(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 159777, 4333] NotebookOptionsPosition[ 153691, 4126] NotebookOutlinePosition[ 154099, 4142] CellTagsIndexPosition[ 154056, 4139] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Example : Fish Diagram", "Title", CellChangeTimes->{{3.7228339539484987`*^9, 3.722834002350849*^9}, { 3.7228340597290087`*^9, 3.722834077093857*^9}, {3.723356536716668*^9, 3.723356552750582*^9}, {3.725016086179448*^9, 3.725016090737365*^9}, { 3.745323393243149*^9, 3.7453234024260883`*^9}}, ExpressionUUID -> "ae0f8528-0738-4422-8dc1-0fe67dae1a3d"], Cell[CellGroupData[{ Cell["Loading packages:", "Section", CellChangeTimes->{{3.722834386267066*^9, 3.722834429274197*^9}, { 3.74532337119553*^9, 3.745323380282126*^9}}, ExpressionUUID -> "d69a460b-6742-4dc9-909f-45091d067231"], Cell[BoxData[{ RowBox[{ RowBox[{"<<", "UF`"}], " "}], "\[IndentingNewLine]", RowBox[{"<<", "NDConvexHull`"}]}], "Input", CellChangeTimes->{{3.722834921427322*^9, 3.7228350044612303`*^9}, { 3.722835040327033*^9, 3.722835050997019*^9}, {3.72335280998143*^9, 3.723352822379871*^9}, {3.736779338342458*^9, 3.736779367196349*^9}, { 3.7414179754399843`*^9, 3.741417975983501*^9}, {3.745224593905141*^9, 3.7452245967463217`*^9}}, ExpressionUUID -> "cb7ceff6-81f7-4999-a81d-15861318efad"] }, Open ]], Cell[CellGroupData[{ Cell["Defining relevant functions:", "Section", CellChangeTimes->{{3.745323470777869*^9, 3.7453234862499123`*^9}, { 3.745323741499222*^9, 3.7453237423475103`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"getMul", "[", RowBox[{"exp_", ",", "vars_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "expArr", "}"}], ",", RowBox[{ RowBox[{"expArr", "=", RowBox[{"ConstantArray", "[", RowBox[{"0", ",", RowBox[{"Length", "[", "vars", "]"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"expArr", "=", RowBox[{ RowBox[{ RowBox[{"Exponent", "[", RowBox[{"exp", ",", RowBox[{"vars", "[", RowBox[{"[", "#", "]"}], "]"}]}], "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", "vars", "]"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"expArr", ",", "Module"}], "]"}]}]}], "]"}]}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"getCoordinates", "[", RowBox[{"listc_", ",", "listpID_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "}"}], ",", RowBox[{"retList", "=", RowBox[{ RowBox[{ RowBox[{"listc", "[", RowBox[{"[", RowBox[{"listpID", "[", RowBox[{"[", "#", "]"}], "]"}], "]"}], "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", "listpID", "]"}], "]"}]}]}]}], "]"}]}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"getNormal", "[", RowBox[{"cList_", ",", "restList_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "dimension", "}"}], ",", RowBox[{ RowBox[{"dimension", "=", RowBox[{ RowBox[{"cList", "[", RowBox[{"[", "1", "]"}], "]"}], "//", "Length"}]}], ";", "\[IndentingNewLine]", RowBox[{"nv", "=", RowBox[{ RowBox[{ RowBox[{"v", "[", "#", "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"dimension", "-", "1"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"nv", "=", RowBox[{"Prepend", "[", RowBox[{"nv", ",", "1"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"planeList", "=", RowBox[{ RowBox[{ RowBox[{"nv", ".", "#"}], "&"}], "/@", "cList"}]}], ";", "\[IndentingNewLine]", RowBox[{"belowList", "=", RowBox[{ RowBox[{ RowBox[{"nv", ".", "#"}], "&"}], "/@", "restList"}]}], ";", "\[IndentingNewLine]", RowBox[{"conditions1", "=", RowBox[{"Join", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"#", "\[Equal]", "c"}], "&"}], "/@", "planeList"}], ",", RowBox[{ RowBox[{ RowBox[{"#", "\[GreaterEqual]", "c"}], "&"}], "/@", "belowList"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"inst1", "=", RowBox[{"FindInstance", "[", RowBox[{"conditions1", ",", RowBox[{"Append", "[", RowBox[{ RowBox[{"Rest", "[", "nv", "]"}], ",", "c"}], "]"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"Length", "[", "inst1", "]"}], ">", "0"}], ",", RowBox[{"AppendTo", "[", RowBox[{ RowBox[{"inst1", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"surf", "\[Rule]", RowBox[{"-", "1"}]}]}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"nv", "=", RowBox[{ RowBox[{ RowBox[{"v", "[", "#", "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"dimension", "-", "1"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"nv", "=", RowBox[{"Prepend", "[", RowBox[{"nv", ",", "1"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"planeList", "=", RowBox[{ RowBox[{ RowBox[{"nv", ".", "#"}], "&"}], "/@", "cList"}]}], ";", "\[IndentingNewLine]", RowBox[{"belowList", "=", RowBox[{ RowBox[{ RowBox[{"nv", ".", "#"}], "&"}], "/@", "restList"}]}], ";", "\[IndentingNewLine]", RowBox[{"conditions2", "=", RowBox[{"Join", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"#", "\[Equal]", "c"}], "&"}], "/@", "planeList"}], ",", RowBox[{ RowBox[{ RowBox[{"#", "\[LessEqual]", "c"}], "&"}], "/@", "belowList"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"inst2", "=", RowBox[{"FindInstance", "[", RowBox[{"conditions2", ",", RowBox[{"Append", "[", RowBox[{ RowBox[{"Rest", "[", "nv", "]"}], ",", "c"}], "]"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"Length", "[", "inst2", "]"}], ">", "0"}], ",", RowBox[{ RowBox[{"AppendTo", "[", RowBox[{ RowBox[{"inst2", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"surf", "\[Rule]", "1"}]}], "]"}], ";"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"normal", "=", "Null"}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Length", "[", "inst1", "]"}], "\[NotEqual]", "0"}], "&&", RowBox[{ RowBox[{"Length", "[", "inst2", "]"}], "\[NotEqual]", "0"}]}], ",", RowBox[{ RowBox[{"normal", "=", RowBox[{"Union", "[", RowBox[{ RowBox[{"inst1", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"inst2", "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}]}], ";"}], ",", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"Length", "[", "inst1", "]"}], "\[NotEqual]", "0"}], ",", RowBox[{ RowBox[{"normal", "=", RowBox[{"inst1", "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], ",", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"Length", "[", "inst2", "]"}], "\[NotEqual]", "0"}], ",", RowBox[{ RowBox[{"normal", "=", RowBox[{"inst2", "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}]}], "]"}]}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"normal", ",", "Module"}], "]"}]}]}], "]"}]}], "\n", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"genCoordinatesNormal", "[", RowBox[{"surfPts_", ",", "allPts_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "}"}], ",", RowBox[{"(*", RowBox[{ "This", " ", "function", " ", "takes", " ", "the", " ", "set", " ", "of", " ", "points", " ", "on", " ", "the", " ", "surface", " ", "and", " ", "finds", " ", "the", " ", "normal"}], "*)"}], RowBox[{ RowBox[{"RegSurfPts", "=", RowBox[{"getCoordinates", "[", RowBox[{"allPts", ",", "surfPts"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"restregSurfList", "=", RowBox[{"Complement", "[", RowBox[{"allPts", ",", "RegSurfPts"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"normale", "=", RowBox[{"getNormal", "[", RowBox[{"RegSurfPts", ",", "restregSurfList"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"normale", ",", "Module"}], "]"}], ";"}]}], "]"}]}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"UniqueRegion", "[", RowBox[{"allRegions_", ",", "length_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "}"}], ",", RowBox[{ RowBox[{"arr", "=", RowBox[{"allRegions", "/.", RowBox[{"Null", "\[Rule]", RowBox[{"Sequence", "[", "]"}]}]}]}], ";", "\[IndentingNewLine]", RowBox[{"vectors", "=", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Array", "[", RowBox[{"v", ",", "length"}], "]"}], ",", "surf"}], "}"}], "//", "Flatten"}]}], ";", "\[IndentingNewLine]", RowBox[{"NumVectors", "=", RowBox[{"vectors", "/.", "arr"}]}], ";", "\[IndentingNewLine]", RowBox[{"scalings", "=", RowBox[{"DeleteDuplicates", "[", RowBox[{"arr", ",", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"t", "=", RowBox[{"(", RowBox[{"#", "-", "#2"}], ")"}]}], ")"}], "\[Equal]", RowBox[{ RowBox[{"t", "[", RowBox[{"[", "1", "]"}], "]"}], " ", RowBox[{"ConstantArray", "[", RowBox[{"1", ",", RowBox[{"Length", "[", "t", "]"}]}], "]"}]}]}], "&"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"scalings", ",", "Module"}], "]"}]}]}], "\[IndentingNewLine]", "]"}]}], "\[IndentingNewLine]"}], "\n", RowBox[{ RowBox[{ RowBox[{"getLOUF", "[", RowBox[{"U_", ",", "F_", ",", "scaling_", ",", "param_", ",", "surf_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "i", ",", "UVarList", ",", "FVarList", ",", "UScalings", ",", "FScalings", ",", "ULOTerms", ",", "FLOTerms", ",", "ULOExp", ",", "FLOExp", ",", "retArr"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"UVarList", "=", RowBox[{"U", "[", RowBox[{"[", "All", "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"surf", "=", RowBox[{ RowBox[{"1", "\[Rule]", RowBox[{"top", " ", "surface", " ", "and", " ", "surf"}]}], "=", RowBox[{ RowBox[{"-", "1"}], "\[Rule]", RowBox[{"bottom", " ", "surface"}]}]}]}], "*)"}], "\[IndentingNewLine]", RowBox[{"FVarList", "=", RowBox[{"F", "[", RowBox[{"[", "All", "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"UScalings", "=", RowBox[{"{", "0", "}"}]}], ";", "\[IndentingNewLine]", RowBox[{"FScalings", "=", RowBox[{"{", "0", "}"}]}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", RowBox[{"i", "\[LessEqual]", RowBox[{"Length", "[", "UVarList", "]"}]}], ",", RowBox[{"i", "++"}], ",", RowBox[{ RowBox[{"AppendTo", "[", RowBox[{"UScalings", ",", RowBox[{"getScaling", "[", RowBox[{"scaling", ",", RowBox[{"UVarList", "[", RowBox[{"[", "i", "]"}], "]"}], ",", "param"}], "]"}]}], "]"}], ";"}]}], "]"}], " ", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", RowBox[{"i", "\[LessEqual]", RowBox[{"Length", "[", "FVarList", "]"}]}], ",", RowBox[{"i", "++"}], ",", RowBox[{ RowBox[{"AppendTo", "[", RowBox[{"FScalings", ",", RowBox[{"getScaling", "[", RowBox[{"scaling", ",", RowBox[{"FVarList", "[", RowBox[{"[", "i", "]"}], "]"}], ",", "param"}], "]"}]}], "]"}], ";"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"UScalings", "=", RowBox[{"Rest", "[", "UScalings", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"FScalings", "=", RowBox[{"Rest", "[", "FScalings", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"", "]"}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", "UScalings"}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", "FScalings"}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ "find", " ", "the", " ", "min", " ", "in", " ", "each", " ", "list", " ", "and", " ", "then", " ", "write", " ", "out", " ", "the", " ", "leading", " ", "expression"}], "*)"}], RowBox[{"If", "[", RowBox[{ RowBox[{"surf", "\[Equal]", RowBox[{"-", "1"}]}], ",", RowBox[{ RowBox[{"ULOTerms", "=", RowBox[{"Position", "[", RowBox[{"UScalings", ",", RowBox[{"Min", "[", "UScalings", "]"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"FLOTerms", "=", RowBox[{"Position", "[", RowBox[{"FScalings", ",", RowBox[{"Min", "[", "FScalings", "]"}]}], "]"}]}], ";"}], ",", RowBox[{ RowBox[{"ULOTerms", "=", RowBox[{"Position", "[", RowBox[{"UScalings", ",", RowBox[{"Max", "[", "UScalings", "]"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"FLOTerms", "=", RowBox[{"Position", "[", RowBox[{"FScalings", ",", RowBox[{"Max", "[", "FScalings", "]"}]}], "]"}]}], ";"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"ULOExp", "=", RowBox[{"Total", "[", RowBox[{"Extract", "[", RowBox[{"U", ",", "ULOTerms"}], "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"FLOExp", "=", RowBox[{"Total", "[", RowBox[{"Extract", "[", RowBox[{"F", ",", "FLOTerms"}], "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"retArr", "=", RowBox[{"{", RowBox[{"ULOExp", ",", "FLOExp"}], "}"}]}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"retArr", ",", "Module"}], "]"}], ";"}]}], "\[IndentingNewLine]", "]"}]}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"getScaling", "[", RowBox[{"scaling_", ",", "exp_", ",", "param_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"scale", ",", "i", ",", "pow", ",", "temp", ",", "pos"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ "param", " ", "is", " ", "a", " ", "list", " ", "of", " ", "variables"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"scale", "=", "0"}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"Head", "[", "exp", "]"}], "\[Equal]", "Power"}], ",", RowBox[{ RowBox[{"pos", "=", RowBox[{ RowBox[{"Flatten", "[", RowBox[{"Position", "[", RowBox[{"param", ",", RowBox[{"exp", "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"pow", "=", RowBox[{"exp", "[", RowBox[{"[", "2", "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"scale", "+=", RowBox[{ RowBox[{"scaling", "[", RowBox[{"[", "pos", "]"}], "]"}], "*", "pow"}]}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"scale", ",", "Module"}], "]"}], ";"}]}], "]"}], " ", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", RowBox[{"i", "\[LessEqual]", RowBox[{"Length", "[", "scaling", "]"}]}], ",", RowBox[{"i", "++"}], ",", RowBox[{ RowBox[{"scale", "+=", RowBox[{ RowBox[{"Count", "[", RowBox[{"exp", ",", RowBox[{"param", "[", RowBox[{"[", "i", "]"}], "]"}]}], "]"}], "*", RowBox[{"scaling", "[", RowBox[{"[", "i", "]"}], "]"}]}]}], ";"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", RowBox[{"i", "\[LessEqual]", RowBox[{"Length", "[", "exp", "]"}]}], ",", RowBox[{"i", "++"}], ",", RowBox[{"(*", RowBox[{"Multiple", " ", "variables", " ", "present"}], "*)"}], RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"Head", "[", RowBox[{"Extract", "[", RowBox[{"exp", ",", "i"}], "]"}], "]"}], "\[Equal]", "Power"}], ",", RowBox[{"(*", RowBox[{"Power", " ", "found"}], "*)"}], RowBox[{ RowBox[{"temp", "=", RowBox[{"Extract", "[", RowBox[{"exp", ",", "i"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"pos", "=", RowBox[{ RowBox[{"Flatten", "[", RowBox[{"Position", "[", RowBox[{"param", ",", RowBox[{"temp", "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"pow", "=", RowBox[{"temp", "[", RowBox[{"[", "2", "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"scale", "+=", RowBox[{ RowBox[{"scaling", "[", RowBox[{"[", "pos", "]"}], "]"}], "*", "pow"}]}], ";"}]}], "]"}], ";"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"scale", ",", "Module"}], "]"}]}]}], "]"}]}], ";"}], "\n", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Scalecheck", "[", RowBox[{"Upoly_", ",", "Fpoly_", ",", "alphaparam_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "a", "}"}], ",", RowBox[{ RowBox[{"alphaCount", "=", RowBox[{"Length", "[", "alphaparam", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"AllSub", "=", RowBox[{"Subsets", "[", RowBox[{"alphaparam", ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"alphaCount", "-", "1"}]}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"AllTran", "=", RowBox[{"a", " ", "AllSub"}]}], ";", "\[IndentingNewLine]", RowBox[{"TabTran", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"AllSub", "[", RowBox[{"[", RowBox[{"Row", ",", "Column"}], "]"}], "]"}], "\[Rule]", RowBox[{"AllTran", "[", RowBox[{"[", RowBox[{"Row", ",", "Column"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"Row", ",", "1", ",", RowBox[{"Length", "[", "AllSub", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"Column", ",", "1", ",", RowBox[{"Length", "[", RowBox[{"AllSub", "[", RowBox[{"[", "Row", "]"}], "]"}], "]"}]}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"list1", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Upoly", "/.", RowBox[{"TabTran", "[", RowBox[{"[", "#", "]"}], "]"}]}], ")"}], "/", RowBox[{"(", RowBox[{"a", "^", RowBox[{"Exponent", "[", RowBox[{ RowBox[{"Upoly", "/.", RowBox[{"TabTran", "[", RowBox[{"[", "#", "]"}], "]"}]}], ",", "a"}], "]"}]}], ")"}]}], "===", "Upoly"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", "TabTran", "]"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"list2", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Fpoly", "/.", RowBox[{"TabTran", "[", RowBox[{"[", "#", "]"}], "]"}]}], ")"}], "/", RowBox[{"(", RowBox[{"a", "^", RowBox[{"Exponent", "[", RowBox[{ RowBox[{"Fpoly", "/.", RowBox[{"TabTran", "[", RowBox[{"[", "#", "]"}], "]"}]}], ",", "a"}], "]"}]}], ")"}]}], "===", "Fpoly"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", "TabTran", "]"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"list", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"list1", "[", RowBox[{"[", "#", "]"}], "]"}], "&&", RowBox[{"list2", "[", RowBox[{"[", "#", "]"}], "]"}]}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"1", ",", RowBox[{"Length", "[", "list1", "]"}]}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"TabTran", "[", RowBox[{"[", RowBox[{"Flatten", "@", RowBox[{"Position", "[", RowBox[{"list", ",", "True"}], "]"}]}], "]"}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"TrueLength", "=", RowBox[{"Length", "[", RowBox[{"Position", "[", RowBox[{"list", ",", "True"}], "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{ RowBox[{"TrueLength", "\[NotEqual]", "0"}], ",", RowBox[{ RowBox[{"Print", "[", "\"\\"", "]"}], ";"}], ",", RowBox[{"Print", "[", "\"\\"", "]"}]}], "]"}], ";"}]}], "\[IndentingNewLine]", "]"}]}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{ 3.745323493680318*^9, {3.745323527685009*^9, 3.745323544735964*^9}, { 3.7453235945686417`*^9, 3.745323606669588*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Algorithm: ASPIRE", "Section", CellChangeTimes->{{3.7453234340113077`*^9, 3.745323447402123*^9}, 3.745323498794016*^9}], Cell[CellGroupData[{ Cell["Finding U and F polynomial:", "Subsection", CellChangeTimes->{3.745323456578875*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ "Changes", " ", "have", " ", "been", " ", "made", " ", "to", " ", "ensure", " ", "that", " ", "U", " ", "polynomial", " ", "has", " ", "the", " ", "correct", " ", "positive", " ", "sign"}], "*)"}], "\[IndentingNewLine]", RowBox[{"uf", "=", RowBox[{"UF", "[", RowBox[{ RowBox[{"{", RowBox[{"k", ",", "l"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["m", "2"], "-", RowBox[{"l", "^", "2"}]}], ")"}], ",", RowBox[{"(", RowBox[{ RowBox[{"m", "^", "2"}], "-", RowBox[{ RowBox[{"(", RowBox[{"k", "-", "l"}], ")"}], "^", "2"}]}], ")"}], ",", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "P", " ", "k"}], "-", SuperscriptBox["k", "2"]}], ")"}], ",", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "p", " ", "k"}], "-", RowBox[{"k", "^", "2"}]}], ")"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"M", "^", "2"}], "\[Rule]", "1"}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "\[Rule]", RowBox[{"x", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", "^", "2"}], "\[Rule]", RowBox[{"M", "^", "2"}]}], ",", RowBox[{ RowBox[{"p", "^", "2"}], "\[Rule]", RowBox[{"m", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", " ", "p"}], "\[Rule]", RowBox[{"M", " ", "m"}]}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", "x"}], ",", RowBox[{ RowBox[{"m", " ", "M"}], "\[Rule]", "x"}]}], "}"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7228349238847847`*^9, 3.722834926112064*^9}, 3.722835100382756*^9, 3.723352892510958*^9, {3.723553865515687*^9, 3.723553872086929*^9}, {3.736229363484263*^9, 3.736229427243569*^9}, { 3.736229563415998*^9, 3.736229643893468*^9}, {3.736229757244121*^9, 3.736229804575006*^9}, {3.7362299019789248`*^9, 3.736229919220683*^9}, { 3.73622997896474*^9, 3.7362299803359823`*^9}, {3.7362300778006783`*^9, 3.736230147929124*^9}, {3.736230273204919*^9, 3.7362302972926826`*^9}, { 3.7362303396170063`*^9, 3.736230368468912*^9}, 3.7363456087480516`*^9, { 3.736345981857995*^9, 3.7363459874369698`*^9}, {3.736346874019369*^9, 3.736346891595656*^9}, {3.736347665439528*^9, 3.7363476671648684`*^9}, { 3.7363478131384487`*^9, 3.736347827088077*^9}, {3.73634939982653*^9, 3.7363494110735598`*^9}, {3.736350279329763*^9, 3.73635028018185*^9}, { 3.736779375229273*^9, 3.736779476362354*^9}, {3.736779525054439*^9, 3.736779630508673*^9}, {3.73677966478057*^9, 3.736779704795961*^9}, { 3.736779736965495*^9, 3.736779744050795*^9}, 3.736780165990054*^9, { 3.736780209140539*^9, 3.736780216592466*^9}, 3.736780246906234*^9, { 3.7414179574942703`*^9, 3.7414179860141373`*^9}, {3.7524179364375143`*^9, 3.7524179652194567`*^9}, {3.752903369070147*^9, 3.752903372259815*^9}, { 3.753983551017684*^9, 3.753983558362657*^9}}, ExpressionUUID -> "0a094a07-2941-4d08-84ed-13160c6dda4e"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", "2"}], "}"}]], "Output", CellChangeTimes->{ 3.7414179867623043`*^9, 3.741418731550791*^9, 3.741425551887168*^9, 3.744809164923111*^9, 3.74480924410168*^9, 3.744809283581683*^9, 3.744809344600088*^9, 3.744809377185514*^9, 3.7448099960085297`*^9, 3.744810098194186*^9, 3.7448101631070347`*^9, 3.744810485937624*^9, { 3.7448145828855133`*^9, 3.7448145948811626`*^9}, 3.745029098274027*^9, 3.7452251777296267`*^9, 3.745226159506935*^9, 3.745226222116439*^9, 3.745310362317052*^9, 3.745323697024385*^9, 3.74564865939117*^9, 3.74564912467341*^9, 3.7456492214552107`*^9, 3.745649312733914*^9, 3.745649419240309*^9, {3.745649718204455*^9, 3.745649739223373*^9}, { 3.7456506743160553`*^9, 3.745650685365304*^9}, 3.752417967974271*^9, 3.753983562500882*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Finding U and F polynomial:", "Subsection", CellChangeTimes->{{3.741418240203156*^9, 3.741418252856059*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ FormBox[ RowBox[{"U", "=", RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"F", "=", RowBox[{"uf", "[", RowBox[{"[", "2", "]"}], "]"}]}], TraditionalForm]}], "Input", CellChangeTimes->{{3.7228353898034077`*^9, 3.722835467534136*^9}}, ExpressionUUID -> "5650dc6d-7544-4443-b0d3-a682b39f8af9"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}]], "Output", CellChangeTimes->{ 3.722835414775017*^9, 3.722835469908021*^9, 3.722837981631256*^9, 3.722838066793128*^9, 3.722843024006673*^9, 3.722928327863433*^9, 3.722929169867307*^9, 3.722929366687823*^9, 3.722929885441618*^9, 3.72318971427802*^9, 3.723352907233295*^9, 3.723356513519358*^9, 3.723553895918307*^9, {3.7356363190959806`*^9, 3.735636348020101*^9}, 3.7356364285223207`*^9, 3.735636468185266*^9, 3.735636521674597*^9, 3.735636573928249*^9, 3.7356367202140923`*^9, 3.7357182420838413`*^9, 3.736227489121874*^9, {3.736227585641735*^9, 3.7362276236710873`*^9}, 3.736230191008759*^9, 3.73634557154879*^9, 3.7363461324936247`*^9, 3.7363469097758913`*^9, 3.7363494330831003`*^9, 3.7363495006284637`*^9, 3.736354865686178*^9, 3.7364136203567038`*^9, 3.736506426440218*^9, 3.736506612702271*^9, 3.736660503595605*^9, 3.736779844737092*^9, 3.736779994098705*^9, 3.736780236389187*^9, 3.741350336390373*^9, { 3.741352374424417*^9, 3.7413523822407837`*^9}, 3.741413718700581*^9, 3.741418731573535*^9, 3.741425551910157*^9, 3.7448091649375*^9, 3.744809244104903*^9, 3.7448092835852737`*^9, 3.744809344615467*^9, 3.744809377199396*^9, 3.744809996043206*^9, 3.744810098212088*^9, 3.744810163123653*^9, 3.744810485973558*^9, {3.7448145828890133`*^9, 3.744814594884541*^9}, 3.7450290983011227`*^9, 3.745225177741008*^9, 3.7452261595113707`*^9, 3.745226222149331*^9, 3.745310362330546*^9, 3.74532369705996*^9, 3.745648659414997*^9, 3.745649124705414*^9, 3.7456492214835567`*^9, 3.74564931276058*^9, 3.745649419267852*^9, { 3.745649718234261*^9, 3.7456497392519484`*^9}, {3.745650674320833*^9, 3.745650685405377*^9}, 3.752417972399439*^9, 3.7539835625269012`*^9}, ExpressionUUID -> "16275b45-ca39-4b12-9e08-6b2001abe07f"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}]], "Output", CellChangeTimes->{ 3.722835414775017*^9, 3.722835469908021*^9, 3.722837981631256*^9, 3.722838066793128*^9, 3.722843024006673*^9, 3.722928327863433*^9, 3.722929169867307*^9, 3.722929366687823*^9, 3.722929885441618*^9, 3.72318971427802*^9, 3.723352907233295*^9, 3.723356513519358*^9, 3.723553895918307*^9, {3.7356363190959806`*^9, 3.735636348020101*^9}, 3.7356364285223207`*^9, 3.735636468185266*^9, 3.735636521674597*^9, 3.735636573928249*^9, 3.7356367202140923`*^9, 3.7357182420838413`*^9, 3.736227489121874*^9, {3.736227585641735*^9, 3.7362276236710873`*^9}, 3.736230191008759*^9, 3.73634557154879*^9, 3.7363461324936247`*^9, 3.7363469097758913`*^9, 3.7363494330831003`*^9, 3.7363495006284637`*^9, 3.736354865686178*^9, 3.7364136203567038`*^9, 3.736506426440218*^9, 3.736506612702271*^9, 3.736660503595605*^9, 3.736779844737092*^9, 3.736779994098705*^9, 3.736780236389187*^9, 3.741350336390373*^9, { 3.741352374424417*^9, 3.7413523822407837`*^9}, 3.741413718700581*^9, 3.741418731573535*^9, 3.741425551910157*^9, 3.7448091649375*^9, 3.744809244104903*^9, 3.7448092835852737`*^9, 3.744809344615467*^9, 3.744809377199396*^9, 3.744809996043206*^9, 3.744810098212088*^9, 3.744810163123653*^9, 3.744810485973558*^9, {3.7448145828890133`*^9, 3.744814594884541*^9}, 3.7450290983011227`*^9, 3.745225177741008*^9, 3.7452261595113707`*^9, 3.745226222149331*^9, 3.745310362330546*^9, 3.74532369705996*^9, 3.745648659414997*^9, 3.745649124705414*^9, 3.7456492214835567`*^9, 3.74564931276058*^9, 3.745649419267852*^9, { 3.745649718234261*^9, 3.7456497392519484`*^9}, {3.745650674320833*^9, 3.745650685405377*^9}, 3.752417972399439*^9, 3.753983562528459*^9}, ExpressionUUID -> "16275b45-ca39-4b12-9e08-6b2001abe07f"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"AlphaVariables", "=", RowBox[{ RowBox[{ RowBox[{"x", "[", "#", "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", RowBox[{"Variables", "[", RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], "]"}], "]"}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.722835122745984*^9, 3.722835201204563*^9}, { 3.722835535178196*^9, 3.722835539626348*^9}, 3.7414182258763123`*^9}, ExpressionUUID -> "1eb5dae4-8c84-4500-85ef-0a0d5a49659f"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]], "Output", CellChangeTimes->{ 3.723553911373274*^9, {3.735636319102849*^9, 3.73563634802497*^9}, 3.735636428527575*^9, 3.735636468189513*^9, 3.735636521678685*^9, 3.735636573933803*^9, 3.735636720219359*^9, 3.735718242139551*^9, 3.736227489127722*^9, {3.736227585661251*^9, 3.7362276236763906`*^9}, 3.736230191013956*^9, 3.7363455715554*^9, 3.736346135652584*^9, 3.73634691525112*^9, 3.736349435801257*^9, 3.73634950295372*^9, 3.7363548686408167`*^9, 3.7364136231422443`*^9, 3.7365064288390093`*^9, 3.7365066127064734`*^9, 3.7366605035994864`*^9, 3.736779844812331*^9, 3.736779994117511*^9, 3.736780236407044*^9, 3.74135034225452*^9, { 3.74135237445571*^9, 3.741352382247058*^9}, 3.741413718705071*^9, 3.741418731577382*^9, 3.7414255519144917`*^9, 3.744809164942587*^9, 3.7448092441203547`*^9, 3.744809283617722*^9, 3.744809344619969*^9, 3.744809377204479*^9, 3.744809996060886*^9, 3.744810098216445*^9, 3.744810163128292*^9, 3.744810485979184*^9, {3.744814582921711*^9, 3.744814594921342*^9}, 3.745029098306343*^9, 3.7452251777456083`*^9, 3.7452261595431223`*^9, 3.7452262221543016`*^9, 3.745310362335401*^9, 3.745323697065691*^9, 3.745648659420248*^9, 3.7456491247111073`*^9, 3.74564922148874*^9, 3.745649312766118*^9, 3.745649419273643*^9, { 3.7456497182401*^9, 3.745649739257352*^9}, {3.7456506743542633`*^9, 3.745650685410452*^9}, 3.752417977814538*^9, 3.7539835625323143`*^9}, ExpressionUUID -> "f9420e7c-8d21-4d4e-b0c7-b35fa23579b3"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "Subtitle", CellChangeTimes->{{3.723189564940287*^9, 3.723189593230688*^9}, 3.74141827255377*^9}, ExpressionUUID -> "6d248a30-2f2e-4ad3-944e-980141a82664"], Cell[CellGroupData[{ Cell["\<\ Finding the derivatives of F w.r.t alpha parameters:\ \>", "Subsection", CellChangeTimes->{{3.741418280474955*^9, 3.7414183095950537`*^9}}], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["F", "1"], "=", RowBox[{"D", "[", RowBox[{"F", ",", RowBox[{"x", "[", "1", "]"}]}], "]"}]}], " ", ";"}], " "}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["F", "2"], "=", RowBox[{"D", "[", RowBox[{"F", ",", RowBox[{"x", "[", "2", "]"}]}], "]"}]}], ";"}], " "}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ SubscriptBox["F", "3"], "=", RowBox[{"D", "[", RowBox[{"F", ",", RowBox[{"x", "[", "3", "]"}]}], "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ SubscriptBox["F", "4"], "=", RowBox[{"D", "[", RowBox[{"F", ",", RowBox[{"x", "[", "4", "]"}]}], "]"}]}], ";"}], TraditionalForm]}], "Input", CellChangeTimes->{ 3.722836118962399*^9, {3.722836480522983*^9, 3.722836827348332*^9}, { 3.722836858115449*^9, 3.722836895554323*^9}, {3.722837107505892*^9, 3.7228371294570093`*^9}, {3.723352946539824*^9, 3.723353047017481*^9}, { 3.723553925954246*^9, 3.723553928380214*^9}, {3.745323613732642*^9, 3.745323621627714*^9}}, ExpressionUUID -> "ce6f112d-fe58-453e-8665-3576844d774e"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Finding the Grobner basis of the Landau equations:\ \>", "Subsection", CellChangeTimes->{{3.7414183703463717`*^9, 3.741418393482442*^9}}], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"Gb", "=", RowBox[{ RowBox[{"GroebnerBasis", "[", RowBox[{ RowBox[{"{", RowBox[{"F", ",", SubscriptBox["F", "1"], ",", SubscriptBox["F", "2"], ",", SubscriptBox["F", "3"], ",", SubscriptBox["F", "4"]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]}], "]"}], "//", "FullSimplify"}]}], TraditionalForm]], "Input", CellChangeTimes->{ 3.722837786198927*^9, {3.722837857323566*^9, 3.722837892422553*^9}, { 3.722837926543071*^9, 3.722837930883264*^9}, {3.723353060071512*^9, 3.723353114496337*^9}, {3.723553937482562*^9, 3.723553943891122*^9}}, ExpressionUUID -> "24550bfe-687b-4ecc-b3a7-371c6af373e8"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "3", "]"}], "+", RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "2", "]"}], "-", RowBox[{"x", "[", "4", "]"}]}], ")"}]}]}], ")"}]}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "3", "]"}], "-", RowBox[{"x", " ", RowBox[{"x", "[", "4", "]"}]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "3", "]"}], "-", RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "2", "]"}], "+", RowBox[{"x", "[", "4", "]"}]}], ")"}]}]}], ")"}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "3", "]"}], "+", RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "2", "]"}], "-", RowBox[{"x", "[", "4", "]"}]}], ")"}]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "2", "]"}], "+", RowBox[{"x", "[", "4", "]"}]}], ")"}]}]}], ")"}]}], ",", RowBox[{ SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"], "+", RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "x"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"x", " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "1", "]"}], "+", RowBox[{"x", "[", "2", "]"}]}], ")"}]}], "-", RowBox[{"x", "[", "3", "]"}]}], ")"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"x", " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ")"}]}]}], ",", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "1", "]"}], "+", RowBox[{"x", "[", "2", "]"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "3", "]"}], "-", RowBox[{"x", " ", RowBox[{"x", "[", "4", "]"}]}]}], ")"}]}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"x", "[", "1", "]"}], "+", RowBox[{"x", "[", "2", "]"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{"x", "[", "3", "]"}], "-", RowBox[{"x", " ", RowBox[{"x", "[", "4", "]"}]}]}], ")"}]}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"x", "[", "1", "]"}], "+", RowBox[{"x", "[", "2", "]"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "3", "]"}], "-", RowBox[{"x", " ", RowBox[{"x", "[", "4", "]"}]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "2", "]"}], "+", RowBox[{"x", "[", "3", "]"}], "+", RowBox[{ RowBox[{"(", RowBox[{"2", "+", "x"}], ")"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ")"}]}], ",", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], "-", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], ")"}]}], ",", RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], "-", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"x", "[", "3", "]"}], "-", RowBox[{"x", " ", RowBox[{"x", "[", "4", "]"}]}]}], ")"}]}]}], "}"}]], "Output", CellChangeTimes->{{3.7228378983948174`*^9, 3.722837931956873*^9}, 3.7228380018793592`*^9, 3.722838066897316*^9, 3.722843024219103*^9, 3.722928328480194*^9, 3.722929170433453*^9, 3.722929366763604*^9, 3.722929885580543*^9, 3.723189714994671*^9, 3.723353117435468*^9, 3.72335651366731*^9, 3.723553945403858*^9, {3.735636319249817*^9, 3.735636348181671*^9}, 3.735636428725937*^9, 3.735636468343289*^9, 3.735636521827306*^9, 3.735636574093093*^9, 3.735636720380892*^9, 3.735718242204273*^9, 3.736227489290917*^9, {3.736227585695882*^9, 3.736227623871866*^9}, 3.736230191142839*^9, 3.7363455716738443`*^9, 3.736346143490884*^9, 3.736346921175308*^9, 3.736349441060014*^9, 3.736349508148026*^9, 3.736354874010232*^9, 3.7364136286971684`*^9, 3.736506433521873*^9, 3.736506612863235*^9, 3.736660503750249*^9, 3.736779846401841*^9, 3.736779995539156*^9, 3.736780236781049*^9, 3.741350353414587*^9, {3.741352374492552*^9, 3.741352382404705*^9}, 3.741413718938838*^9, 3.741418731720421*^9, 3.741425552073921*^9, 3.744809165133679*^9, 3.744809244144568*^9, 3.7448092837844353`*^9, 3.744809344771377*^9, 3.744809377352077*^9, 3.744809996207041*^9, 3.744810098371204*^9, 3.744810163281456*^9, 3.744810486156219*^9, { 3.744814582978858*^9, 3.744814594966865*^9}, 3.745029098468398*^9, 3.7452251779335957`*^9, 3.745226159583675*^9, 3.745226222319347*^9, 3.745310362484408*^9, 3.745323697236058*^9, 3.745648659580944*^9, 3.7456491248790617`*^9, 3.7456492216516733`*^9, 3.745649312927095*^9, 3.7456494194352083`*^9, {3.745649718402214*^9, 3.745649739418769*^9}, { 3.745650674386414*^9, 3.745650685585577*^9}, 3.752417985554723*^9, 3.7539835627003317`*^9}, ExpressionUUID -> "e8039a35-7bd3-43f6-908e-6c664d010d30"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Finding the linear transformations from the analysis of Grobner basis \ elements:\ \>", "Subsection", CellChangeTimes->{{3.7414184138971148`*^9, 3.741418451769496*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{ SubscriptBox["T", "1"], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{"x", "[", "1", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", RowBox[{"x", "[", "2", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{"x", "[", "4", "]"}]}]}], "}"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ SubscriptBox["T", "2"], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{ RowBox[{"x", "[", "1", "]"}], "+", FractionBox[ RowBox[{"x", "[", "2", "]"}], "2"]}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{"x", "[", "4", "]"}]}]}], "}"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ SubscriptBox["T", "3"], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "1", "]"}], "2"]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", RowBox[{ RowBox[{"x", "[", "2", "]"}], "+", FractionBox[ RowBox[{"x", "[", "1", "]"}], "2"]}]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{"x", "[", "4", "]"}]}]}], "}"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ SubscriptBox["T", "4"], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{"x", "[", "1", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", RowBox[{ RowBox[{"x", "[", "2", "]"}], "+", FractionBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], "}"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ SubscriptBox["T", "5"], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{"x", "[", "1", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{ RowBox[{"x", "[", "4", "]"}], "+", FractionBox[ RowBox[{"x", "[", "2", "]"}], "2"]}]}]}], "}"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"T", "=", RowBox[{"{", RowBox[{ SubscriptBox["T", "1"], ",", SubscriptBox["T", "2"], ",", SubscriptBox["T", "3"], ",", SubscriptBox["T", "4"], ",", SubscriptBox["T", "5"]}], "}"}]}], TraditionalForm]}], "Input", CellChangeTimes->{{3.7228397371985683`*^9, 3.722839806812768*^9}, { 3.7228398447469397`*^9, 3.722840028894867*^9}, {3.723353627575676*^9, 3.723353667759509*^9}, {3.723353840146206*^9, 3.723354294372462*^9}, { 3.723554254765249*^9, 3.723554524806182*^9}}, ExpressionUUID -> "436ce362-346e-4fd9-baaa-ddb4eda48757"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{"x", "[", "1", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", RowBox[{"x", "[", "2", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{"x", "[", "4", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{ RowBox[{"x", "[", "1", "]"}], "+", FractionBox[ RowBox[{"x", "[", "2", "]"}], "2"]}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{"x", "[", "4", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "1", "]"}], "2"]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", RowBox[{ FractionBox[ RowBox[{"x", "[", "1", "]"}], "2"], "+", RowBox[{"x", "[", "2", "]"}]}]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{"x", "[", "4", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{"x", "[", "1", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", RowBox[{ RowBox[{"x", "[", "2", "]"}], "+", FractionBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{"x", "[", "1", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{ FractionBox[ RowBox[{"x", "[", "2", "]"}], "2"], "+", RowBox[{"x", "[", "4", "]"}]}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.722840031890696*^9, 3.72284302430401*^9, 3.722928328569485*^9, 3.722929170512382*^9, 3.72292936679013*^9, 3.722929885621268*^9, 3.72318971510907*^9, 3.723354296520091*^9, 3.723356513708697*^9, { 3.723554504348995*^9, 3.723554526171424*^9}, {3.7356363192627296`*^9, 3.735636348204144*^9}, 3.735636428775029*^9, 3.735636468362013*^9, 3.7356365218414593`*^9, 3.735636574117169*^9, 3.735636720422071*^9, 3.7357182422519627`*^9, 3.736227489321007*^9, {3.7362275857117853`*^9, 3.736227623901767*^9}, 3.736230191166092*^9, 3.736345571688816*^9, 3.736346360663072*^9, 3.736346925011094*^9, 3.736349443833365*^9, 3.736349513142264*^9, 3.736354876892453*^9, 3.736413632211318*^9, 3.7365064357734203`*^9, 3.736506612885407*^9, 3.736660503766713*^9, 3.736779846568619*^9, 3.736779995697904*^9, 3.736780236823932*^9, 3.741350361941778*^9, {3.741352374520849*^9, 3.741352382431251*^9}, 3.7414137189691257`*^9, 3.741418731738793*^9, 3.7414255520977573`*^9, 3.744809165149088*^9, 3.7448092441612043`*^9, 3.744809283814465*^9, 3.744809344816291*^9, 3.744809377366118*^9, 3.7448099962204723`*^9, 3.74481009841875*^9, 3.74481016329753*^9, 3.7448104861935987`*^9, { 3.744814583007802*^9, 3.744814595009549*^9}, 3.745029098492322*^9, 3.745225177947641*^9, 3.74522615961353*^9, 3.745226222389464*^9, 3.745310362539736*^9, 3.745323697270027*^9, 3.745648659602436*^9, 3.7456491249494257`*^9, 3.7456492217221813`*^9, 3.74564931299223*^9, 3.7456494194996967`*^9, {3.745649718429173*^9, 3.7456497394827557`*^9}, { 3.745650674407701*^9, 3.7456506856767063`*^9}, 3.752417991764358*^9, 3.753983562724661*^9}, ExpressionUUID -> "eb929d79-6583-4337-b322-59af3b4a521b"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Finding the G=U+F polynomials with the above transformations:\ \>", "Subsection", CellChangeTimes->{{3.741418483737836*^9, 3.741418536598106*^9}}], Cell["", "Text", CellChangeTimes->{{3.72293006770977*^9, 3.722930074631317*^9}, { 3.722930209352237*^9, 3.722930491448426*^9}, {3.722931831391452*^9, 3.722931850565148*^9}, 3.744810258291854*^9}, ExpressionUUID -> "00a5531c-aeab-4fcd-a9cc-377d60207f20"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"G", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"U", "+", "F"}], "/.", " ", "#"}], " ", "&"}], " ", "/@", " ", "T"}], " ", "//", " ", "Expand"}]}], ";"}], TraditionalForm]], "Input",\ CellChangeTimes->{{3.722840154992169*^9, 3.722840189918486*^9}, { 3.722840445652952*^9, 3.722840452420834*^9}, 3.722931863935188*^9, { 3.723354326243998*^9, 3.723354363677651*^9}, 3.723554554458895*^9, 3.73622829742093*^9}, ExpressionUUID -> "f1ca1d83-901a-47ee-9309-1d029e2171b3"] }, Open ]], Cell[CellGroupData[{ Cell["Finding the support of G polynomials:", "Subsection", CellChangeTimes->{{3.741418559194498*^9, 3.741418581866418*^9}, 3.745029277564086*^9}], Cell["", "Text", CellChangeTimes->{{3.722930541907394*^9, 3.722930565338584*^9}, { 3.72293059712787*^9, 3.722930897609713*^9}, {3.722930980082925*^9, 3.722931188695549*^9}, {3.722931617682831*^9, 3.722931622934147*^9}, 3.722931875647881*^9, 3.722936667323109*^9, {3.72318925857035*^9, 3.723189259595693*^9}, 3.74481024478799*^9}, ExpressionUUID -> "1a12cfc8-31d7-496b-bb04-f3ebe822aa3f"], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{"terms1", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"G", "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "#", "]"}], "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", RowBox[{"G", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], "]"}]}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"terms2", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"G", "[", RowBox[{"[", "2", "]"}], "]"}], "[", RowBox[{"[", "#", "]"}], "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", RowBox[{"G", "[", RowBox[{"[", "2", "]"}], "]"}], "]"}], "]"}]}]}], ";"}], TraditionalForm], "\n", FormBox[ RowBox[{ RowBox[{"terms3", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"G", "[", RowBox[{"[", "3", "]"}], "]"}], "[", RowBox[{"[", "#", "]"}], "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", RowBox[{"G", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}], "]"}]}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"terms4", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"G", "[", RowBox[{"[", "4", "]"}], "]"}], "[", RowBox[{"[", "#", "]"}], "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", RowBox[{"G", "[", RowBox[{"[", "4", "]"}], "]"}], "]"}], "]"}]}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ RowBox[{"terms5", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"G", "[", RowBox[{"[", "5", "]"}], "]"}], "[", RowBox[{"[", "#", "]"}], "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", RowBox[{"G", "[", RowBox[{"[", "5", "]"}], "]"}], "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"vecExp1", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"getMul", "[", RowBox[{ RowBox[{"terms1", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"Prepend", "[", RowBox[{"AlphaVariables", ",", "x"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", RowBox[{"Length", "[", "terms1", "]"}]}], "}"}]}], "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"vecExp2", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"getMul", "[", RowBox[{ RowBox[{"terms2", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"Prepend", "[", RowBox[{"AlphaVariables", ",", "x"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", RowBox[{"Length", "[", "terms2", "]"}]}], "}"}]}], "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"vecExp3", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"getMul", "[", RowBox[{ RowBox[{"terms3", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"Prepend", "[", RowBox[{"AlphaVariables", ",", "x"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", RowBox[{"Length", "[", "terms3", "]"}]}], "}"}]}], "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"vecExp4", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"getMul", "[", RowBox[{ RowBox[{"terms4", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"Prepend", "[", RowBox[{"AlphaVariables", ",", "x"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", RowBox[{"Length", "[", "terms4", "]"}]}], "}"}]}], "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"vecExp5", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"getMul", "[", RowBox[{ RowBox[{"terms5", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"Prepend", "[", RowBox[{"AlphaVariables", ",", "x"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", RowBox[{"Length", "[", "terms5", "]"}]}], "}"}]}], "]"}]}], ";"}], TraditionalForm]}], "Input", CellChangeTimes->{{3.72284031178369*^9, 3.7228403117848663`*^9}, { 3.722840363937941*^9, 3.722840401432641*^9}, {3.7228404644240093`*^9, 3.722840468458436*^9}, {3.7228405851220617`*^9, 3.722840609143409*^9}, { 3.7228406653121977`*^9, 3.722840672269648*^9}, {3.722841028958622*^9, 3.722841227573945*^9}, {3.7228412784446983`*^9, 3.7228412975394287`*^9}, { 3.722841355879231*^9, 3.7228413784988127`*^9}, {3.722841415808712*^9, 3.7228414501630297`*^9}, {3.722841706950453*^9, 3.722841722058918*^9}, { 3.722841790741061*^9, 3.722841891800474*^9}, {3.722841930696952*^9, 3.722842076272503*^9}, 3.722929201935166*^9, {3.722931911127454*^9, 3.722932031319484*^9}, {3.723354379097821*^9, 3.723354624437169*^9}, { 3.7356363969614162`*^9, 3.735636401440143*^9}, {3.745323637714281*^9, 3.745323640658998*^9}}, ExpressionUUID -> "21650b68-763a-4436-88fd-fc89f7a91801"] }, Open ]], Cell[CellGroupData[{ Cell["Finding the Newton polytopes:", "Subsection", CellChangeTimes->{{3.7414185984902687`*^9, 3.7414186224890537`*^9}}], Cell["", "Text", CellChangeTimes->{{3.722936720424366*^9, 3.72293677996474*^9}, { 3.722936860347929*^9, 3.722937128856309*^9}, 3.7450293116610413`*^9}, ExpressionUUID -> "caef5d13-b8e8-447d-b1e7-9c0855c6648d"], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{"NP1", "=", RowBox[{"CHNQuickHull", "[", "vecExp1", "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"NP2", "=", RowBox[{"CHNQuickHull", "[", "vecExp2", "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"NP3", "=", RowBox[{"CHNQuickHull", "[", "vecExp3", "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"NP4", "=", RowBox[{"CHNQuickHull", "[", "vecExp4", "]"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"NP5", "=", RowBox[{"CHNQuickHull", "[", "vecExp5", "]"}]}], ";"}], TraditionalForm]}], "Input", CellChangeTimes->{{3.722842265966642*^9, 3.722842365130946*^9}, { 3.723354635006041*^9, 3.723354669313227*^9}, {3.7356364055658703`*^9, 3.735636410076684*^9}, {3.735636533371415*^9, 3.7356365387973127`*^9}, { 3.736346937321815*^9, 3.73634694177483*^9}}, ExpressionUUID -> "dadf3903-4b68-46c2-8520-8c6327d98630"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Finding the normal vectors from the Newton polytopes:\ \>", "Subsection", CellChangeTimes->{{3.7414186485066557`*^9, 3.7414186714352827`*^9}}], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{"NormalReg", "=", "0"}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"regions", " ", "=", " ", RowBox[{"{", RowBox[{"NP1", ",", "NP2", ",", "NP3", ",", "NP4", ",", "NP5"}], "}"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"listPts", " ", "=", " ", RowBox[{"{", RowBox[{ "vecExp1", ",", "vecExp2", ",", "vecExp3", ",", "vecExp4", ",", "vecExp5"}], "}"}]}], ";"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"AllRegions", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"NormalReg", " ", "=", " ", RowBox[{"genCoordinatesNormal", "[", RowBox[{ RowBox[{"regions", "[", RowBox[{"[", RowBox[{"reg", ",", "2", ",", "surface"}], "]"}], "]"}], ",", RowBox[{"listPts", "[", RowBox[{"[", "reg", "]"}], "]"}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"reg", ",", "1", ",", RowBox[{"Length", "[", "regions", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"surface", ",", "1", ",", RowBox[{"Length", "[", RowBox[{ RowBox[{"regions", "[", RowBox[{"[", "reg", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], "]"}]}], "}"}]}], "]"}]}], ";"}], TraditionalForm]}], "Input", CellChangeTimes->{{3.722842724833445*^9, 3.722842797699345*^9}, { 3.722842866423526*^9, 3.7228428932945423`*^9}, {3.7228456568012457`*^9, 3.722845703771682*^9}, {3.722845954969739*^9, 3.722845965294837*^9}, { 3.722846187400817*^9, 3.722846228648136*^9}, {3.7228462590238457`*^9, 3.72284628231117*^9}, {3.722846630771235*^9, 3.722846641447033*^9}, { 3.722846743577209*^9, 3.722846796049019*^9}, {3.722850395595558*^9, 3.7228504077337093`*^9}, {3.7228504449088078`*^9, 3.722850446718305*^9}, { 3.7228530696290627`*^9, 3.722853137005579*^9}, {3.722853876600852*^9, 3.722853975395844*^9}, {3.7228540294379683`*^9, 3.722854075011505*^9}, { 3.722854157218721*^9, 3.72285419089915*^9}, 3.722867247372451*^9, { 3.722929236291676*^9, 3.72292927331601*^9}, {3.722929328105791*^9, 3.722929359915688*^9}, {3.722929414864398*^9, 3.722929443931379*^9}, { 3.722929651723666*^9, 3.722929658529353*^9}, 3.722929924342893*^9, { 3.723354685847056*^9, 3.723354706543962*^9}, 3.72335572945968*^9, 3.723554617140009*^9, 3.7356361952073402`*^9, {3.735636239238522*^9, 3.7356363394927397`*^9}, {3.735636453464355*^9, 3.7356364576989107`*^9}, { 3.735636497050445*^9, 3.7356365109622602`*^9}, 3.7356365636173162`*^9, { 3.735636636624177*^9, 3.73563669702312*^9}, {3.736227420679302*^9, 3.736227467565662*^9}, {3.736227596530821*^9, 3.736227614763626*^9}, { 3.7362281812086363`*^9, 3.736228210832589*^9}, {3.73622831421634*^9, 3.7362283191307163`*^9}, {3.736229959609495*^9, 3.736229960330411*^9}, 3.736346386643071*^9, {3.736347633819375*^9, 3.7363476560143127`*^9}, 3.7363479161479673`*^9, {3.7363479500729513`*^9, 3.736347990648054*^9}, { 3.736348049233798*^9, 3.736348078267968*^9}, {3.736350448657349*^9, 3.7363504489752398`*^9}, {3.736351562894649*^9, 3.736351723991246*^9}, { 3.736352209178067*^9, 3.736352227679021*^9}, {3.736352281142191*^9, 3.736352325247677*^9}, {3.736413693142253*^9, 3.736413699319018*^9}, { 3.736413858313674*^9, 3.7364138595660067`*^9}, {3.736413956658547*^9, 3.736413991531698*^9}, {3.7364141499621887`*^9, 3.736414181186857*^9}, { 3.736506463837138*^9, 3.736506463987649*^9}, 3.7414489585037518`*^9, 3.741449013918083*^9, {3.7448092635462017`*^9, 3.7448092750146112`*^9}, { 3.7448093275267143`*^9, 3.744809364006647*^9}, {3.744809975788425*^9, 3.744809983755307*^9}, {3.744810061226096*^9, 3.744810082581725*^9}, { 3.744810133276598*^9, 3.744810140731596*^9}, {3.7452250790226088`*^9, 3.745225154303043*^9}, {3.745323652322509*^9, 3.7453236767746143`*^9}, { 3.7456486089881372`*^9, 3.7456486142219057`*^9}, {3.745648647373575*^9, 3.7456486534058437`*^9}, {3.745649107595848*^9, 3.745649112314011*^9}, { 3.7456492042036657`*^9, 3.7456492102044*^9}, {3.745649295675919*^9, 3.7456493034832973`*^9}, {3.745649403766279*^9, 3.745649409020603*^9}, { 3.745649507462098*^9, 3.745649532731145*^9}}, ExpressionUUID -> "17f53719-5348-4f92-b336-f8e8a48fa318"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Selection of the unique regions :\ \>", "Subsection", CellChangeTimes->{{3.741418694379607*^9, 3.7414187185210123`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{"regAllsectors", "=", RowBox[{"Flatten", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"UniqueRegion", "[", RowBox[{ RowBox[{"AllRegions", "[", RowBox[{"[", "#", "]"}], "]"}], ",", RowBox[{"Length", "[", RowBox[{"Variables", "[", "AlphaVariables", "]"}], "]"}]}], "]"}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"Length", "[", "AllRegions", "]"}], "]"}]}], ",", "1"}], "]"}]}], ";"}], TraditionalForm], "\n", FormBox[ RowBox[{"Regions", "=", RowBox[{ RowBox[{"DeleteDuplicates", "[", RowBox[{"regAllsectors", ",", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"t", "=", RowBox[{"#", "-", "#2"}]}], ")"}], "\[Equal]", RowBox[{ RowBox[{"t", "[", RowBox[{"[", "1", "]"}], "]"}], "*", RowBox[{"ConstantArray", "[", RowBox[{"1", ",", RowBox[{"Length", "[", "t", "]"}]}], "]"}]}]}], "&"}]}], "]"}], "//", "MatrixForm"}]}], TraditionalForm]}], "Input", CellChangeTimes->{{3.722929636392969*^9, 3.722929638065078*^9}, { 3.722929670481582*^9, 3.722929672023403*^9}, {3.722929703332988*^9, 3.722929777560493*^9}, {3.722929815929144*^9, 3.722929823267868*^9}, 3.722929871030781*^9, {3.723356077190268*^9, 3.723356079265293*^9}, { 3.736351825806391*^9, 3.736351860052367*^9}, {3.736351891458617*^9, 3.7363519204133673`*^9}, {3.7363520595680647`*^9, 3.736352069585557*^9}, { 3.736352719398344*^9, 3.7363527218539886`*^9}, {3.736352873862075*^9, 3.736352898567492*^9}, {3.736353052632923*^9, 3.736353083547068*^9}, { 3.736354565100601*^9, 3.736354578418481*^9}, 3.736354665871067*^9, { 3.736354708873444*^9, 3.736354728095366*^9}, {3.7363547637352448`*^9, 3.7363547651565027`*^9}, 3.736354833508964*^9, {3.73635490543716*^9, 3.736354914638555*^9}, {3.736414141483656*^9, 3.736414142775543*^9}, { 3.741350378383891*^9, 3.7413503821429577`*^9}, {3.745310326080912*^9, 3.7453103513000593`*^9}, {3.74532365501991*^9, 3.745323673221223*^9}, { 3.7456496678231363`*^9, 3.745649703834058*^9}}, ExpressionUUID -> "13df9abd-74cc-4238-b44d-0076f11b13d0"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{"v", "[", "1", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "2", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "3", "]"}], "\[Rule]", RowBox[{"-", "1"}]}], RowBox[{ RowBox[{"v", "[", "4", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{"c", "\[Rule]", RowBox[{"-", "4"}]}], RowBox[{"surf", "\[Rule]", RowBox[{"-", "1"}]}]}, { RowBox[{ RowBox[{"v", "[", "1", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "2", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "3", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "4", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{"c", "\[Rule]", RowBox[{"-", "4"}]}], RowBox[{"surf", "\[Rule]", "1"}]}, { RowBox[{ RowBox[{"v", "[", "1", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "2", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "3", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "4", "]"}], "\[Rule]", "0"}], RowBox[{"c", "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{"surf", "\[Rule]", RowBox[{"-", "1"}]}]}, { RowBox[{ RowBox[{"v", "[", "1", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "2", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "3", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "4", "]"}], "\[Rule]", "0"}], RowBox[{"c", "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{"surf", "\[Rule]", RowBox[{"-", "1"}]}]}, { RowBox[{ RowBox[{"v", "[", "1", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "2", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "3", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "4", "]"}], "\[Rule]", "0"}], RowBox[{"c", "\[Rule]", "0"}], RowBox[{"surf", "\[Rule]", RowBox[{"-", "1"}]}]}, { RowBox[{ RowBox[{"v", "[", "1", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "2", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "3", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "4", "]"}], "\[Rule]", RowBox[{"-", "1"}]}], RowBox[{"c", "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{"surf", "\[Rule]", "1"}]}, { RowBox[{ RowBox[{"v", "[", "1", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "2", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{ RowBox[{"v", "[", "3", "]"}], "\[Rule]", "0"}], RowBox[{ RowBox[{"v", "[", "4", "]"}], "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{"c", "\[Rule]", RowBox[{"-", "2"}]}], RowBox[{"surf", "\[Rule]", "1"}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.736354906366325*^9, 3.7363549149024897`*^9}, 3.736413670857642*^9, 3.736413826723941*^9, 3.736413868377396*^9, 3.7364140096291924`*^9, 3.7364141438106213`*^9, 3.7364141927294893`*^9, 3.7365064703727283`*^9, 3.7365066154519453`*^9, 3.736660506186694*^9, 3.736779855861756*^9, 3.736780004037899*^9, 3.736780242899638*^9, 3.7413503857813463`*^9, {3.741352376992189*^9, 3.741352384953052*^9}, 3.741413721845792*^9, 3.741418734178426*^9, 3.741425554531464*^9, 3.744809167607099*^9, 3.7448092465904293`*^9, 3.7448092844773827`*^9, 3.744809345424286*^9, 3.744809378009156*^9, 3.744809997132423*^9, { 3.744810076290657*^9, 3.744810099060717*^9}, 3.744810164216902*^9, 3.7448104871497097`*^9, {3.744814583891418*^9, 3.7448145959088497`*^9}, 3.7450290994169693`*^9, 3.745225180406972*^9, 3.745226162037446*^9, 3.745226224886507*^9, 3.745310365029339*^9, 3.745323699755769*^9, 3.745648660204198*^9, 3.745649125580888*^9, 3.745649222341773*^9, 3.745649313898638*^9, 3.745649420134419*^9, {3.7456496796787643`*^9, 3.745649741984326*^9}, {3.745650676880186*^9, 3.7456506882274237`*^9}, 3.75241801168611*^9, 3.7539835652421627`*^9}, ExpressionUUID -> "1c3517d1-4917-4f97-a389-5c577fbe79f7"] }, Open ]] }, Open ]], Cell["\<\ Analyzing the Scalings : whether Scaleful or Scaleless? \ \>", "Section", CellChangeTimes->{{3.7456501389802322`*^9, 3.745650156946774*^9}}], Cell[CellGroupData[{ Cell["The scalings from top facets:", "Section", CellChangeTimes->{{3.745650166978891*^9, 3.745650180690297*^9}}], Cell[CellGroupData[{ Cell["The scaling {-2,-2,-2,-2}:", "Subsection", CellChangeTimes->{{3.7456504029615183`*^9, 3.745650435138033*^9}, 3.7456504797461653`*^9}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"uf", "=", RowBox[{ RowBox[{ RowBox[{"UF", "[", RowBox[{ RowBox[{"{", RowBox[{"k", ",", "l"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["m", "2"], "-", RowBox[{"l", "^", "2"}]}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "-", RowBox[{ RowBox[{"(", RowBox[{"k", "-", "l"}], ")"}], "^", "2"}]}], ",", RowBox[{ RowBox[{"2", " ", "P", " ", "k"}], "-", SuperscriptBox["k", "2"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "p", " ", "k"}], "-", RowBox[{"k", "^", "2"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"M", "^", "2"}], "\[Rule]", "1"}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "\[Rule]", RowBox[{"x", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", "^", "2"}], "\[Rule]", RowBox[{"M", "^", "2"}]}], ",", RowBox[{ RowBox[{"p", "^", "2"}], "\[Rule]", RowBox[{"m", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", " ", "p"}], "\[Rule]", RowBox[{"M", " ", "m"}]}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", "x"}], ",", RowBox[{ RowBox[{"m", " ", "M"}], "\[Rule]", "x"}]}], "}"}]}], "]"}], "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], "\[Rule]", RowBox[{"x", "[", "1", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "2", "]"}], "\[Rule]", RowBox[{ RowBox[{"x", "[", "2", "]"}], "+", FractionBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", RowBox[{ RowBox[{"x", "[", "3", "]"}], "\[Rule]", RowBox[{"x", "[", "3", "]"}]}], ",", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", FractionBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], "}"}]}], "//", "Expand"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Reg", "=", RowBox[{"getLOUF", "[", RowBox[{ RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"uf", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}], ",", "1"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Scalecheck", "[", RowBox[{ RowBox[{"Reg", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"Reg", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]}], "]"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.745650213002811*^9, 3.745650321639789*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", FractionBox[ SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"], "4"]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ FractionBox["3", "4"], " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "-", RowBox[{ FractionBox["1", "2"], " ", "x", " ", RowBox[{"x", "[", "3", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ FractionBox["1", "4"], " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "3", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ FractionBox["1", "4"], " ", SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "3"]}]}], ",", "2"}], "}"}]], "Output", CellChangeTimes->{ 3.7456502699109697`*^9, 3.745650325340431*^9, 3.745650456969049*^9, { 3.745650676946954*^9, 3.745650688276237*^9}, 3.753983565278475*^9}], Cell[CellGroupData[{ Cell[BoxData["\<\"Scalings obtained :\"\>"], "Print", CellChangeTimes->{ 3.7456502699124613`*^9, 3.7456503253419323`*^9, 3.7456504569705963`*^9, { 3.7456506769516363`*^9, 3.745650688277775*^9}, 3.753983565279977*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"U: \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}]}], "}"}]}], SequenceForm["U: ", {-4, -4, -4, -4, -4, -4, -4}], Editable->False]], "Print", CellChangeTimes->{ 3.7456502699124613`*^9, 3.7456503253419323`*^9, 3.7456504569705963`*^9, { 3.7456506769516363`*^9, 3.745650688277775*^9}, 3.753983565281003*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"F \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "6"}], ",", RowBox[{"-", "6"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "5"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "5"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "6"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "5"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}]}], "}"}]}], SequenceForm[ "F ", {-4, -4, -4, -4, -4, -6, -6, -4, -4, -4, -5, -4, -5, -4, -6, -4, -4, \ -5, -4, -4}], Editable->False]], "Print", CellChangeTimes->{ 3.7456502699124613`*^9, 3.7456503253419323`*^9, 3.7456504569705963`*^9, { 3.7456506769516363`*^9, 3.745650688277775*^9}, 3.753983565282199*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"List of transformations :\"\>", "\[InvisibleSpace]", RowBox[{"{", "}"}]}], SequenceForm["List of transformations :", {}], Editable->False]], "Print", CellChangeTimes->{ 3.7456502699124613`*^9, 3.7456503253419323`*^9, 3.7456504569705963`*^9, { 3.7456506769516363`*^9, 3.745650688277775*^9}, 3.753983565283156*^9}], Cell[BoxData["\<\"Scaleful\"\>"], "Print", CellChangeTimes->{ 3.7456502699124613`*^9, 3.7456503253419323`*^9, 3.7456504569705963`*^9, { 3.7456506769516363`*^9, 3.745650688277775*^9}, 3.753983565284007*^9}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The scaling {-2,-2,0,-1}:", "Subsection", CellChangeTimes->{{3.7456504876661673`*^9, 3.745650502385984*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"uf", "=", RowBox[{ RowBox[{"UF", "[", RowBox[{ RowBox[{"{", RowBox[{"k", ",", "l"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["m", "2"], "-", RowBox[{"l", "^", "2"}]}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "-", RowBox[{ RowBox[{"(", RowBox[{"k", "-", "l"}], ")"}], "^", "2"}]}], ",", RowBox[{ RowBox[{"2", " ", "P", " ", "k"}], "-", SuperscriptBox["k", "2"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "p", " ", "k"}], "-", RowBox[{"k", "^", "2"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"M", "^", "2"}], "\[Rule]", "1"}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "\[Rule]", RowBox[{"x", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", "^", "2"}], "\[Rule]", RowBox[{"M", "^", "2"}]}], ",", RowBox[{ RowBox[{"p", "^", "2"}], "\[Rule]", RowBox[{"m", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", " ", "p"}], "\[Rule]", RowBox[{"M", " ", "m"}]}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", "x"}], ",", RowBox[{ RowBox[{"m", " ", "M"}], "\[Rule]", "x"}]}], "}"}]}], "]"}], "//", "Expand"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Reg", "=", RowBox[{"getLOUF", "[", RowBox[{ RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"uf", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", "0", ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}], ",", "1"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Scalecheck", "[", RowBox[{ RowBox[{"Reg", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"Reg", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]}], "]"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.745650335697624*^9, 3.7456503560467587`*^9}, { 3.7456504427542543`*^9, 3.7456504471397867`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", "2"}], "}"}]], "Output", CellChangeTimes->{ 3.74565035742636*^9, 3.745650451591742*^9, {3.745650677026772*^9, 3.7456506883227997`*^9}, 3.7539835653125753`*^9}], Cell[CellGroupData[{ Cell[BoxData["\<\"Scalings obtained :\"\>"], "Print", CellChangeTimes->{ 3.745650357427312*^9, 3.7456504515955877`*^9, {3.745650677028945*^9, 3.7456506883240833`*^9}, 3.7539835653137913`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"U: \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "3"}]}], "}"}]}], SequenceForm["U: ", {-4, -2, -2, -3, -3}], Editable->False]], "Print", CellChangeTimes->{ 3.745650357427312*^9, 3.7456504515955877`*^9, {3.745650677028945*^9, 3.7456506883240833`*^9}, 3.753983565314542*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"F \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}]}], "}"}]}], SequenceForm["F ", {-4, -4, -2, -2, -2, -2, -2, -3, -3, -3, -2, -2, -2, -2}], Editable->False]], "Print", CellChangeTimes->{ 3.745650357427312*^9, 3.7456504515955877`*^9, {3.745650677028945*^9, 3.7456506883240833`*^9}, 3.753983565315815*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"List of transformations :\"\>", "\[InvisibleSpace]", RowBox[{"{", "}"}]}], SequenceForm["List of transformations :", {}], Editable->False]], "Print", CellChangeTimes->{ 3.745650357427312*^9, 3.7456504515955877`*^9, {3.745650677028945*^9, 3.7456506883240833`*^9}, 3.7539835653167152`*^9}], Cell[BoxData["\<\"Scaleful\"\>"], "Print", CellChangeTimes->{ 3.745650357427312*^9, 3.7456504515955877`*^9, {3.745650677028945*^9, 3.7456506883240833`*^9}, 3.7539835653175097`*^9}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The scaling {-2,-2,0,-2}:", "Subsection", CellChangeTimes->{{3.745650511298132*^9, 3.7456505262581244`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"uf", "=", RowBox[{ RowBox[{"UF", "[", RowBox[{ RowBox[{"{", RowBox[{"k", ",", "l"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["m", "2"], "-", RowBox[{"l", "^", "2"}]}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "-", RowBox[{ RowBox[{"(", RowBox[{"k", "-", "l"}], ")"}], "^", "2"}]}], ",", RowBox[{ RowBox[{"2", " ", "P", " ", "k"}], "-", SuperscriptBox["k", "2"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "p", " ", "k"}], "-", RowBox[{"k", "^", "2"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"M", "^", "2"}], "\[Rule]", "1"}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "\[Rule]", RowBox[{"x", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", "^", "2"}], "\[Rule]", RowBox[{"M", "^", "2"}]}], ",", RowBox[{ RowBox[{"p", "^", "2"}], "\[Rule]", RowBox[{"m", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", " ", "p"}], "\[Rule]", RowBox[{"M", " ", "m"}]}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", "x"}], ",", RowBox[{ RowBox[{"m", " ", "M"}], "\[Rule]", "x"}]}], "}"}]}], "]"}], "//", "Expand"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Reg", "=", RowBox[{"getLOUF", "[", RowBox[{ RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"uf", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", "0", ",", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}], ",", "1"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Scalecheck", "[", RowBox[{ RowBox[{"Reg", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"Reg", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]}], "]"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.745650360409058*^9, 3.745650373330071*^9}, { 3.745650465016142*^9, 3.745650466614995*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", "2"}], "}"}]], "Output", CellChangeTimes->{ 3.745650374891787*^9, 3.745650467513173*^9, {3.745650677068768*^9, 3.745650688367441*^9}, 3.753983565349263*^9}], Cell[CellGroupData[{ Cell[BoxData["\<\"Scalings obtained :\"\>"], "Print", CellChangeTimes->{ 3.745650374893057*^9, 3.745650467519537*^9, {3.745650677070321*^9, 3.745650688370421*^9}, 3.753983565350412*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"U: \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}]}], "}"}]}], SequenceForm["U: ", {-4, -2, -2, -4, -4}], Editable->False]], "Print", CellChangeTimes->{ 3.745650374893057*^9, 3.745650467519537*^9, {3.745650677070321*^9, 3.745650688370421*^9}, 3.7539835653509283`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"F \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}]}], "}"}]}], SequenceForm["F ", {-4, -4, -2, -2, -2, -2, -2, -4, -4, -4, -3, -3, -4, -4}], Editable->False]], "Print", CellChangeTimes->{ 3.745650374893057*^9, 3.745650467519537*^9, {3.745650677070321*^9, 3.745650688370421*^9}, 3.753983565351796*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"List of transformations :\"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "4", "]"}], "\[Rule]", RowBox[{"a$4432", " ", RowBox[{"x", "[", "4", "]"}]}]}], "}"}], "}"}]}], SequenceForm[ "List of transformations :", {{$CellContext`x[ 4] -> $CellContext`a$4432 $CellContext`x[4]}}], Editable->False]], "Print", CellChangeTimes->{ 3.745650374893057*^9, 3.745650467519537*^9, {3.745650677070321*^9, 3.745650688370421*^9}, 3.7539835653526087`*^9}], Cell[BoxData["\<\"Scaleless\"\>"], "Print", CellChangeTimes->{ 3.745650374893057*^9, 3.745650467519537*^9, {3.745650677070321*^9, 3.745650688370421*^9}, 3.7539835653536386`*^9}] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The scalings from the bottom facets:\ \>", "Section", CellChangeTimes->{{3.745650536337635*^9, 3.745650561777787*^9}}], Cell[CellGroupData[{ Cell["The Scaling {0,-2,0,0}:", "Subsubsection", CellChangeTimes->{{3.7413506354878883`*^9, 3.741350657950529*^9}, { 3.745480321331505*^9, 3.745480330737618*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"uf", "=", RowBox[{ RowBox[{"UF", "[", RowBox[{ RowBox[{"{", RowBox[{"k", ",", "l"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["m", "2"], "-", RowBox[{"l", "^", "2"}]}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "-", RowBox[{ RowBox[{"(", RowBox[{"k", "-", "l"}], ")"}], "^", "2"}]}], ",", RowBox[{ RowBox[{"2", " ", "P", " ", "k"}], "-", SuperscriptBox["k", "2"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "p", " ", "k"}], "-", RowBox[{"k", "^", "2"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"M", "^", "2"}], "\[Rule]", "1"}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "\[Rule]", RowBox[{"x", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", "^", "2"}], "\[Rule]", RowBox[{"M", "^", "2"}]}], ",", RowBox[{ RowBox[{"p", "^", "2"}], "\[Rule]", RowBox[{"m", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", " ", "p"}], "\[Rule]", RowBox[{"M", " ", "m"}]}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", "x"}], ",", RowBox[{ RowBox[{"m", " ", "M"}], "\[Rule]", "x"}]}], "}"}]}], "]"}], "//", "Expand"}]}], "\[IndentingNewLine]", RowBox[{"Reg", "=", RowBox[{"getLOUF", "[", RowBox[{ RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"uf", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"1", ",", "0", ",", RowBox[{"-", "2"}], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}], ",", RowBox[{"-", "1"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Scalecheck", "[", RowBox[{ RowBox[{"Reg", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"Reg", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]}], "]"}], "\n"}], "Input", CellChangeTimes->{{3.736227986958395*^9, 3.736228040948349*^9}, { 3.736228104630562*^9, 3.736228166567049*^9}, 3.7362282581268883`*^9, { 3.736228370099218*^9, 3.73622840307705*^9}, {3.7362288498241568`*^9, 3.736228884976407*^9}, {3.736230170722567*^9, 3.736230171907753*^9}, 3.736347592333672*^9, 3.73634769528644*^9, {3.736352512693059*^9, 3.7363525256909637`*^9}, {3.73667241463191*^9, 3.73667243029454*^9}, 3.741350673100073*^9, {3.7448119687289543`*^9, 3.744811985882162*^9}, { 3.7448120221188*^9, 3.744812151409843*^9}, {3.744812222828816*^9, 3.7448123109507017`*^9}, {3.744812728480544*^9, 3.744812791179357*^9}, 3.744813014871441*^9, {3.744813398020352*^9, 3.74481340723494*^9}, { 3.744813449524076*^9, 3.744813449924798*^9}, {3.744813487552657*^9, 3.7448134919455137`*^9}, {3.744813530961583*^9, 3.744813555242429*^9}, { 3.744813597989683*^9, 3.744813693964039*^9}, {3.7448137263302402`*^9, 3.744813727854704*^9}, {3.744813762214346*^9, 3.744813783073045*^9}, { 3.744814223226302*^9, 3.744814261025413*^9}, {3.744814516842648*^9, 3.744814567291285*^9}, {3.744814622941985*^9, 3.74481464349818*^9}, { 3.745029092985146*^9, 3.74502909355835*^9}, {3.745226007633923*^9, 3.745226020192116*^9}, {3.745650609214279*^9, 3.745650610440092*^9}}, ExpressionUUID -> "5f65034b-3fa4-4dfc-b4e5-95b419b9f17b"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", "2"}], "}"}]], "Output", CellChangeTimes->{{3.7362280513375397`*^9, 3.736228058054715*^9}, { 3.73622814830698*^9, 3.736228167706648*^9}, {3.736228373591454*^9, 3.736228403785161*^9}, {3.73622885678279*^9, 3.736228885711123*^9}, { 3.7362301728305807`*^9, 3.736230191982188*^9}, 3.736345572427074*^9, 3.7363464224785833`*^9, 3.736347593646658*^9, 3.736347696325601*^9, 3.736350630529052*^9, {3.736352514065001*^9, 3.736352526561583*^9}, 3.7365066155455647`*^9, 3.736660506219058*^9, {3.736672420019587*^9, 3.73667243141154*^9}, 3.736779856321772*^9, 3.736780004173499*^9, 3.736780242993995*^9, 3.741350425844493*^9, 3.741350673801344*^9, { 3.741352377026087*^9, 3.7413523849852343`*^9}, 3.741413721883759*^9, 3.741418734209671*^9, 3.741425554835751*^9, 3.744809167629682*^9, 3.744809246613791*^9, 3.7448092845603933`*^9, 3.744809345454095*^9, 3.744809378037086*^9, 3.7448099971596403`*^9, 3.7448100990892267`*^9, 3.7448101643513927`*^9, 3.744810487250403*^9, 3.7448119930990343`*^9, { 3.744812023723461*^9, 3.7448121427150383`*^9}, {3.744812223651263*^9, 3.7448122742424994`*^9}, 3.744812318301358*^9, {3.744812729920842*^9, 3.744812791884445*^9}, 3.7448130175706244`*^9, {3.744813398909337*^9, 3.744813408496286*^9}, 3.744813450822166*^9, 3.744813495254528*^9, { 3.744813541903969*^9, 3.744813556035754*^9}, {3.744813598904379*^9, 3.744813660532961*^9}, 3.7448136946253023`*^9, {3.744813764384399*^9, 3.7448137870160017`*^9}, 3.744814375771401*^9, {3.7448145683658447`*^9, 3.74481459596587*^9}, 3.744814644997308*^9, {3.745029095142646*^9, 3.745029099456881*^9}, 3.745225180491116*^9, 3.7452261620753613`*^9, 3.745226224925943*^9, 3.745310365056035*^9, 3.745323699801358*^9, 3.7456486602369213`*^9, 3.745649125618772*^9, 3.745649222377303*^9, 3.745649313933868*^9, 3.7456494201689568`*^9, {3.7456497209318123`*^9, 3.7456497420222883`*^9}, {3.745650677114457*^9, 3.745650688409902*^9}, 3.753983565570326*^9}, ExpressionUUID -> "ed378e74-93cb-407f-bde1-4f9607939b4c"], Cell[CellGroupData[{ Cell[BoxData["\<\"Scalings obtained :\"\>"], "Print", CellChangeTimes->{ 3.745029099454927*^9, 3.7452251804886417`*^9, 3.7452261620733957`*^9, 3.7452262249237337`*^9, 3.745310365054171*^9, 3.7453236997994623`*^9, 3.745648660234746*^9, 3.745649125616683*^9, 3.745649222375705*^9, 3.745649313931876*^9, 3.745649420167397*^9, {3.745649720929656*^9, 3.745649742020615*^9}, {3.745650677115705*^9, 3.745650688412567*^9}, 3.7539835655714817`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"U: \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0", ",", RowBox[{"-", "2"}], ",", "0", ",", RowBox[{"-", "2"}]}], "}"}]}], SequenceForm["U: ", {-2, 0, -2, 0, -2}], Editable->False]], "Print", CellChangeTimes->{ 3.745029099454927*^9, 3.7452251804886417`*^9, 3.7452261620733957`*^9, 3.7452262249237337`*^9, 3.745310365054171*^9, 3.7453236997994623`*^9, 3.745648660234746*^9, 3.745649125616683*^9, 3.745649222375705*^9, 3.745649313931876*^9, 3.745649420167397*^9, {3.745649720929656*^9, 3.745649742020615*^9}, {3.745650677115705*^9, 3.745650688412567*^9}, 3.753983565571899*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"F \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "2"}], ",", "2", ",", "0", ",", RowBox[{"-", "2"}], ",", "0", ",", RowBox[{"-", "2"}], ",", "2", ",", "0", ",", RowBox[{"-", "2"}], ",", "1", ",", RowBox[{"-", "1"}], ",", "2", ",", "0"}], "}"}]}], SequenceForm["F ", {0, -2, 2, 0, -2, 0, -2, 2, 0, -2, 1, -1, 2, 0}], Editable->False]], "Print", CellChangeTimes->{ 3.745029099454927*^9, 3.7452251804886417`*^9, 3.7452261620733957`*^9, 3.7452262249237337`*^9, 3.745310365054171*^9, 3.7453236997994623`*^9, 3.745648660234746*^9, 3.745649125616683*^9, 3.745649222375705*^9, 3.745649313931876*^9, 3.745649420167397*^9, {3.745649720929656*^9, 3.745649742020615*^9}, {3.745650677115705*^9, 3.745650688412567*^9}, 3.753983565572846*^9}] }, Open ]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}]}]}], "}"}]], "Output", CellChangeTimes->{{3.7362280513375397`*^9, 3.736228058054715*^9}, { 3.73622814830698*^9, 3.736228167706648*^9}, {3.736228373591454*^9, 3.736228403785161*^9}, {3.73622885678279*^9, 3.736228885711123*^9}, { 3.7362301728305807`*^9, 3.736230191982188*^9}, 3.736345572427074*^9, 3.7363464224785833`*^9, 3.736347593646658*^9, 3.736347696325601*^9, 3.736350630529052*^9, {3.736352514065001*^9, 3.736352526561583*^9}, 3.7365066155455647`*^9, 3.736660506219058*^9, {3.736672420019587*^9, 3.73667243141154*^9}, 3.736779856321772*^9, 3.736780004173499*^9, 3.736780242993995*^9, 3.741350425844493*^9, 3.741350673801344*^9, { 3.741352377026087*^9, 3.7413523849852343`*^9}, 3.741413721883759*^9, 3.741418734209671*^9, 3.741425554835751*^9, 3.744809167629682*^9, 3.744809246613791*^9, 3.7448092845603933`*^9, 3.744809345454095*^9, 3.744809378037086*^9, 3.7448099971596403`*^9, 3.7448100990892267`*^9, 3.7448101643513927`*^9, 3.744810487250403*^9, 3.7448119930990343`*^9, { 3.744812023723461*^9, 3.7448121427150383`*^9}, {3.744812223651263*^9, 3.7448122742424994`*^9}, 3.744812318301358*^9, {3.744812729920842*^9, 3.744812791884445*^9}, 3.7448130175706244`*^9, {3.744813398909337*^9, 3.744813408496286*^9}, 3.744813450822166*^9, 3.744813495254528*^9, { 3.744813541903969*^9, 3.744813556035754*^9}, {3.744813598904379*^9, 3.744813660532961*^9}, 3.7448136946253023`*^9, {3.744813764384399*^9, 3.7448137870160017`*^9}, 3.744814375771401*^9, {3.7448145683658447`*^9, 3.74481459596587*^9}, 3.744814644997308*^9, {3.745029095142646*^9, 3.745029099456881*^9}, 3.745225180491116*^9, 3.7452261620753613`*^9, 3.745226224925943*^9, 3.745310365056035*^9, 3.745323699801358*^9, 3.7456486602369213`*^9, 3.745649125618772*^9, 3.745649222377303*^9, 3.745649313933868*^9, 3.7456494201689568`*^9, {3.7456497209318123`*^9, 3.7456497420222883`*^9}, {3.745650677114457*^9, 3.745650688409902*^9}, 3.753983565573781*^9}, ExpressionUUID -> "ed378e74-93cb-407f-bde1-4f9607939b4c"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"List of transformations :\"\>", "\[InvisibleSpace]", RowBox[{"{", "}"}]}], SequenceForm["List of transformations :", {}], Editable->False]], "Print", CellChangeTimes->{ 3.745029099454927*^9, 3.7452251804886417`*^9, 3.7452261620733957`*^9, 3.7452262249237337`*^9, 3.745310365054171*^9, 3.7453236997994623`*^9, 3.745648660234746*^9, 3.745649125616683*^9, 3.745649222375705*^9, 3.745649313931876*^9, 3.745649420167397*^9, {3.745649720929656*^9, 3.745649742020615*^9}, {3.745650677115705*^9, 3.745650688412567*^9}, 3.753983565574574*^9}], Cell[BoxData["\<\"Scaleful\"\>"], "Print", CellChangeTimes->{ 3.745029099454927*^9, 3.7452251804886417`*^9, 3.7452261620733957`*^9, 3.7452262249237337`*^9, 3.745310365054171*^9, 3.7453236997994623`*^9, 3.745648660234746*^9, 3.745649125616683*^9, 3.745649222375705*^9, 3.745649313931876*^9, 3.745649420167397*^9, {3.745649720929656*^9, 3.745649742020615*^9}, {3.745650677115705*^9, 3.745650688412567*^9}, 3.753983565575098*^9}] }, Open ]] }, Open ]] }, Open ]], Cell["", "Subsubsection"], Cell[CellGroupData[{ Cell["The Scaling {-2,-2,-1,-2}:", "Subsubsection", CellChangeTimes->{{3.741350923405739*^9, 3.741350953644639*^9}, 3.7454803408192043`*^9}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"uf", "=", RowBox[{ RowBox[{"UF", "[", RowBox[{ RowBox[{"{", RowBox[{"k", ",", "l"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["m", "2"], "-", RowBox[{"l", "^", "2"}]}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "-", RowBox[{ RowBox[{"(", RowBox[{"k", "-", "l"}], ")"}], "^", "2"}]}], ",", RowBox[{ RowBox[{"2", " ", "P", " ", "k"}], "-", SuperscriptBox["k", "2"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "p", " ", "k"}], "-", RowBox[{"k", "^", "2"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"M", "^", "2"}], "\[Rule]", "1"}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "\[Rule]", RowBox[{"x", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", "^", "2"}], "\[Rule]", RowBox[{"M", "^", "2"}]}], ",", RowBox[{ RowBox[{"p", "^", "2"}], "\[Rule]", RowBox[{"m", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", " ", "p"}], "\[Rule]", RowBox[{"M", " ", "m"}]}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", "x"}], ",", RowBox[{ RowBox[{"m", " ", "M"}], "\[Rule]", "x"}]}], "}"}]}], "]"}], "//", "Expand"}]}], "\[IndentingNewLine]", RowBox[{"Reg", "=", RowBox[{"getLOUF", "[", RowBox[{ RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"uf", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "1"}], ",", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}], ",", RowBox[{"-", "1"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Scalecheck", "[", RowBox[{ RowBox[{"Reg", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"Reg", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]}], "]"}], "\n"}], "Input", CellChangeTimes->{{3.741350956386169*^9, 3.741350984795392*^9}, { 3.7413523946938353`*^9, 3.741352399959025*^9}, 3.741352471236677*^9, 3.741441401387541*^9, {3.745226026751049*^9, 3.745226033906555*^9}, { 3.745226211087761*^9, 3.745226211616579*^9}, {3.745650615858491*^9, 3.7456506168647223`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", "2"}], "}"}]], "Output", CellChangeTimes->{ 3.7413509875368834`*^9, 3.7413515910184603`*^9, {3.741352377053347*^9, 3.741352400790822*^9}, 3.741352471807282*^9, 3.741413721919364*^9, 3.7414187342351522`*^9, 3.7414255548640547`*^9, 3.744809167645749*^9, 3.744809246631474*^9, 3.744809284595241*^9, 3.744809345473063*^9, 3.74480937805617*^9, 3.744809997178838*^9, 3.7448100991084337`*^9, 3.7448101644145927`*^9, 3.7448104874054003`*^9, {3.744814583969048*^9, 3.7448145960082483`*^9}, 3.745029099486171*^9, 3.745225180516844*^9, 3.745226162117271*^9, 3.745226224956273*^9, 3.74531036507627*^9, 3.745323699843567*^9, 3.745648660263811*^9, 3.74564912565231*^9, 3.745649222409515*^9, 3.745649314046135*^9, 3.7456494201990843`*^9, { 3.7456497209638243`*^9, 3.745649742058436*^9}, {3.7456506771588373`*^9, 3.7456506884524508`*^9}, 3.7539835656070013`*^9}], Cell[CellGroupData[{ Cell[BoxData["\<\"Scalings obtained :\"\>"], "Print", CellChangeTimes->{ 3.741350987517672*^9, 3.7413515909930897`*^9, {3.741352377050441*^9, 3.741352400764812*^9}, 3.7413524717920637`*^9, 3.7414137219173107`*^9, 3.741418734233398*^9, 3.741425554861588*^9, 3.744809167642877*^9, 3.744809246628459*^9, 3.744809284592642*^9, 3.7448093454705877`*^9, 3.744809378053632*^9, 3.744809997175724*^9, 3.744810099106101*^9, 3.744810164412651*^9, 3.744810487362529*^9, {3.744814583965921*^9, 3.74481459600561*^9}, 3.745029099481997*^9, 3.745225180514279*^9, 3.745226162112945*^9, 3.7452262249527817`*^9, 3.7453103650742493`*^9, 3.7453236998417187`*^9, 3.745648660261302*^9, 3.745649125649213*^9, 3.745649222406583*^9, 3.745649314043117*^9, 3.745649420195099*^9, { 3.745649720960679*^9, 3.745649742054481*^9}, {3.7456506771605673`*^9, 3.745650688455195*^9}, 3.7539835656081963`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"U: \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}]}], "}"}]}], SequenceForm["U: ", {-4, -3, -3, -4, -4}], Editable->False]], "Print", CellChangeTimes->{ 3.741350987517672*^9, 3.7413515909930897`*^9, {3.741352377050441*^9, 3.741352400764812*^9}, 3.7413524717920637`*^9, 3.7414137219173107`*^9, 3.741418734233398*^9, 3.741425554861588*^9, 3.744809167642877*^9, 3.744809246628459*^9, 3.744809284592642*^9, 3.7448093454705877`*^9, 3.744809378053632*^9, 3.744809997175724*^9, 3.744810099106101*^9, 3.744810164412651*^9, 3.744810487362529*^9, {3.744814583965921*^9, 3.74481459600561*^9}, 3.745029099481997*^9, 3.745225180514279*^9, 3.745226162112945*^9, 3.7452262249527817`*^9, 3.7453103650742493`*^9, 3.7453236998417187`*^9, 3.745648660261302*^9, 3.745649125649213*^9, 3.745649222406583*^9, 3.745649314043117*^9, 3.745649420195099*^9, { 3.745649720960679*^9, 3.745649742054481*^9}, {3.7456506771605673`*^9, 3.745650688455195*^9}, 3.7539835656087523`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"F \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "4"}]}], "}"}]}], SequenceForm["F ", {-4, -4, -3, -3, -3, -4, -4, -4, -4, -4, -4, -4, -4, -4}], Editable->False]], "Print", CellChangeTimes->{ 3.741350987517672*^9, 3.7413515909930897`*^9, {3.741352377050441*^9, 3.741352400764812*^9}, 3.7413524717920637`*^9, 3.7414137219173107`*^9, 3.741418734233398*^9, 3.741425554861588*^9, 3.744809167642877*^9, 3.744809246628459*^9, 3.744809284592642*^9, 3.7448093454705877`*^9, 3.744809378053632*^9, 3.744809997175724*^9, 3.744810099106101*^9, 3.744810164412651*^9, 3.744810487362529*^9, {3.744814583965921*^9, 3.74481459600561*^9}, 3.745029099481997*^9, 3.745225180514279*^9, 3.745226162112945*^9, 3.7452262249527817`*^9, 3.7453103650742493`*^9, 3.7453236998417187`*^9, 3.745648660261302*^9, 3.745649125649213*^9, 3.745649222406583*^9, 3.745649314043117*^9, 3.745649420195099*^9, { 3.745649720960679*^9, 3.745649742054481*^9}, {3.7456506771605673`*^9, 3.745650688455195*^9}, 3.753983565609655*^9}] }, Open ]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.7413509875368834`*^9, 3.7413515910184603`*^9, {3.741352377053347*^9, 3.741352400790822*^9}, 3.741352471807282*^9, 3.741413721919364*^9, 3.7414187342351522`*^9, 3.7414255548640547`*^9, 3.744809167645749*^9, 3.744809246631474*^9, 3.744809284595241*^9, 3.744809345473063*^9, 3.74480937805617*^9, 3.744809997178838*^9, 3.7448100991084337`*^9, 3.7448101644145927`*^9, 3.7448104874054003`*^9, {3.744814583969048*^9, 3.7448145960082483`*^9}, 3.745029099486171*^9, 3.745225180516844*^9, 3.745226162117271*^9, 3.745226224956273*^9, 3.74531036507627*^9, 3.745323699843567*^9, 3.745648660263811*^9, 3.74564912565231*^9, 3.745649222409515*^9, 3.745649314046135*^9, 3.7456494201990843`*^9, { 3.7456497209638243`*^9, 3.745649742058436*^9}, {3.7456506771588373`*^9, 3.7456506884524508`*^9}, 3.753983565610642*^9}], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"List of transformations :\"\>", "\[InvisibleSpace]", RowBox[{"{", "}"}]}], SequenceForm["List of transformations :", {}], Editable->False]], "Print", CellChangeTimes->{ 3.741350987517672*^9, 3.7413515909930897`*^9, {3.741352377050441*^9, 3.741352400764812*^9}, 3.7413524717920637`*^9, 3.7414137219173107`*^9, 3.741418734233398*^9, 3.741425554861588*^9, 3.744809167642877*^9, 3.744809246628459*^9, 3.744809284592642*^9, 3.7448093454705877`*^9, 3.744809378053632*^9, 3.744809997175724*^9, 3.744810099106101*^9, 3.744810164412651*^9, 3.744810487362529*^9, {3.744814583965921*^9, 3.74481459600561*^9}, 3.745029099481997*^9, 3.745225180514279*^9, 3.745226162112945*^9, 3.7452262249527817`*^9, 3.7453103650742493`*^9, 3.7453236998417187`*^9, 3.745648660261302*^9, 3.745649125649213*^9, 3.745649222406583*^9, 3.745649314043117*^9, 3.745649420195099*^9, { 3.745649720960679*^9, 3.745649742054481*^9}, {3.7456506771605673`*^9, 3.745650688455195*^9}, 3.7539835656114902`*^9}], Cell[BoxData["\<\"Scaleful\"\>"], "Print", CellChangeTimes->{ 3.741350987517672*^9, 3.7413515909930897`*^9, {3.741352377050441*^9, 3.741352400764812*^9}, 3.7413524717920637`*^9, 3.7414137219173107`*^9, 3.741418734233398*^9, 3.741425554861588*^9, 3.744809167642877*^9, 3.744809246628459*^9, 3.744809284592642*^9, 3.7448093454705877`*^9, 3.744809378053632*^9, 3.744809997175724*^9, 3.744810099106101*^9, 3.744810164412651*^9, 3.744810487362529*^9, {3.744814583965921*^9, 3.74481459600561*^9}, 3.745029099481997*^9, 3.745225180514279*^9, 3.745226162112945*^9, 3.7452262249527817`*^9, 3.7453103650742493`*^9, 3.7453236998417187`*^9, 3.745648660261302*^9, 3.745649125649213*^9, 3.745649222406583*^9, 3.745649314043117*^9, 3.745649420195099*^9, { 3.745649720960679*^9, 3.745649742054481*^9}, {3.7456506771605673`*^9, 3.745650688455195*^9}, 3.753983565611969*^9}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["the Scaling {0,0,0,0}:", "Subsection", CellChangeTimes->{{3.741351086365046*^9, 3.741351089547419*^9}, { 3.741351143996175*^9, 3.7413511539322166`*^9}, {3.7454803470890617`*^9, 3.745480349345104*^9}, {3.7456505699526854`*^9, 3.745650578338235*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"uf", "=", RowBox[{ RowBox[{"UF", "[", RowBox[{ RowBox[{"{", RowBox[{"k", ",", "l"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["m", "2"], "-", RowBox[{"l", "^", "2"}]}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "-", RowBox[{ RowBox[{"(", RowBox[{"k", "-", "l"}], ")"}], "^", "2"}]}], ",", RowBox[{ RowBox[{"2", " ", "P", " ", "k"}], "-", SuperscriptBox["k", "2"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "p", " ", "k"}], "-", RowBox[{"k", "^", "2"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"M", "^", "2"}], "\[Rule]", "1"}], ",", RowBox[{ RowBox[{"m", "^", "2"}], "\[Rule]", RowBox[{"x", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", "^", "2"}], "\[Rule]", RowBox[{"M", "^", "2"}]}], ",", RowBox[{ RowBox[{"p", "^", "2"}], "\[Rule]", RowBox[{"m", "^", "2"}]}], ",", RowBox[{ RowBox[{"P", " ", "p"}], "\[Rule]", RowBox[{"M", " ", "m"}]}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", "x"}], ",", RowBox[{ RowBox[{"m", " ", "M"}], "\[Rule]", "x"}]}], "}"}]}], "]"}], "//", "Expand"}]}], "\[IndentingNewLine]", RowBox[{"Reg", "=", RowBox[{"getLOUF", "[", RowBox[{ RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"uf", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"1", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}], ",", RowBox[{"-", "1"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Scalecheck", "[", RowBox[{ RowBox[{"Reg", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"Reg", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]}], "]"}], "\n"}], "\n"}], "Input",\ CellChangeTimes->{{3.741351159770712*^9, 3.741351181567595*^9}, 3.7414413988092937`*^9, {3.745226059440254*^9, 3.745226066703833*^9}, 3.745650204237441*^9, {3.745650625770074*^9, 3.7456506338452263`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "1", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox[ RowBox[{"x", "[", "2", "]"}], "2"], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "-", RowBox[{"2", " ", "x", " ", RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox["x", "2"], " ", RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "4", "]"}], "2"]}]}], ",", "2"}], "}"}]], "Output", CellChangeTimes->{{3.745650677206044*^9, 3.74565068849504*^9}, 3.7539835656445217`*^9}], Cell[CellGroupData[{ Cell[BoxData["\<\"Scalings obtained :\"\>"], "Print", CellChangeTimes->{ 3.741351182491699*^9, {3.741352377077724*^9, 3.7413523850293407`*^9}, 3.741413721951516*^9, 3.741418734263607*^9, 3.7414255548901653`*^9, 3.7448091676604357`*^9, 3.744809246646922*^9, 3.74480928461091*^9, 3.744809345490922*^9, 3.744809378074109*^9, 3.7448099971966257`*^9, 3.744810099126513*^9, 3.744810164430029*^9, 3.744810487457201*^9, { 3.7448145840028133`*^9, 3.7448145960457573`*^9}, 3.7450290995158033`*^9, 3.7452251805365353`*^9, 3.7452261623836803`*^9, 3.745226224986932*^9, 3.7453103650957403`*^9, 3.7453236998869123`*^9, 3.745648660288494*^9, 3.7456491256865997`*^9, 3.7456492224368563`*^9, 3.7456493140756607`*^9, 3.745649420225185*^9, {3.7456497209930353`*^9, 3.745649742089055*^9}, { 3.745650677208529*^9, 3.745650688496192*^9}, 3.753983565645754*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"U: \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}], SequenceForm["U: ", {0, 0, 0, 0, 0}], Editable->False]], "Print", CellChangeTimes->{ 3.741351182491699*^9, {3.741352377077724*^9, 3.7413523850293407`*^9}, 3.741413721951516*^9, 3.741418734263607*^9, 3.7414255548901653`*^9, 3.7448091676604357`*^9, 3.744809246646922*^9, 3.74480928461091*^9, 3.744809345490922*^9, 3.744809378074109*^9, 3.7448099971966257`*^9, 3.744810099126513*^9, 3.744810164430029*^9, 3.744810487457201*^9, { 3.7448145840028133`*^9, 3.7448145960457573`*^9}, 3.7450290995158033`*^9, 3.7452251805365353`*^9, 3.7452261623836803`*^9, 3.745226224986932*^9, 3.7453103650957403`*^9, 3.7453236998869123`*^9, 3.745648660288494*^9, 3.7456491256865997`*^9, 3.7456492224368563`*^9, 3.7456493140756607`*^9, 3.745649420225185*^9, {3.7456497209930353`*^9, 3.745649742089055*^9}, { 3.745650677208529*^9, 3.745650688496192*^9}, 3.753983565646275*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"F \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ "2", ",", "2", ",", "2", ",", "2", ",", "2", ",", "0", ",", "0", ",", "2", ",", "2", ",", "2", ",", "1", ",", "1", ",", "2", ",", "2"}], "}"}]}], SequenceForm["F ", {2, 2, 2, 2, 2, 0, 0, 2, 2, 2, 1, 1, 2, 2}], Editable->False]], "Print", CellChangeTimes->{ 3.741351182491699*^9, {3.741352377077724*^9, 3.7413523850293407`*^9}, 3.741413721951516*^9, 3.741418734263607*^9, 3.7414255548901653`*^9, 3.7448091676604357`*^9, 3.744809246646922*^9, 3.74480928461091*^9, 3.744809345490922*^9, 3.744809378074109*^9, 3.7448099971966257`*^9, 3.744810099126513*^9, 3.744810164430029*^9, 3.744810487457201*^9, { 3.7448145840028133`*^9, 3.7448145960457573`*^9}, 3.7450290995158033`*^9, 3.7452251805365353`*^9, 3.7452261623836803`*^9, 3.745226224986932*^9, 3.7453103650957403`*^9, 3.7453236998869123`*^9, 3.745648660288494*^9, 3.7456491256865997`*^9, 3.7456492224368563`*^9, 3.7456493140756607`*^9, 3.745649420225185*^9, {3.7456497209930353`*^9, 3.745649742089055*^9}, { 3.745650677208529*^9, 3.745650688496192*^9}, 3.753983565647155*^9}] }, Open ]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "2", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "3", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", RowBox[{"x", "[", "4", "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}], "+", RowBox[{ RowBox[{"x", "[", "2", "]"}], " ", SuperscriptBox[ RowBox[{"x", "[", "3", "]"}], "2"]}]}]}], "}"}]], "Output", CellChangeTimes->{{3.745650677206044*^9, 3.74565068849504*^9}, 3.7539835656480837`*^9}], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"List of transformations :\"\>", "\[InvisibleSpace]", RowBox[{"{", "}"}]}], SequenceForm["List of transformations :", {}], Editable->False]], "Print", CellChangeTimes->{ 3.741351182491699*^9, {3.741352377077724*^9, 3.7413523850293407`*^9}, 3.741413721951516*^9, 3.741418734263607*^9, 3.7414255548901653`*^9, 3.7448091676604357`*^9, 3.744809246646922*^9, 3.74480928461091*^9, 3.744809345490922*^9, 3.744809378074109*^9, 3.7448099971966257`*^9, 3.744810099126513*^9, 3.744810164430029*^9, 3.744810487457201*^9, { 3.7448145840028133`*^9, 3.7448145960457573`*^9}, 3.7450290995158033`*^9, 3.7452251805365353`*^9, 3.7452261623836803`*^9, 3.745226224986932*^9, 3.7453103650957403`*^9, 3.7453236998869123`*^9, 3.745648660288494*^9, 3.7456491256865997`*^9, 3.7456492224368563`*^9, 3.7456493140756607`*^9, 3.745649420225185*^9, {3.7456497209930353`*^9, 3.745649742089055*^9}, { 3.745650677208529*^9, 3.745650688496192*^9}, 3.753983565648802*^9}], Cell[BoxData["\<\"Scaleful\"\>"], "Print", CellChangeTimes->{ 3.741351182491699*^9, {3.741352377077724*^9, 3.7413523850293407`*^9}, 3.741413721951516*^9, 3.741418734263607*^9, 3.7414255548901653`*^9, 3.7448091676604357`*^9, 3.744809246646922*^9, 3.74480928461091*^9, 3.744809345490922*^9, 3.744809378074109*^9, 3.7448099971966257`*^9, 3.744810099126513*^9, 3.744810164430029*^9, 3.744810487457201*^9, { 3.7448145840028133`*^9, 3.7448145960457573`*^9}, 3.7450290995158033`*^9, 3.7452251805365353`*^9, 3.7452261623836803`*^9, 3.745226224986932*^9, 3.7453103650957403`*^9, 3.7453236998869123`*^9, 3.745648660288494*^9, 3.7456491256865997`*^9, 3.7456492224368563`*^9, 3.7456493140756607`*^9, 3.745649420225185*^9, {3.7456497209930353`*^9, 3.745649742089055*^9}, { 3.745650677208529*^9, 3.745650688496192*^9}, 3.753983565649218*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The Scaling {-2,0,0,0}:", "Subsubsection", CellChangeTimes->{{3.741351223643197*^9, 3.74135123513242*^9}, { 3.745480354130352*^9, 3.745480354847868*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"Reg", "=", RowBox[{"getLOUF", "[", RowBox[{ RowBox[{"uf", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"uf", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "2"}], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}], ",", RowBox[{"-", "1"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Scalecheck", "[", RowBox[{ RowBox[{"Reg", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"Reg", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "1", "]"}], ",", RowBox[{"x", "[", "2", "]"}], ",", RowBox[{"x", "[", "3", "]"}], ",", RowBox[{"x", "[", "4", "]"}]}], "}"}]}], "]"}], "\n"}], "Input", CellChangeTimes->{{3.741351239194406*^9, 3.741351251943563*^9}, 3.741441396235579*^9, {3.745226077231739*^9, 3.745226082571856*^9}}], Cell[CellGroupData[{ Cell[BoxData["\<\"Scalings obtained :\"\>"], "Print", CellChangeTimes->{ 3.741351256519195*^9, {3.741352377105689*^9, 3.74135238505509*^9}, 3.7414137219822397`*^9, 3.7414187342928543`*^9, 3.7414255549171333`*^9, 3.744809167677113*^9, 3.744809246664177*^9, 3.74480928462808*^9, 3.7448093455094767`*^9, 3.744809378093*^9, 3.744809997216243*^9, 3.744810099146199*^9, 3.744810164447756*^9, 3.744810487496963*^9, { 3.744814584038282*^9, 3.7448145960830097`*^9}, 3.7450290995482883`*^9, 3.7452251805545692`*^9, 3.74522616242284*^9, 3.745226225022263*^9, 3.745310365117715*^9, 3.7453236999294367`*^9, 3.745648660314818*^9, 3.7456491257239656`*^9, 3.745649222468156*^9, 3.745649314106741*^9, 3.7456494202539787`*^9, {3.7456497210245857`*^9, 3.745649742121464*^9}, { 3.7456506772479486`*^9, 3.7456506885374603`*^9}, 3.753983565677916*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"U: \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}], ",", "0", ",", RowBox[{"-", "2"}], ",", "0"}], "}"}]}], SequenceForm["U: ", {-2, -2, 0, -2, 0}], Editable->False]], "Print", CellChangeTimes->{ 3.741351256519195*^9, {3.741352377105689*^9, 3.74135238505509*^9}, 3.7414137219822397`*^9, 3.7414187342928543`*^9, 3.7414255549171333`*^9, 3.744809167677113*^9, 3.744809246664177*^9, 3.74480928462808*^9, 3.7448093455094767`*^9, 3.744809378093*^9, 3.744809997216243*^9, 3.744810099146199*^9, 3.744810164447756*^9, 3.744810487496963*^9, { 3.744814584038282*^9, 3.7448145960830097`*^9}, 3.7450290995482883`*^9, 3.7452251805545692`*^9, 3.74522616242284*^9, 3.745226225022263*^9, 3.745310365117715*^9, 3.7453236999294367`*^9, 3.745648660314818*^9, 3.7456491257239656`*^9, 3.745649222468156*^9, 3.745649314106741*^9, 3.7456494202539787`*^9, {3.7456497210245857`*^9, 3.745649742121464*^9}, { 3.7456506772479486`*^9, 3.7456506885374603`*^9}, 3.753983565679381*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"F \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0", ",", RowBox[{"-", "2"}], ",", "0", ",", "2", ",", RowBox[{"-", "2"}], ",", "0", ",", RowBox[{"-", "2"}], ",", "0", ",", "2", ",", RowBox[{"-", "1"}], ",", "1", ",", "0", ",", "2"}], "}"}]}], SequenceForm["F ", {-2, 0, -2, 0, 2, -2, 0, -2, 0, 2, -1, 1, 0, 2}], Editable->False]], "Print", CellChangeTimes->{ 3.741351256519195*^9, {3.741352377105689*^9, 3.74135238505509*^9}, 3.7414137219822397`*^9, 3.7414187342928543`*^9, 3.7414255549171333`*^9, 3.744809167677113*^9, 3.744809246664177*^9, 3.74480928462808*^9, 3.7448093455094767`*^9, 3.744809378093*^9, 3.744809997216243*^9, 3.744810099146199*^9, 3.744810164447756*^9, 3.744810487496963*^9, { 3.744814584038282*^9, 3.7448145960830097`*^9}, 3.7450290995482883`*^9, 3.7452251805545692`*^9, 3.74522616242284*^9, 3.745226225022263*^9, 3.745310365117715*^9, 3.7453236999294367`*^9, 3.745648660314818*^9, 3.7456491257239656`*^9, 3.745649222468156*^9, 3.745649314106741*^9, 3.7456494202539787`*^9, {3.7456497210245857`*^9, 3.745649742121464*^9}, { 3.7456506772479486`*^9, 3.7456506885374603`*^9}, 3.753983565680814*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"List of transformations :\"\>", "\[InvisibleSpace]", RowBox[{"{", "}"}]}], SequenceForm["List of transformations :", {}], Editable->False]], "Print", CellChangeTimes->{ 3.741351256519195*^9, {3.741352377105689*^9, 3.74135238505509*^9}, 3.7414137219822397`*^9, 3.7414187342928543`*^9, 3.7414255549171333`*^9, 3.744809167677113*^9, 3.744809246664177*^9, 3.74480928462808*^9, 3.7448093455094767`*^9, 3.744809378093*^9, 3.744809997216243*^9, 3.744810099146199*^9, 3.744810164447756*^9, 3.744810487496963*^9, { 3.744814584038282*^9, 3.7448145960830097`*^9}, 3.7450290995482883`*^9, 3.7452251805545692`*^9, 3.74522616242284*^9, 3.745226225022263*^9, 3.745310365117715*^9, 3.7453236999294367`*^9, 3.745648660314818*^9, 3.7456491257239656`*^9, 3.745649222468156*^9, 3.745649314106741*^9, 3.7456494202539787`*^9, {3.7456497210245857`*^9, 3.745649742121464*^9}, { 3.7456506772479486`*^9, 3.7456506885374603`*^9}, 3.753983565681525*^9}], Cell[BoxData["\<\"Scaleful\"\>"], "Print", CellChangeTimes->{ 3.741351256519195*^9, {3.741352377105689*^9, 3.74135238505509*^9}, 3.7414137219822397`*^9, 3.7414187342928543`*^9, 3.7414255549171333`*^9, 3.744809167677113*^9, 3.744809246664177*^9, 3.74480928462808*^9, 3.7448093455094767`*^9, 3.744809378093*^9, 3.744809997216243*^9, 3.744810099146199*^9, 3.744810164447756*^9, 3.744810487496963*^9, { 3.744814584038282*^9, 3.7448145960830097`*^9}, 3.7450290995482883`*^9, 3.7452251805545692`*^9, 3.74522616242284*^9, 3.745226225022263*^9, 3.745310365117715*^9, 3.7453236999294367`*^9, 3.745648660314818*^9, 3.7456491257239656`*^9, 3.745649222468156*^9, 3.745649314106741*^9, 3.7456494202539787`*^9, {3.7456497210245857`*^9, 3.745649742121464*^9}, { 3.7456506772479486`*^9, 3.7456506885374603`*^9}, 3.753983565682003*^9}] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{1863, 1056}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"9.0 for Linux x86 (64-bit) (November 20, 2012)", StyleDefinitions->FrontEnd`FileName[{"Report"}, "StandardReport.nb", CharacterEncoding -> "UTF-8"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 363, 5, 90, "Title"], Cell[CellGroupData[{ Cell[968, 31, 209, 3, 82, "Section"], Cell[1180, 36, 497, 9, 64, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[1714, 50, 166, 2, 82, "Section"], Cell[1883, 54, 22011, 588, 1912, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[23931, 647, 130, 2, 82, "Section"], Cell[CellGroupData[{ Cell[24086, 653, 91, 1, 32, "Subsection"], Cell[CellGroupData[{ Cell[24202, 658, 3217, 74, 67, "Input"], Cell[27422, 734, 3572, 97, 71, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[31043, 837, 115, 1, 32, "Subsection"], Cell[CellGroupData[{ Cell[31183, 842, 396, 10, 73, "Input"], Cell[31582, 854, 2285, 42, 41, "Output"], Cell[33870, 898, 3928, 93, 67, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[37835, 996, 542, 14, 45, "Input"], Cell[38380, 1012, 1717, 28, 41, "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[40158, 1047, 170, 3, 30, "Subtitle"], Cell[CellGroupData[{ Cell[40353, 1054, 150, 3, 32, "Subsection"], Cell[40506, 1059, 1325, 41, 127, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[41868, 1105, 148, 3, 32, "Subsection"], Cell[CellGroupData[{ Cell[42041, 1112, 892, 23, 45, "Input"], Cell[42936, 1137, 5781, 158, 71, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[48766, 1301, 179, 4, 32, "Subsection"], Cell[CellGroupData[{ Cell[48970, 1309, 4038, 125, 306, "Input"], Cell[53011, 1436, 4575, 114, 98, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[57635, 1556, 157, 3, 32, "Subsection"], Cell[57795, 1561, 262, 4, 30, "Text"], Cell[58060, 1567, 578, 15, 45, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[58675, 1587, 151, 2, 32, "Subsection"], Cell[58829, 1591, 404, 6, 30, "Text"], Cell[59236, 1599, 5441, 159, 316, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[64714, 1763, 121, 1, 32, "Subsection"], Cell[64838, 1766, 214, 3, 30, "Text"], Cell[65055, 1771, 1072, 30, 154, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[66164, 1806, 153, 3, 32, "Subsection"], Cell[66320, 1811, 4416, 82, 127, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[70773, 1898, 131, 3, 32, "Subsection"], Cell[CellGroupData[{ Cell[70929, 1905, 2263, 50, 73, "Input"], Cell[73195, 1957, 5001, 136, 163, "Output"] }, Open ]] }, Open ]], Cell[78223, 2097, 151, 4, 117, "Section"], Cell[CellGroupData[{ Cell[78399, 2105, 114, 1, 52, "Section"], Cell[CellGroupData[{ Cell[78538, 2110, 144, 2, 32, "Subsection"], Cell[CellGroupData[{ Cell[78707, 2116, 3219, 98, 161, "Input"], Cell[81929, 2216, 4305, 129, 141, "Output"], Cell[CellGroupData[{ Cell[86259, 2349, 222, 3, 23, "Print"], Cell[86484, 2354, 584, 16, 23, "Print"], Cell[87071, 2372, 1042, 31, 23, "Print"], Cell[88116, 2405, 375, 8, 23, "Print"], Cell[88494, 2415, 211, 3, 23, "Print"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[88766, 2425, 115, 1, 32, "Subsection"], Cell[CellGroupData[{ Cell[88906, 2430, 2600, 79, 112, "Input"], Cell[91509, 2511, 2950, 88, 71, "Output"], Cell[CellGroupData[{ Cell[94484, 2603, 198, 3, 23, "Print"], Cell[94685, 2608, 488, 14, 23, "Print"], Cell[95176, 2624, 801, 23, 23, "Print"], Cell[95980, 2649, 351, 8, 23, "Print"], Cell[96334, 2659, 187, 3, 23, "Print"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[96582, 2669, 115, 1, 32, "Subsection"], Cell[CellGroupData[{ Cell[96722, 2674, 2594, 79, 112, "Input"], Cell[99319, 2755, 2947, 88, 71, "Output"], Cell[CellGroupData[{ Cell[102291, 2847, 192, 3, 23, "Print"], Cell[102486, 2852, 486, 14, 23, "Print"], Cell[102975, 2868, 797, 23, 23, "Print"], Cell[103775, 2893, 576, 15, 23, "Print"], Cell[104354, 2910, 184, 3, 23, "Print"] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[104611, 2921, 129, 3, 82, "Section"], Cell[CellGroupData[{ Cell[104765, 2928, 165, 2, 31, "Subsubsection"], Cell[CellGroupData[{ Cell[104955, 2934, 3763, 92, 112, "Input"], Cell[108721, 3028, 4853, 114, 71, "Output"], Cell[CellGroupData[{ Cell[113599, 3146, 462, 7, 23, "Print"], Cell[114064, 3155, 708, 16, 23, "Print"], Cell[114775, 3173, 867, 18, 23, "Print"] }, Open ]], Cell[115657, 3194, 3028, 61, 46, "Output"], Cell[CellGroupData[{ Cell[118710, 3259, 613, 12, 23, "Print"], Cell[119326, 3273, 449, 7, 23, "Print"] }, Open ]] }, Open ]] }, Open ]], Cell[119814, 3285, 25, 0, 31, "Subsubsection"], Cell[CellGroupData[{ Cell[119864, 3289, 145, 2, 31, "Subsubsection"], Cell[CellGroupData[{ Cell[120034, 3295, 2777, 81, 112, "Input"], Cell[122814, 3378, 3676, 98, 71, "Output"], Cell[CellGroupData[{ Cell[126515, 3480, 915, 13, 23, "Print"], Cell[127433, 3495, 1207, 24, 23, "Print"], Cell[128643, 3521, 1518, 33, 23, "Print"] }, Open ]], Cell[130176, 3557, 2961, 77, 69, "Output"], Cell[CellGroupData[{ Cell[133162, 3638, 1068, 18, 23, "Print"], Cell[134233, 3658, 902, 13, 23, "Print"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[135196, 3678, 260, 3, 32, "Subsection"], Cell[CellGroupData[{ Cell[135481, 3685, 2621, 77, 134, "Input"], Cell[138105, 3764, 2901, 87, 71, "Output"], Cell[CellGroupData[{ Cell[141031, 3855, 879, 12, 23, "Print"], Cell[141913, 3869, 1061, 18, 23, "Print"], Cell[142977, 3889, 1187, 20, 23, "Print"] }, Open ]], Cell[144179, 3912, 909, 29, 46, "Output"], Cell[CellGroupData[{ Cell[145113, 3945, 1032, 17, 23, "Print"], Cell[146148, 3964, 868, 12, 23, "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[147065, 3982, 162, 2, 31, "Subsubsection"], Cell[CellGroupData[{ Cell[147252, 3988, 1167, 32, 86, "Input"], Cell[CellGroupData[{ Cell[148444, 4024, 870, 12, 23, "Print"], Cell[149317, 4038, 1118, 21, 23, "Print"], Cell[150438, 4061, 1277, 23, 23, "Print"], Cell[151718, 4086, 1023, 17, 23, "Print"], Cell[152744, 4105, 859, 12, 23, "Print"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, Open ]] } ] *)