 254, 7, 59, "Input",
      CellTags->"Examples"],
    Cell[24966, 756, 239, 6, 59, "Input",
      CellTags->"Examples"],
    Cell[25208, 764, 185, 6, 62, "Input",
      CellTags->"Examples"],
    Cell[25396, 772, 209, 7, 62, "Input",
      CellTags->"Examples"],
    Cell[25608, 781, 166, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[25777, 787, 166, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[25980, 796, 112, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[26095, 801, 407, 10, 85, "Input",
      CellTags->"Examples"],
    Cell[26505, 813, 180, 5, 59, "Input",
      CellTags->"Examples"],
    Cell[26722, 823, 129, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[26854, 828, 362, 8, 85, "Input",
      CellTags->"Examples"],
    Cell[27219, 838, 157, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[27379, 844, 203, 7, 62, "Input",
      CellTags->"Examples"],
    Cell[27619, 856, 124, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[27746, 861, 381, 8, 85, "Input",
      CellTags->"Examples"],
    Cell[28130, 871, 157, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[28290, 877, 193, 6, 62, "Input",
      CellTags->"Examples"],
    Cell[28520, 888, 116, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[28639, 893, 332, 8, 110, "Input",
      CellTags->"Examples"],
    Cell[28974, 903, 155, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[29132, 909, 191, 6, 62, "Input",
      CellTags->"Examples"],
    Cell[29326, 917, 206, 7, 62, "Input",
      CellTags->"Examples"],
    Cell[29569, 929, 109, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[29681, 934, 262, 7, 59, "Input",
      CellTags->"Examples"],
    Cell[29946, 943, 149, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[30098, 949, 160, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[30295, 958, 128, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[30426, 963, 437, 10, 59, "Input",
      CellTags->"Examples"],
    Cell[30866, 975, 147, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[31016, 981, 190, 6, 62, "Input",
      CellTags->"Examples"],
    Cell[31209, 989, 142, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[31354, 995, 189, 6, 62, "Input",
      CellTags->"Examples"],
    Cell[31580, 1006, 114, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[31697, 1011, 486, 11, 85, "Input",
      CellTags->"Examples"],
    Cell[32186, 1024, 168, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[32357, 1030, 197, 6, 59, "Input",
      CellTags->"Examples"],
    Cell[32591, 1041, 105, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[32699, 1046, 314, 8, 85, "Input",
      CellTags->"Examples"],
    Cell[33016, 1056, 199, 6, 59, "Input",
      CellTags->"Examples"],
    Cell[33218, 1064, 169, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[33424, 1073, 114, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[33541, 1078, 224, 7, 59, "Input",
      CellTags->"Examples"],
    Cell[33768, 1087, 179, 5, 59, "Input",
      CellTags->"Examples"],
    Cell[33950, 1094, 152, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[34105, 1100, 166, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[34308, 1109, 142, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[34453, 1114, 461, 11, 85, "Input",
      CellTags->"Examples"],
    Cell[34917, 1127, 153, 4, 62, "Input",
      CellTags->"Examples"],
    Cell[35073, 1133, 223, 6, 59, "Input",
      CellTags->"Examples"],
    Cell[35333, 1144, 119, 3, 77, "Subsection",
      CellTags->"Examples"],
    Cell[35455, 1149, 333, 10, 85, "Input",
      CellTags->"Examples"],
    Cell[35791, 1161, 196, 6, 59, "Input",
      CellTags->"Examples"],
    Cell[35990, 1169, 165, 4, 62, "Input",
      CellTags->"Examples"]},
  "Scope"->{
    Cell[36204, 1179, 235, 5, 114, "Section",
      Evaluatable->False,
      CellTags->"Scope"],
    Cell[36442, 1186, 497, 13, 75, "Text",
      Evaluatable->False,
      CellTags->"Scope"],
    Cell[36942, 1201, 493, 12, 75, "Text",
      Evaluatable->False,
      CellTags->"Scope"],
    Cell[37438, 1215, 817, 17, 119, "Text",
      Evaluatable->False,
      CellTags->"Scope"],
    Cell[38258, 1234, 657, 14, 97, "Text",
      Evaluatable->False,
      CellTags->"Scope"],
    Cell[38918, 1250, 958, 18, 141, "Text",
      Evaluatable->False,
      CellTags->"Scope"],
    Cell[39879, 1270, 560, 13, 75, "Text",
      Evaluatable->False,
      CellTags->"Scope"]},
  "Acknowledgements"->{
    Cell[40476, 1288, 175, 4, 114, "Section",
      Evaluatable->False,
      CellTags->"Acknowledgements"],
    Cell[40654, 1294, 733, 20, 75, "Text",
      Evaluatable->False,
      CellTags->"Acknowledgements"]},
  "References"->{
    Cell[41424, 1319, 277, 8, 114, "Section",
      Evaluatable->False,
      CellTags->"References"],
    Cell[41704, 1329, 2181, 49, 463, "Text",
      Evaluatable->False,
      CellTags->"References"]},
  "Contact"->{
    Cell[43922, 1383, 169, 4, 114, "Section",
      Evaluatable->False,
      CellTags->"Contact"],
    Cell[44094, 1389, 899, 23, 119, "Text",
      Evaluatable->False,
      CellTags->"Contact"]}
  }
*)

(*CellTagsIndex
CellTagsIndex->{
  {"Copyright", 46299, 1450},
  {"Introduction", 46515, 1457},
  {"Organization", 46945, 1470},
  {"Loading", 47872, 1497},
  {"Palettes", 48651, 1521},
  {"Examples", 48839, 1527},
  {"Scope", 53043, 1644},
  {"Acknowledgements", 53738, 1666},
  {"References", 53972, 1673},
  {"Contact", 54193, 1680}
  }
*)

(*NotebookFileOutline
Notebook[{
Cell[1754, 51, 188, 3, 66, "Input"],
Cell[1945, 56, 98, 2, 40, "Input"],
Cell[2046, 60, 203, 4, 66, "Input"],
Cell[2252, 66, 90, 2, 40, "Input"],
Cell[2345, 70, 335, 6, 92, "Input"],

Cell[CellGroupData[{
Cell[2705, 80, 247, 5, 114, "Section",
  Evaluatable->False,
  CellTags->"Copyright"],
Cell[2955, 87, 870, 23, 97, "Text",
  Evaluatable->False,
  CellTags->"Copyright"]
}, Open  ]],

Cell[CellGroupData[{
Cell[3862, 115, 386, 12, 114, "Section",
  Evaluatable->False,
  CellTags->"Introduction"],
Cell[4251, 129, 1131, 24, 141, "Text",
  Evaluatable->False,
  CellTags->"Introduction"],
Cell[5385, 155, 852, 21, 135, "Text",
  Evaluatable->False,
  CellTags->"Introduction"],
Cell[6240, 178, 1075, 28, 119, "Text",
  Evaluatable->False,
  CellTags->"Introduction"]
}, Open  ]],

Cell[CellGroupData[{
Cell[7352, 211, 266, 5, 114, "Section",
  Evaluatable->False,
  CellTags->"Organization"],
Cell[7621, 218, 451, 15, 53, "Text",
  Evaluatable->False,
  CellTags->"Organization"],
Cell[8075, 235, 714, 22, 107, "Text",
  Evaluatable->False,
  CellTags->"Organization"],
Cell[8792, 259, 409, 12, 52, "Text",
  CellTags->"Organization"],
Cell[9204, 273, 2768, 82, 347, "Text",
  Evaluatable->False,
  CellTags->"Organization"],
Cell[11975, 357, 420, 12, 52, "Text",
  CellTags->"Organization"],
Cell[12398, 371, 2795, 83, 347, "Text",
  Evaluatable->False,
  CellTags->"Organization"],
Cell[15196, 456, 420, 12, 52, "Text",
  CellTags->"Organization"],
Cell[15619, 470, 1964, 48, 285, "Text",
  Evaluatable->False,
  CellTags->"Organization"],
Cell[17586, 520, 406, 12, 52, "Text",
  CellTags->"Organization"],
Cell[17995, 534, 100, 2, 66, "Input"],
Cell[18098, 538, 709, 11, 170, "Input"]
}, Open  ]],

Cell[CellGroupData[{
Cell[18844, 554, 235, 5, 114, "Section",
  Evaluatable->False,
  CellTags->"Loading"],
Cell[19082, 561, 585, 18, 53, "Text",
  Evaluatable->False,
  CellTags->"Loading"],
Cell[19670, 581, 208, 4, 62, "Input",
  CellTags->"Loading"],
Cell[19881, 587, 245, 5, 75, "Text",
  CellTags->"Loading"],
Cell[20129, 594, 300, 9, 61, "Input",
  CellTags->"Loading"],
Cell[20432, 605, 633, 17, 75, "Text",
  Evaluatable->False,
  CellTags->"Loading"],
Cell[21068, 624, 798, 25, 75, "Text",
  Evaluatable->False,
  CellTags->"Loading"],
Cell[21869, 651, 215, 4, 62, "Input",
  CellTags->"Loading"],
Cell[22087, 657, 825, 21, 97, "Text",
  Evaluatable->False,
  CellTags->"Loading"]
}, Open  ]],

Cell[CellGroupData[{
Cell[22949, 683, 170, 4, 114, "Section",
  CellTags->"Palettes"],
Cell[23122, 689, 407, 12, 53, "Text",
  Evaluatable->False,
  CellTags->"Palettes"]
}, Open  ]],

Cell[CellGroupData[{
Cell[23566, 706, 272, 7, 114, "Section",
  Evaluatable->False,
  CellTags->"Examples"],
Cell[23841, 715, 364, 10, 53, "Text",
  Evaluatable->False,
  CellTags->"Examples"],
Cell[24208, 727, 348, 11, 61, "Input",
  CellTags->"Examples"],

Cell[CellGroupData[{
Cell[24581, 742, 125, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[24709, 747, 254, 7, 59, "Input",
  CellTags->"Examples"],
Cell[24966, 756, 239, 6, 59, "Input",
  CellTags->"Examples"],
Cell[25208, 764, 185, 6, 62, "Input",
  CellTags->"Examples"],
Cell[25396, 772, 209, 7, 62, "Input",
  CellTags->"Examples"],
Cell[25608, 781, 166, 4, 62, "Input",
  CellTags->"Examples"],
Cell[25777, 787, 166, 4, 62, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[25980, 796, 112, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[26095, 801, 407, 10, 85, "Input",
  CellTags->"Examples"],
Cell[26505, 813, 180, 5, 59, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[26722, 823, 129, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[26854, 828, 362, 8, 85, "Input",
  CellTags->"Examples"],
Cell[27219, 838, 157, 4, 62, "Input",
  CellTags->"Examples"],
Cell[27379, 844, 203, 7, 62, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[27619, 856, 124, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[27746, 861, 381, 8, 85, "Input",
  CellTags->"Examples"],
Cell[28130, 871, 157, 4, 62, "Input",
  CellTags->"Examples"],
Cell[28290, 877, 193, 6, 62, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[28520, 888, 116, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[28639, 893, 332, 8, 110, "Input",
  CellTags->"Examples"],
Cell[28974, 903, 155, 4, 62, "Input",
  CellTags->"Examples"],
Cell[29132, 909, 191, 6, 62, "Input",
  CellTags->"Examples"],
Cell[29326, 917, 206, 7, 62, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[29569, 929, 109, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[29681, 934, 262, 7, 59, "Input",
  CellTags->"Examples"],
Cell[29946, 943, 149, 4, 62, "Input",
  CellTags->"Examples"],
Cell[30098, 949, 160, 4, 62, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[30295, 958, 128, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[30426, 963, 437, 10, 59, "Input",
  CellTags->"Examples"],
Cell[30866, 975, 147, 4, 62, "Input",
  CellTags->"Examples"],
Cell[31016, 981, 190, 6, 62, "Input",
  CellTags->"Examples"],
Cell[31209, 989, 142, 4, 62, "Input",
  CellTags->"Examples"],
Cell[31354, 995, 189, 6, 62, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[31580, 1006, 114, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[31697, 1011, 486, 11, 85, "Input",
  CellTags->"Examples"],
Cell[32186, 1024, 168, 4, 62, "Input",
  CellTags->"Examples"],
Cell[32357, 1030, 197, 6, 59, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[32591, 1041, 105, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[32699, 1046, 314, 8, 85, "Input",
  CellTags->"Examples"],
Cell[33016, 1056, 199, 6, 59, "Input",
  CellTags->"Examples"],
Cell[33218, 1064, 169, 4, 62, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[33424, 1073, 114, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[33541, 1078, 224, 7, 59, "Input",
  CellTags->"Examples"],
Cell[33768, 1087, 179, 5, 59, "Input",
  CellTags->"Examples"],
Cell[33950, 1094, 152, 4, 62, "Input",
  CellTags->"Examples"],
Cell[34105, 1100, 166, 4, 62, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[34308, 1109, 142, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[34453, 1114, 461, 11, 85, "Input",
  CellTags->"Examples"],
Cell[34917, 1127, 153, 4, 62, "Input",
  CellTags->"Examples"],
Cell[35073, 1133, 223, 6, 59, "Input",
  CellTags->"Examples"]
}, Open  ]],

Cell[CellGroupData[{
Cell[35333, 1144, 119, 3, 77, "Subsection",
  CellTags->"Examples"],
Cell[35455, 1149, 333, 10, 85, "Input",
  CellTags->"Examples"],
Cell[35791, 1161, 196, 6, 59, "Input",
  CellTags->"Examples"],
Cell[35990, 1169, 165, 4, 62, "Input",
  CellTags->"Examples"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[36204, 1179, 235, 5, 114, "Section",
  Evaluatable->False,
  CellTags->"Scope"],
Cell[36442, 1186, 497, 13, 75, "Text",
  Evaluatable->False,
  CellTags->"Scope"],
Cell[36942, 1201, 493, 12, 75, "Text",
  Evaluatable->False,
  CellTags->"Scope"],
Cell[37438, 1215, 817, 17, 119, "Text",
  Evaluatable->False,
  CellTags->"Scope"],
Cell[38258, 1234, 657, 14, 97, "Text",
  Evaluatable->False,
  CellTags->"Scope"],
Cell[38918, 1250, 958, 18, 141, "Text",
  Evaluatable->False,
  CellTags->"Scope"],
Cell[39879, 1270, 560, 13, 75, "Text",
  Evaluatable->False,
  CellTags->"Scope"]
}, Open  ]],

Cell[CellGroupData[{
Cell[40476, 1288, 175, 4, 114, "Section",
  Evaluatable->False,
  CellTags->"Acknowledgements"],
Cell[40654, 1294, 733, 20, 75, "Text",
  Evaluatable->False,
  CellTags->"Acknowledgements"]
}, Open  ]],

Cell[CellGroupData[{
Cell[41424, 1319, 277, 8, 114, "Section",
  Evaluatable->False,
  CellTags->"References"],
Cell[41704, 1329, 2181, 49, 463, "Text",
  Evaluatable->False,
  CellTags->"References"]
}, Open  ]],

Cell[CellGroupData[{
Cell[43922, 1383, 169, 4, 114, "Section",
  Evaluatable->False,
  CellTags->"Contact"],
Cell[44094, 1389, 899, 23, 119, "Text",
  Evaluatable->False,
  CellTags->"Contact"]
}, Open  ]]
}
]
*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)