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\>"], "HeaderCell",
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  Evaluatable->False,
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Cell[TextData[{
  "Creating New Notations in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"]
}], "Title"],

Cell["Jason Harris", "Author",
  Evaluatable->False,
  FontFamily->"Times",
  FontSize->16,
  FontWeight->"Plain",
  FontSlant->"Italic"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " 3.0 and 4.0 understand most standard mathematical notations such as \
integrals, sums, products, etc. But in many mathematical fields, particularly \
pure mathematics, a variety of additional specific notations are used. The \
programmability of input and output in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " makes it possible to introduce these kinds of notations, and to integrate \
them into your work."
}], "Text"],

Cell[TextData[{
  "The ",
  StyleBox["Notation",
    FontSlant->"Italic"],
  " package provides a variety of utilities allowing you to flexibly and \
extensively create and add your own notations to a ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " session. The full documentation for the package can be found ",
  ButtonBox["here",
    ButtonData:>{"Notation Documentation"},
    ButtonStyle->"AddOnsLink"],
  ". Without the ",
  StyleBox["Notation",
    FontSlant->"Italic"],
  " package creating and defining your own notations can be laborious and \
problematic."
}], "Text"],

Cell[TextData[{
  "This loads the ",
  StyleBox["Notation",
    FontSlant->"Italic"],
  " package  and launches a palette for setting up notations."
}], "Commentary"],

Cell[BoxData[
    \(<< Utilities`Notation`\)], "InputOnly",
  CellLabel->"In[1]:="]
}, Closed]],

Cell[CellGroupData[{

Cell["A Simple Example", "Section"],

Cell[TextData[{
  "The following declares a new notation for a function ",
  StyleBox["gplus", "MR"],
  "."
}], "Commentary"],

Cell[BoxData[
    RowBox[{"Notation", "[", 
      RowBox[{
        TagBox[\(\(\ \)\(x_\(\[CirclePlus]\_n_\)y_\)\(\ \)\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle",
          Editable->True], " ", "\[DoubleLongLeftRightArrow]", " ", 
        TagBox[\(\(\ \)\(gplus[x_, y_, n_]\)\(\ \)\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle",
          Editable->True]}], "]"}]], "InputOnly",
  CellLabel->"In[2]:="],

Cell[TextData[{
  "Any input matching  ",
  Cell[BoxData[
      \(TraditionalForm\`x_\ \(\[CirclePlus]\_n_\ y_\)\)]],
  " is now interpreted as ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          StyleBox["gplus",
            "MR"], 
          StyleBox["[",
            "MR"], 
          RowBox[{
            StyleBox["x",
              "MR",
              FontSlant->"Plain"], 
            StyleBox[",",
              "MR"], 
            StyleBox["y",
              "MR",
              FontSlant->"Plain"], 
            StyleBox[",",
              "MR"], 
            StyleBox["n",
              "MR",
              FontSlant->"Plain"]}], 
          StyleBox["]",
            "MR"]}], TraditionalForm]]],
  "."
}], "Commentary"],

Cell[CellGroupData[{

Cell[BoxData[
    \(a\(\[CirclePlus]\_\(n + m\)\)b\  // \ FullForm\)], "Input",
  CellLabel->"In[3]:="],

Cell[BoxData[
    TagBox[
      StyleBox[\(gplus[a, b, Plus[m, n]]\),
        ShowStringCharacters->True],
      FullForm]], "Output",
  CellLabel->"Out[3]="]
}, Open  ]],

Cell[TextData[{
  "Any ",
  StyleBox["gplus", "MR"],
  " expression is now formatted in the new notation."
}], "Commentary"],

Cell[CellGroupData[{

Cell[BoxData[
    \(gplus[\[Alpha], \ \[Beta], \ \[CapitalGamma]]\)], "Input",
  CellLabel->"In[4]:="],

Cell[BoxData[
    \(\(\(\ \)\(\[Alpha]\(\[CirclePlus]\_\[CapitalGamma]\)\[Beta]\)\(\ \
\)\)\)], "Output",
  CellLabel->"Out[4]="]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[{
  "Notation for the Wigner 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " Symbol"
}], "Section"],

Cell[TextData[{
  "The Wigner 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " Symbol is used in the coupling of angular momenta in quantum mechanics. \
The 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbol is a more symmetric form of a corresponding Clebsch-Gordon \
coefficient. This section creates a notation for 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbols and illustrates the use of this notation with a few simple \
examples."
}], "Text"],

Cell["Define the notation for the Wigner 3-j symbol.", "Commentary"],

Cell[BoxData[
    RowBox[{"Notation", "[", 
      RowBox[{
        TagBox[
          RowBox[{" ", 
            SubscriptBox[
              RowBox[{"(", GridBox[{
                    {"j1_", "j2_", "j3_"},
                    {"m1_", "m2_", "m3_"}
                    }], ")"}], \(3  \[ScriptJ]\)], " "}],
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle",
          Editable->True], " ", "\[DoubleLongLeftRightArrow]", " ", 
        TagBox[\(\(\ \)\(ThreeJSymbol[{j1_, m1_}, {j2_, m2_}, {j3_, 
              m3_}]\)\(\ \)\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle",
          Editable->True]}], "]"}]], "InputOnly",
  CellLabel->"In[5]:="],

Cell[TextData[{
  "This notation consists of a 3\[Cross]2 GridBox with round braces \
subscripted by 3",
  StyleBox["j",
    FontSlant->"Italic"],
  ". There are many other notational forms that could have been chosen \
including ones with hidden ",
  StyleBox["TagBoxes", "MR"],
  "."
}], "Text"],

Cell["This is a typical calculation of a Wigner 3-j symbol.", "Commentary"],

Cell[CellGroupData[{

Cell[BoxData[
    SubscriptBox[
      RowBox[{"(", GridBox[{
            {"j", \(j + 1\/2\), \(1\/2\)},
            {"m", \(\(-m\) - 1\/2\), \(1\/2\)}
            }], ")"}], \(3  \[ScriptJ]\)]], "Input",
  CellLabel->"In[6]:="],

Cell[BoxData[
    \(\(-\(\(\((\(-1\))\)\^\(\(-j\) + m\)\ \@\(1 + j + m\)\)\/\(\@\(1 + 2\ \
j\)\ \@\(2 + 2\ j\)\)\)\)\)], "Output",
  CellLabel->"Out[6]="]
}, Open  ]],

Cell[TextData[{
  "The notation for 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbols both parses and formats correctly."
}], "Commentary"],

Cell[CellGroupData[{

Cell[BoxData[
    \(HoldForm\ @\ 
      ThreeJSymbol[{j, m}, {j + 1/2, \(-m\) - 1/2}, {1/2, 1/2}]\)], "Input",
  CellLabel->"In[7]:="],

Cell[BoxData[
    TagBox[
      RowBox[{" ", 
        SubscriptBox[
          RowBox[{"(", GridBox[{
                {"j", \(j + 1\/2\), \(1\/2\)},
                {"m", \(\(-m\) - 1\/2\), \(1\/2\)}
                }], ")"}], \(3  \[ScriptJ]\)], " "}],
      HoldForm]], "Output",
  CellLabel->"Out[7]="]
}, Open  ]],

Cell[TextData[{
  "This notation for the Wigner 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbol can now be freely used in calculations, greatly increasing \
readability in certain cases. Furthermore, for ease of input we can add an \
input-alias for the Wigner 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbol to the current notebook."
}], "Text"],

Cell[TextData[{
  "Add an input alias for the Wigner 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbol."
}], "Commentary"],

Cell[BoxData[
    RowBox[{"AddInputAlias", "[", 
      RowBox[{
        TagBox[
          SubscriptBox[
            RowBox[{"(", GridBox[{
                  {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]"},
                  {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}
                  }], ")"}], \(3  \[ScriptJ]\)],
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], ",", "\"\<3j\>\""}], 
      "]"}]], "InputOnly",
  CellLabel->"In[8]:="],

Cell[TextData[{
  "A blank template of a Wigner 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbol can now be created by typing \[EscapeKey]3j\[EscapeKey] in any \
input cell."
}], "Text"],

Cell[TextData[{
  "The following is a short example illustrating an identity involving the \
Wigner 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbol."
}], "Text"],

Cell[TextData[{
  "This defines the vectors ",
  Cell[BoxData[
      \(u\&\[RightVector]\)]],
  ", ",
  Cell[BoxData[
      \(v\&\[RightVector]\)]],
  " and ",
  Cell[BoxData[
      \(w\&\[RightVector]\)]],
  "."
}], "Commentary"],

Cell[BoxData[{
    \(\(u\&\[RightVector] = {u\_x, u\_y, u\_z};\)\), "\n", 
    \(\(v\&\[RightVector] = {v\_x, v\_y, v\_z};\)\), "\n", 
    \(\(w\&\[RightVector] = {w\_x, w\_y, w\_z};\)\)}], "InputOnly",
  CellLabel->"In[9]:="],

Cell[TextData[{
  "This calculates the triple scalar product of ",
  Cell[BoxData[
      \(u\&\[RightVector]\)]],
  ", ",
  Cell[BoxData[
      \(v\&\[RightVector]\)]],
  ", and ",
  Cell[BoxData[
      \(w\&\[RightVector]\)]],
  "."
}], "Commentary"],

Cell[CellGroupData[{

Cell[BoxData[
    \(u\&\[RightVector] . v\&\[RightVector]\ \[Cross]\ w\&\[RightVector]\  // 
      Expand\)], "Input",
  CellLabel->"In[12]:="],

Cell[BoxData[
    \(\(-u\_z\)\ v\_y\ w\_x + u\_y\ v\_z\ w\_x + u\_z\ v\_x\ w\_y - 
      u\_x\ v\_z\ w\_y - u\_y\ v\_x\ w\_z + u\_x\ v\_y\ w\_z\)], "Output",
  CellLabel->"Out[12]="]
}, Open  ]],

Cell["\<\
There are many ways to represent spherical tensors. The following is a \
simple, though not general approach.\
\>", "Text"],

Cell[TextData[{
  "This defines the spherical tensor components of ",
  Cell[BoxData[
      \(u\&\[RightVector]\)]],
  ", ",
  Cell[BoxData[
      \(v\&\[RightVector]\)]],
  ", and ",
  Cell[BoxData[
      \(w\&\[RightVector]\)]],
  "."
}], "Commentary"],

Cell[BoxData[{
    \(u\_1 = \(-\(\(u\_x + \[ImaginaryI]\ u\_y\)\/\@2\)\); 
    u\_0 = u\_z; \(u\_\(-1\) = \(u\_x - \[ImaginaryI]\ u\_y\)\/\@2;\)\), "\n\
", 
    \(v\_1 = \(-\(\(v\_x + \[ImaginaryI]\ v\_y\)\/\@2\)\); 
    v\_0 = v\_z; \(v\_\(-1\) = \(v\_x - \[ImaginaryI]\ v\_y\)\/\@2;\)\), "\n\
", 
    \(w\_1 = \(-\(\(w\_x + \[ImaginaryI]\ w\_y\)\/\@2\)\); w\_0 = w\_z; 
    w\_\(-1\) = \(w\_x - \[ImaginaryI]\ w\_y\)\/\@2;\)}], "InputOnly",
  CellLabel->"In[13]:="],

Cell[TextData[{
  "This turns off the message that the ",
  StyleBox["Clebsch-Gordan", "MR"],
  " coefficient is unphysical."
}], "Commentary"],

Cell[BoxData[
    \(Off[ClebschGordan::phy]\)], "InputOnly",
  CellLabel->"In[16]:="],

Cell[TextData[{
  "This demonstrates an identity of the Wigner 3",
  StyleBox["j",
    FontSlant->"Italic"],
  " symbol."
}], "Commentary"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{
      RowBox[{"(", 
        RowBox[{\(u\&\[RightVector] . 
            v\&\[RightVector]\ \[Cross]\ w\&\[RightVector]\), "  ", "==", " ",
           
          RowBox[{\(-\[ImaginaryI]\), \(\@6\), 
            RowBox[{\(\[Sum]\+\(i = \(-1\)\)\%1\), 
              RowBox[{\(\[Sum]\+\(j = \(-1\)\)\%1\), 
                RowBox[{" ", 
                  RowBox[{\(\[Sum]\+\(k = \(-1\)\)\%1\), 
                    RowBox[{\(u\_i\), \(v\_j\), \(w\_k\), "  ", 
                      SubscriptBox[
                        RowBox[{"(", GridBox[{
                              {"1", "1", "1"},
                              {"i", "j", "k"}
                              }], ")"}], \(3  \[ScriptJ]\)]}]}]}]}]}]}]}], 
        " ", ")"}], "//", "Simplify"}]], "Input",
  CellLabel->"In[17]:="],

Cell[BoxData[
    \(True\)], "Output",
  CellLabel->"Out[17]="]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Bras, Kets and Operators in Quantum Mechanics", "Section"],

Cell["\<\
The following illustrates a notational system for the Dirac bra-ket notation \
used in quantum mechanics. \
\>", "Text"],

Cell[CellGroupData[{

Cell[TextData[{
  "Modify ",
  StyleBox["Parenthesize", "MR"],
  " and ",
  StyleBox["FullForm", "MR"]
}], "Subsubsection"],

Cell["\<\
The following code is used to fix the formatting of tag boxes and full form. \
This correction is not central to the points being illustrated in this \
notebook, but is necessary for correct formatting.\
\>", "Text"],

Cell[BoxData[{
    \(\(Unprotect\ @\ RowBox;\)\), "\n", 
    \(\(RowBox[{"\<(\>", 
            TagBox[\[ScriptA]\[ScriptR]\[ScriptG]_, 
              Ket, \[ScriptO]\[ScriptP]\[ScriptT]\[ScriptS]___], "\<)\>"}] \
:= \n\tTagBox[\[ScriptA]\[ScriptR]\[ScriptG], 
          Ket, \[ScriptO]\[ScriptP]\[ScriptT]\[ScriptS]];\)\), "\n", 
    \(\(RowBox[{"\<(\>", 
            TagBox[\[ScriptA]\[ScriptR]\[ScriptG]_, 
              Bra, \[ScriptO]\[ScriptP]\[ScriptT]\[ScriptS]___], "\<)\>"}] \
:= \n\tTagBox[\[ScriptA]\[ScriptR]\[ScriptG], 
          Bra, \[ScriptO]\[ScriptP]\[ScriptT]\[ScriptS]];\)\), "\n", 
    \(\(RowBox[{"\<(\>", 
            TagBox[\[ScriptA]\[ScriptR]\[ScriptG]_, 
              BraKet, \[ScriptO]\[ScriptP]\[ScriptT]\[ScriptS]___], "\<)\>"}] \
:= \n\tTagBox[\[ScriptA]\[ScriptR]\[ScriptG], 
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  ", ",
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  " and ",
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}], "Subsection"],

Cell[TextData[{
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  " and ",
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  "."
}], "Commentary"],

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Cell["\<\
The following notations for bras, kets, and brakets depend on specific styles \
that have been added to the style sheet of this notebook. These styles are \
necessary to give the bras, kets and brakets the correct visual and \
structural properties.\
\>", "Text"],

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The full forms of these new objects are uniform and intuitive.\
\>", "Commentary"],

Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell["Entering Bras and Kets", "Subsection"],

Cell["\<\
To enter complex templates consisting of tag boxes and other underlying \
structures it is desirable to add input aliases to the current notebook. \
Creating input aliases for bras and kets is a perfect illustration of this.\
\>", "Text"],

Cell["Create input aliases for bras, kets and brakets.", "Commentary"],

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Cell["\<\
Bras and kets can now be easily entered with  \[EscapeKey]bra\[EscapeKey]  \
and \[EscapeKey]ket\[EscapeKey].\
\>", "Commentary"],

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Cell["Angular Momentum", "Subsection"],

Cell["\<\
This subsection contains some simple definitions and calculations to \
illustrate angular momentum and raising and lowering operators in quantum \
mechanics.\
\>", "Text"],

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Cell["\<\
This defines how raising and lowering operators act on eigenstates of angular \
momentum.\
\>", "Commentary"],

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  "To progress further we must add some basic properties to our ",
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factored out of a ",
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Cell["Now our calculations work correctly.", "Commentary"],

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  "In angular momentum theory the operator ",
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  "For those that know enough about quantized angular momentum it is evident \
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Let us extend our implementation of angular momentum a little further and \
note that the eigenstates of angular momentum are orthogonal.\
\>", "Text"],

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Finally, for illustration let us calculate an expression involving products \
of operators between eigenstates.\
\>", "Text"],

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Cell["Category Theory Diagrams", "Section"],

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  ". The following example comes from the section on natural transformations \
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\
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    Evaluatable->False,
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  }, Closed]]
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  Cell[CellGroupData[{
  
  Cell["Styles for Headings", "Section"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Title"],
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    FontSize->14,
    FontSlant->"Italic",
    FontColor->GrayLevel[0]],
  
  Cell[StyleData["Subtitle", "Printout"],
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    ParagraphIndent->-85,
    FontSize->16]
  }, Closed]],
  
  Cell[CellGroupData[{
  
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    CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}},
    FontFamily->"Helvetica",
    FontSize->12,
    FontSlant->"Italic"],
  
  Cell[StyleData["Subsubtitle", "Printout"],
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    FontSize->14]
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  Cell[CellGroupData[{
  
  Cell[StyleData["Section"],
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    CellDingbat->None,
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    FontFamily->"Helvetica",
    FontSize->20,
    FontWeight->"Bold",
    FontColor->RGBColor[0.571389, 0.19675, 0.570504]],
  
  Cell[StyleData["Section", "Printout"],
    CellMargins->{{9, 0}, {2, 50}},
    CellFrameMargins->{{5, 4}, {0, 4}},
    FontSize->14,
    FontTracking->"Plain",
    FontColor->GrayLevel[0]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Subsection"],
    CellMargins->{{60, 0}, {8, 10}},
    CellGroupingRules->{"SectionGrouping", 40},
    PageBreakBelow->False,
    CounterIncrements->"Subsection",
    CounterAssignments->{{"Subsubsection", 0}},
    FontFamily->"Helvetica",
    FontSize->16,
    FontWeight->"Bold"],
  
  Cell[StyleData["Subsection", "Printout"],
    CellMargins->{{9, 0}, {6, 40}},
    FontSize->12]
  }, Closed]],
  
  Cell[CellGroupData[{
  
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    CellDingbat->"\[FilledSquare]",
    CellMargins->{{60, Inherited}, {8, 12}},
    CellGroupingRules->{"SectionGrouping", 50},
    PageBreakBelow->False,
    CellFrameLabelMargins->6,
    CounterIncrements->"Subsubsection",
    FontFamily->"Times",
    FontSize->13,
    FontWeight->"Bold"],
  
  Cell[StyleData["Subsubsection", "Printout"],
    CellMargins->{{22, 0}, {4, 20}},
    FontSize->11]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Styles for Body Text", "Section"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Text"],
    CellMargins->{{60, 20}, {10, 5}},
    LineSpacing->{1, 0},
    ParagraphSpacing->{0, 12},
    FontFamily->"Times"],
  
  Cell[StyleData["Text", "Printout"],
    CellMargins->{{9, 0}, {4, 4}},
    ParagraphSpacing->{0, 6}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Commentary"],
    CellMargins->{{60, 20}, {2, 8}},
    PageBreakBelow->False,
    TextJustification->1,
    LineSpacing->{1, 2},
    FontFamily->"Helvetica",
    FontSize->10,
    FontColor->RGBColor[0, 0, 0.4]],
  
  Cell[StyleData["Commentary", "Printout"],
    CellMargins->{{9, 0}, {4, 4}},
    LineSpacing->{1, 3},
    FontSize->8]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Author"],
    ShowCellBracket->False,
    CellMargins->{{60, 4}, {30, 10}},
    PageBreakBelow->False,
    CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}},
    FontFamily->"Helvetica",
    FontSize->12,
    FontSlant->"Italic"],
  
  Cell[StyleData["Author", "Printout"],
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    FontSize->14]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Styles for Input/Output", "Section"],
  
  Cell["\<\
The cells in this section define styles used for input and output to the \
kernel.  Be careful when modifying, renaming, or removing these styles, \
because the front end associates special meanings with these style names.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
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    Background->RGBColor[1, 0.700008, 0.4]],
  
  Cell[StyleData["Input", "Printout"],
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  Cell[CellGroupData[{
  
  Cell[StyleData["InputOnly"],
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    LinebreakAdjustments->{0.85, 2, 10, 0, 1},
    CounterIncrements->"Input",
    StyleMenuListing->None,
    FontWeight->"Bold",
    Background->RGBColor[1, 0.700008, 0.4]],
  
  Cell[StyleData["InputOnly", "Printout"],
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    CellMargins->{{37, 0}, {6, 6}},
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  Cell[StyleData["InlineInput"],
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  Cell[CellGroupData[{
  
  Cell[StyleData["Output"],
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    GeneratedCell->True,
    CellAutoOverwrite->True,
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  Cell[StyleData["Output", "Printout"],
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  Cell[CellGroupData[{
  
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    GeneratedCell->True,
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    CellFrameColor->RGBColor[0, 0.6, 0.6],
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    FormatType->InputForm,
    CounterIncrements->"Graphics",
    ImageSize->{250, 250},
    Background->RGBColor[1, 0.900008, 0.900008]],
  
  Cell[StyleData["Graphics", "Printout"],
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    ImageSize->{250, 250},
    ImageMargins->{{28, Inherited}, {Inherited, 0}}]
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  Cell[CellGroupData[{
  
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    StyleMenuListing->None,
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  Cell[CellGroupData[{
  
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in palette buttons.\
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Notebooks.\
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styles serve to make visible otherwise invisible tagboxes.\
\>", "Text",
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*)




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

