(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' 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). 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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[ 132115, 4571]*) (*NotebookOutlinePosition[ 133004, 4599]*) (* CellTagsIndexPosition[ 132960, 4595]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ SIAM Meeting, Montreal, Quebec, Canada\ \>", "Text", FontSize->72, FontWeight->"Bold", Background->RGBColor[0, 1, 1]], Cell["Thursday June 19, 2003", "Text", FontSize->36, FontWeight->"Bold"], Cell["Speaker: Earl Glen Whitehead, Jr. ", "Text", FontSize->36, FontWeight->"Bold"], Cell[CellGroupData[{ Cell["Boiling Point Models of Alkanes", "Section", FontSize->36, Background->RGBColor[1, 1, 0]], Cell[TextData[{ "Larry Geller (Expert Mathematical Modeling, 8:45 AM this morning) ", StyleBox["Data-driven modeling", FontVariations->{"Underline"->True}], " vs. Theory-driven modeling." }], "Text", FontSize->36, FontWeight->"Bold", Background->RGBColor[1, 0.450019, 0.74342]], Cell[CellGroupData[{ Cell["Publish by MATCH in January, 2003", "Subsection", FontSize->24, FontWeight->"Bold", Background->RGBColor[0, 1, 0]], Cell["\<\ Kimberly Jordan Burch, Diane Karen Wakefield, and Earl Glen Whitehead, Jr., Boiling point models of alkanes, MATCH \ (Communications in mathematical and in computer chemistry) 47 (2003) 25-52.\ \ \>", "Text", FontSize->36, FontWeight->"Bold"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Introduction.", "Section", FontSize->36, Background->RGBColor[1, 1, 0]], Cell[TextData[{ "The molecules called alkanes have the chemical formula ", Cell[BoxData[ \(TraditionalForm\`C\_n\)]], Cell[BoxData[ FormBox[ SubscriptBox["H", RowBox[{\(2 n\), "+", StyleBox["2", FontSlant->"Italic"]}]], TraditionalForm]]], ". Each carbon atom ", StyleBox["C", FontSlant->"Italic"], " has four chemical bonds (", StyleBox["deg(C) = 4", FontSlant->"Italic"], ") and each hydrogen atom H has one chemical bond (", StyleBox["deg(H) = 1", FontSlant->"Italic"], "). Because of these bonding properties of carbon and hydrogen, we know \ that each n-carbon alkane is a tree which is an acyclic graph containing ", StyleBox["3n+1", FontSlant->"Italic"], " edges (chemical bonds)." }], "Text", FontSize->36, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[{{n, alkanes}, {1, 1}, {2, 1}, {3, 1}, {4, 2}, {5, 3}, {6, 5}, {7, 9}, {8, 18}, {9, 35}, {10, 75}, {11, 159}, {12, 355}, {13, 802}, {14, 1858}, {15, 4347}}]\)], "Input", CellOpen->False], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"n", "alkanes"}, {"1", "1"}, {"2", "1"}, {"3", "1"}, {"4", "2"}, {"5", "3"}, {"6", "5"}, {"7", "9"}, {"8", "18"}, {"9", "35"}, {"10", "75"}, {"11", "159"}, {"12", "355"}, {"13", "802"}, {"14", "1858"}, {"15", "4347"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", FontSize->36] }, Open ]], Cell["\<\ The various alkanes can be characterized by their physical \ properties including boiling points. All boiling point data used in this \ paper is measure in degrees C (centigrade) at standard pressure (760 Torr). \ Indices from graph theory and from geometric structure, provide a basis for \ developing boiling point models.\ \>", "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Indices and Error Vector.", "Section", FontSize->36, Background->RGBColor[1, 1, 0]], Cell["\<\ Our notation for these indices is in agreement with the notation \ used by Todeschini and Consonni [Handbook for Molecular Descriptors, \ Wiley-VCH, Weinheim, Germany, 2000].\ \>", "Text", FontSize->36, FontWeight->"Bold", Background->RGBColor[0, 1, 1]], Cell[CellGroupData[{ Cell["Molecular weight", "Subsection", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "Molecular weight (", StyleBox["MW", FontSlant->"Italic"], ") is the weight in atomic mass units (", StyleBox["amu", FontSlant->"Italic"], ") of all the atoms in a given molecule. Let n denote the number of \ carbons in the given alkane, the molecular weight is given by the formula\n", StyleBox["MW(n) = 12.01115 n + 1.00797(2n+2)", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "For example, the molecular weight of 2-methylbutane is ", StyleBox["72.1514 amu since n = 5", FontSlant->"Italic"], ". 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A ", StyleBox["k-matching", FontSlant->"Italic"], " is a matching with exactly ", StyleBox["k", FontSlant->"Italic"], " edges. Let ", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " denote the number of k-matchings; the simple matching polynomial of a \ graph with ", StyleBox["p", FontSlant->"Italic"], " vertices is given by the formula\n", Cell[BoxData[ \(TraditionalForm\`\[Sum]\_\(k\ = \ 0\)\%\([p/2]\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^\(p - k\)\)]], "." }], "Text", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "For example, the simple matching polynomial for \n2-methylbutane is ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^5\)]], " + 4", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^4\)]], " + 2", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^3\)]], ". The Hosoya index is 7." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Wiener number", "Subsection", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "Wiener number (", StyleBox["W", FontSlant->"Italic"], ") is sum of the distances between all pairs of vertices in a graph. It \ can also be defined as\nW", StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`1\/2\)], FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\+i\)]], " ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\+j\)]], " ", Cell[BoxData[ FormBox[ SubscriptBox["d", StyleBox["ij", FontSlant->"Italic"]], TraditionalForm]]], " where ", Cell[BoxData[ FormBox[ SubscriptBox["d", StyleBox["ij", FontSlant->"Italic"]], TraditionalForm]]], " is the ", StyleBox["ij", FontSlant->"Italic"], StyleBox["th", FontSlant->"Italic", FontVariations->{"Underline"->True}], " entry in the distance matrix, ", StyleBox["D(G)", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"], Cell["For example, the distance matrix for 2-methylbutane is", "Text", FontSize->36, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[{{0, 1, 2, 3, 2}, {1, 0, 1, 2, 1}, {2, 1, 0, 1, 2}, {3, 2, 1, 0, 3}, {2, 1, 2, 3, 0}}]\)], "Input", CellOpen->False], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "2", "3", "2"}, {"1", "0", "1", "2", "1"}, {"2", "1", "0", "1", "2"}, {"3", "2", "1", "0", "3"}, {"2", "1", "2", "3", "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", FontSize->36] }, Open ]], Cell[TextData[{ "The Wiener number ", StyleBox["W", FontSlant->"Italic"], " is ", StyleBox["18", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Wiener Path numbers", "Subsection", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "Wiener ", "Path", " numbers are the numbers ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^i\)P\)]], " (", StyleBox["1 <= i <= 9", FontSlant->"Italic"], ") where ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^i\)P\)]], " is the number of pairs of vertices in the graph separated by ", StyleBox["i", FontSlant->"Italic"], " edges (chemical bonds). Again using the distance matrix, ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^i\)P\)]], " can be obtained by counting the number of times ", StyleBox["i", FontSlant->"Italic"], " appears in the upper triangular part of the matrix." }], "Text", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "For example, 2-methylbutane has\n", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^1\)P\)]], " = ", StyleBox["4", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^2\)P\)]], " = ", StyleBox["4", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)P\)]], " = ", StyleBox["2", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^4\)P\)]], " = ", StyleBox["0", FontSlant->"Italic"], ", . . . , ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^9\)P\)]], " = ", StyleBox["0.", FontSlant->"Italic"] }], "Text", FontSize->36, FontWeight->"Bold"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "2", "3", "2"}, {"1", "0", "1", "2", "1"}, {"2", "1", "0", "1", "2"}, {"3", "2", "1", "0", "3"}, {"2", "1", "2", "3", "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Methyl", "Subsection", FontSize->36, FontWeight->"Bold"], Cell["\<\ Methyl (Mth) is defined to be the number of degree one vertices \ which are adjacent to a vertex of degree three or greater. \ \>", "Text", FontSize->36, FontWeight->"Bold"], Cell["For example, 2,2-dimethylbutane has Mth = 3. ", "Text", FontSize->36, FontWeight->"Bold"], Cell[GraphicsData["PICT", "\<\ 1i@0802M0:h1NP0A0_l<0?ooool0W@000200005j0000[P000000002Q0<006S4^ <30`<20a;S0`<30PLfEdK6U^IGMYI7AX0:40`0013@00X@3001P^@0jGoo05d0hP3T0>T0jgoo 05h0h03R0>/0kGoo0600g`3P0>d0kWoo0680gP3O0>h0kgoo06L0gP3O0>h0kgoo 06T0g`3P0>d0kWoo06/0h03R0>/0kGoo06`0hP3T0>T0jgoo06d0i03YOomoo`0Y @04b000100X0802M0:h1NP1B05/1901^0CL0F00107`0G04U06d1=P1L0B/1<7oo 05d1:@4[0C01@0jGoo0280hP3T0>T0jgoo02<0h03R0>/0kGoo 02D0g`3P0>d0kWoo02L0gP3O0>h0kgoo02`0gP3O0>h0kgoo02h0g`3P0>d0kWoo 0300h03R0>/0kGoo0340hP3T0>T0jgoo0380i03YOomoo`0X02h0i04e00402P0P 09d0[P5j0580V`3M0:h0l01H0040O02L0=h0[@3_09`0i03YOol0W@3R0>@0j@3[ Ool0WP3P0>80j`3]Ool0X03O0>00k@3^Ool0XP3N0=l0kP3_Ool0Y`3N0=l0kP3_ Ool0Z@3O0>00k@3^Ool0Z`3P0>80j`3]Ool0[03R0>@0j@3[Ool0[@3T0>Uoogoo 02Yk0CH00:40`00J"], "Graphics", ImageSize->{283, 181.812}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell["Ethyl", "Subsection", FontSize->36, FontWeight->"Bold"], Cell["\<\ Ethyl (Eth) is defined to be the number of degree one vertices \ which are adjacent to a vertex of degree two which is adjacent to a vertex of \ degree three or greater. \ \>", "Text", FontSize->36, FontWeight->"Bold"], Cell["For example, 2,2-dimethylbutane has Eth = 1. ", "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Error Vector.", "Subsection", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "Error Vector (", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], StyleBox[", number of data points, standard devidation", FontSlant->"Italic"], ") where ", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], " is the coefficient of determination, the number of data points is the \ number of boiling point data used in the model, and the standard deviation is \ the square root of the sum of squares divided by the number of data points \ minus the number of indices used in the model." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Correlations between Wiener Path Numbers.", "Section", FontSize->36, Background->RGBColor[1, 1, 0]], Cell[TextData[{ "Wiener discusses a theory of alternate polarities and claims that the only \ significant Wiener Path Number is ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)P\)]], ". In this paper, we give the Pearson's sample correlation coefficients \ between ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^1\)P\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^2\)P\)]], ", . . . , and ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^9\)P\)]], ". To save space we give a part of this table of correlation \ coefficients." }], "Text", FontSize->36, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[{{_, \*"\"\<\!\(\[InvisiblePrefixScriptBase]\^2\)P\>\"", \*"\ \"\<\!\(\[InvisiblePrefixScriptBase]\^3\)P\>\"", \*"\"\<\!\(\ \[InvisiblePrefixScriptBase]\^4\)P\>\"", \*"\"\<\!\(\ \[InvisiblePrefixScriptBase]\^5\)P\>\"", \*"\"\<\!\(\ \[InvisiblePrefixScriptBase]\^6\)P\>\""}, {\*"\"\<\!\(\ \[InvisiblePrefixScriptBase]\^1\)P\>\"", 0.747, 0.665, 0.764, 0.730, 0.500}, {\*"\"\<\!\(\[InvisiblePrefixScriptBase]\^2\)P\>\"", _, 0.791, 0.692, 0.349, 0.037}, {\*"\"\<\!\(\[InvisiblePrefixScriptBase]\^3\)P\>\"", _, _, 0.764, 0.193, \(-0.166\)}, {\*"\"\<\!\(\[InvisiblePrefixScriptBase]\ \^4\)P\>\"", _, _, _, 0.429, \(-0.005\)}, \ {\*"\"\<\!\(\[InvisiblePrefixScriptBase]\^5\)P\>\"", _, _, _, _, 0.597}}]\)], "Input", CellOpen->False], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { "_", "\<\"\\!\\(\[InvisiblePrefixScriptBase]\\^2\\)P\"\>", \ "\<\"\\!\\(\[InvisiblePrefixScriptBase]\\^3\\)P\"\>", "\<\"\\!\\(\ \[InvisiblePrefixScriptBase]\\^4\\)P\"\>", "\<\"\\!\\(\ \[InvisiblePrefixScriptBase]\\^5\\)P\"\>", "\<\"\\!\\(\ \[InvisiblePrefixScriptBase]\\^6\\)P\"\>"}, {"\<\"\\!\\(\[InvisiblePrefixScriptBase]\\^1\\)P\"\>", "0.747`", "0.665`", "0.764`", "0.73`", "0.5`"}, {"\<\"\\!\\(\[InvisiblePrefixScriptBase]\\^2\\)P\"\>", "_", "0.791`", "0.692`", "0.349`", "0.037`"}, {"\<\"\\!\\(\[InvisiblePrefixScriptBase]\\^3\\)P\"\>", "_", "_", "0.764`", "0.193`", \(-0.166`\)}, {"\<\"\\!\\(\[InvisiblePrefixScriptBase]\\^4\\)P\"\>", "_", "_", "_", "0.429`", \(-0.005`\)}, {"\<\"\\!\\(\[InvisiblePrefixScriptBase]\\^5\\)P\"\>", "_", "_", "_", "_", "0.597`"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", FontSize->24] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Regression Techniques.", "Section", FontSize->36, Background->RGBColor[1, 1, 0]], Cell["\<\ The objective functions for our models were written in AMPL which \ is a modeling language for mathematical programming. We solved these \ mathematical programming problems over the Internet using NEOS solvers \ available through Argonne National Laboratory. We used the FILTER solver \ which is one of the NEOS solvers.\ \>", "Text", FontSize->36, FontWeight->"Bold"], Cell["\<\ Mathematical programming algorithms such as FILTER do not stay in \ the feasible region when optimizing the objective function. This means that \ the resulting powers in the objective function may be slightly negative. \ This causes problems when data values are zero.\ \>", "Text", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "Many of the Wiener path numbers, the methyl indices, and the ethyl indices \ were zero for the alkanes we modelled. We therefore replaced these zero data \ values with ", Cell[BoxData[ \(TraditionalForm\`10\^\(-\ 4\)\)]], ". We then restored these zero data values and used the powers FILTER \ computed to determine the actual models using the Regress command in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["One Variable Models for Families of Alkanes.", "Section", FontSize->36, Background->RGBColor[1, 1, 0]], Cell[CellGroupData[{ Cell["Model 5.1", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "This model uses ", Cell[BoxData[ \(TraditionalForm\`MW\^0.345\)], FontSlant->"Italic"], " of the normal-alkanes (branchless chains) where 1 <= ", StyleBox["n", FontSlant->"Italic"], " <= 25.\n", Cell[BoxData[ \(TraditionalForm\`f\_5.1\)], FontSlant->"Italic"], StyleBox["(MW) = -464.424 + 115.349 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`MW\^0.345\)], FontSlant->"Italic"], ". The error \nvector is (0.999794, 25, 2.3)." }], "Text", FontSize->36, FontWeight->"Bold"], Cell["In Model 5.2, the error is reduced by changing the data set.", "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Model 5.2", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "This model uses ", Cell[BoxData[ \(TraditionalForm\`MW\^0.274\)], FontSlant->"Italic"], " of the normal-alkanes where 5 <= ", StyleBox["n", FontSlant->"Italic"], " <= 25.\n", Cell[BoxData[ \(TraditionalForm\`f\_5.2\)], FontSlant->"Italic"], StyleBox["(MW) = -642.362 + 210.137 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`MW\^0.274\)], FontSlant->"Italic"], ". \nThe error vector is (0.999863, 21, 1.3)." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Model 5.3", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "This model uses three powers of ", 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FontSlant->"Plain"], StyleBox["Hosoya", FontSlant->"Plain"], StyleBox[" ", FontSlant->"Plain"], StyleBox["index", FontSlant->"Plain"], StyleBox[" ", FontSlant->"Plain"], StyleBox["Z", FontSlant->"Plain"]}], TraditionalForm]], FontSlant->"Italic"], " of the normal-alkanes where 1 <= ", StyleBox["n", FontSlant->"Italic"], " <= 25.\n", Cell[BoxData[ \(TraditionalForm\`f\_5.4\)], FontSlant->"Italic"], StyleBox["(Z) = -163.231 + 1202.27 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\(\((ln\ Z)\)\^1.285\)\(\ \)\)\)], FontSlant->"Italic"], StyleBox["- 1800.57 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\((ln\ Z)\)\^1.385\ + \ 707.922\ \((ln\ Z)\)\^1.473\)], FontSlant->"Italic"], StyleBox[".", FontSlant->"Italic"], " \nThe error vector is (0.999879, 25, 1.8)." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Variations on Wiener's Model.", "Section", FontSize->36, Background->RGBColor[1, 1, 0]], Cell["\<\ Wiener stated that the equation in Model 6.1 was obtained by the \ method of least squares fitting to the boiling point data of 37 alkanes from \ C4 to C8. In this section we use boiling point data for the alkanes from C6 \ to C10. In Models 6.1 and 6.2, the normal-alkanes are not fitted because \[CapitalDelta]t = \[CapitalDelta]W = \[CapitalDelta]P = 0.\ \>", "Text", FontSize->36, FontWeight->"Bold"], Cell[CellGroupData[{ Cell["Model 6.1", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "This model fits \[CapitalDelta]t using the change in the Wiener number \ \[CapitalDelta]W, the number of carbon atoms n, and the change in the Wiener \ Path number ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)P\)]], ", denoted \[CapitalDelta]P. The formula is\n", Cell[BoxData[ \(TraditionalForm\`f\_6.1\)]], StyleBox["(\[CapitalDelta]W, n, \[CapitalDelta]P) = 98", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]W\/n\^2\)]], "+ 5.5 \[CapitalDelta]P.\nwhere ", StyleBox["\[CapitalDelta]W = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`W\_normal\)], FontSlant->"Italic"], StyleBox["- ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`W\_isomer\)], FontSlant->"Italic"], ", ", StyleBox["\[CapitalDelta]P = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`P\_normal\)], FontSlant->"Italic"], StyleBox["- ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`P\_isomer\)], FontSlant->"Italic"], ", and P is an abbreviation for ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)P\)]], ". \nThe error vector is ", StyleBox["(0.76492, 138, 3.1)", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "In 1947 and 1948, Wiener used the following model.\n ", StyleBox["\[CapitalDelta]t = 98 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]W\/n\^2\)], FontSlant->"Italic"], StyleBox["+ 5.5 \[CapitalDelta]P\n", FontSlant->"Italic"], "where ", StyleBox["\[CapitalDelta]t = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`t\_normal\)], FontSlant->"Italic"], StyleBox["- ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`t\_isomer\)], FontSlant->"Italic"], ". Model 6.3 was obtained by starting with Wiener's model and making \ several improvements." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Model 6.2", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell["\<\ Model 6.2 was obtained from Model 6.1 by changing the power on n \ from 2 to 2.21 and by allowing a nonzero intercept, namely 1.79342. \ \>", \ "Text", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "This model fits \[CapitalDelta]t by using the change in the Wiener number \ \[CapitalDelta]W, the number of carbon atoms n, and the change in the Wiener \ Path number ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)P\)]], ", denoted \[CapitalDelta]P. The formula is\n", Cell[BoxData[ \(TraditionalForm\`f\_6.2\)]], StyleBox["(\[CapitalDelta]W, n, \[CapitalDelta]P) = \n", FontSlant->"Italic"], "1.79342", StyleBox[" ", FontSlant->"Italic"], "+", StyleBox[" 136.908 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]W\/n\^2.21\)], FontSlant->"Italic"], " + 5.30056 ", StyleBox["\[CapitalDelta]P", FontSlant->"Italic"], "\nwhere ", StyleBox["\[CapitalDelta]W = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`W\_normal\)], FontSlant->"Italic"], StyleBox["- ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`W\_isomer\)], FontSlant->"Italic"], ", ", StyleBox["\[CapitalDelta]P = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`P\_normal\)], FontSlant->"Italic"], StyleBox["- ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`P\_isomer\)], FontSlant->"Italic"], ", and ", StyleBox["P", FontSlant->"Italic"], " is an abbreviation for ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)P\)]], ". \nThe error vector is ", StyleBox["(0.86914, 138, 2.3)", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Model 6.3", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell["\<\ The transition from Model 6.2 to Model 6.3 involved some algebra.\ \ \>", "Text", FontSize->36, FontWeight->"Bold"], Cell[TextData[{ "This model uses the Wiener number ", StyleBox["W", FontSlant->"Italic"], ", the number of carbon atoms ", StyleBox["n", FontSlant->"Italic"], ", and the Wiener path number ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)P\)]], ", denoted as ", StyleBox["P", FontSlant->"Italic"], ". The formula is\n", Cell[BoxData[ \(TraditionalForm\`f\_6.3\)], FontSlant->"Italic"], StyleBox["(W, n, P) = - 180.043 + 225.64 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`W\/n\^2.43001\)], FontSlant->"Italic"], StyleBox[" +\n5.24302 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`P\^1.00409\)]], StyleBox[" + 67.4542 ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\((ln\ n)\)\^1.15584\)]], StyleBox[" .", FontSlant->"Italic"], "\nThe error vector is ", StyleBox["(0.99351, 142, 2.2)", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Multivariable Models.", "Section", FontSize->36, Background->RGBColor[1, 1, 0]], Cell["\<\ The boiling point models in this section use data from the \ Beilstein and NIST databases.\ \>", "Text", FontSize->36, FontWeight->"Bold"], Cell[CellGroupData[{ Cell["Model 7.1", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "This model uses the Wiener Path numbers ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^i\)P\)]], " (1<= i <= 8), and the Mth index. The formula is\n", Cell[BoxData[ \(TraditionalForm\`f\_7.1\)]], StyleBox["(", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^1\)P\)]], StyleBox[", ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^2\)P\)]], StyleBox[", . . . , ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^8\)P\)]], StyleBox[", Mth) = - 167.49997 ", FontSlant->"Italic"], "+ 84.28344 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^1\)P\), "TraditionalForm"], ")"}], "0.46072"], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], " + 15.94534 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^2\)P\), "TraditionalForm"], ")"}], "0.00348"], TraditionalForm]]], " + 17.42198 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^3\)P\), "TraditionalForm"], ")"}], "0.53517"], TraditionalForm]]], "\n+ 11.16515 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^4\)P\), "TraditionalForm"], ")"}], "0.00164"], TraditionalForm]]], " + 4.74582 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^5\)P\), "TraditionalForm"], ")"}], "0.00089"], TraditionalForm]]], "+ 5.23270 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^6\)P\), "TraditionalForm"], ")"}], "0.00143"], TraditionalForm]]], " + 6.67018 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^7\)P\), "TraditionalForm"], ")"}], "0.14687"], TraditionalForm]]], " + 5.27829 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^8\)P\), "TraditionalForm"], ")"}], "0.96677"], TraditionalForm]]], " - 5.53723 ", Cell[BoxData[ \(TraditionalForm\`\((Mth)\)\^0.00072\)]], ".\nThe error vector is ", StyleBox["(0.995322, 195, 4.3)", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Model 7.2", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "This model uses the Wiener Path numbers ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^i\)P\)]], " (1<= i <= 8), the Mth index, and Hosoya index Z. The formula is\n", Cell[BoxData[ \(TraditionalForm\`f\_7.2\)]], StyleBox["(", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^1\)P\)]], StyleBox[", ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^2\)P\)]], StyleBox[", . . . , ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^8\)P\)]], StyleBox[", Mth, Z) =\n - 157.39801 -", FontSlant->"Italic"], " 1.71578 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^1\)P\), "TraditionalForm"], ")"}], "2.06741"], TraditionalForm]]], StyleBox[" +", FontSlant->"Italic"], " 0.38684 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^2\)P\), "TraditionalForm"], ")"}], "1.98738"], TraditionalForm]]], " - 3.53937 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^3\)P\), "TraditionalForm"], ")"}], "0.73904"], TraditionalForm]]], "- 4.69814 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^4\)P\), "TraditionalForm"], ")"}], "0.04224"], TraditionalForm]]], " - \n 2.31936 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^5\)P\), "TraditionalForm"], ")"}], "0.85500"], TraditionalForm]]], "- 0.00004 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^6\)P\), "TraditionalForm"], ")"}], "5.13758"], TraditionalForm]]], " + \n 1.57097 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^7\)P\), "TraditionalForm"], ")"}], "0.57937"], TraditionalForm]]], " + 0.23048 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^8\)P\), "TraditionalForm"], ")"}], "2.70854"], TraditionalForm]]], " + \n 1.20993 ", Cell[BoxData[ \(TraditionalForm\`\((Mth)\)\^1.32159\ + \ 108.36449\ \((ln\ Z)\)\^1.00972\)]], ".\nThe error vector is ", StyleBox["(0.997648, 195, 3.0)", FontSlant->"Italic"], "." }], "Text", FontSize->36, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Model 7.3", "Subsection", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "This model uses the Wiener Path numbers ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^i\)P\)]], " (1<= i <= 6), the Mth index, and Hosoya index Z. The formula is\n", Cell[BoxData[ \(TraditionalForm\`f\_4.3\)], FontSlant->"Italic"], StyleBox["(", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^1\)P\)]], StyleBox[", ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^2\)P\)]], StyleBox[", . . . , ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^6\)P\)]], StyleBox[", Mth, Z) = \n874.41474 + 221", FontSlant->"Italic"], ".61698 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^1\)P\), "TraditionalForm"], ")"}], "0.49420"], TraditionalForm]]], StyleBox[" -\n", FontSlant->"Italic"], "1182.20853 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^2\)P\), "TraditionalForm"], ")"}], "0.03689"], TraditionalForm]]], " + 0.00125 ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\(\[InvisiblePrefixScriptBase]\^3\)P\), 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