(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 32002, 869] NotebookOptionsPosition[ 29327, 818] NotebookOutlinePosition[ 29734, 835] CellTagsIndexPosition[ 29691, 832] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[{ StyleBox[ RowBox[{"Blockchain", " ", RowBox[{"Model", "--"}], "VM", " ", "leader"}], "Subtitle"], "\n", StyleBox[ RowBox[{"The", " ", RowBox[{"SO", "'"}], "s", " ", "Effort", " ", "Under", " ", "Insurance", " ", "Model"}], "Section"]}], "Input", CellChangeTimes->{{3.859451469465022*^9, 3.8594515050588007`*^9}},ExpressionUUID->"75a74793-df95-44ed-a4ee-\ 4109b7b80cc1"], Cell[BoxData[{ RowBox[{"Profitsb", ":=", RowBox[{ RowBox[{ RowBox[{"-", "\[Alpha]"}], " ", "pb"}], "+", RowBox[{"eb", RowBox[{"(", RowBox[{"f", "-", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Alpha]"}], ")"}], "pb"}]}], ")"}]}], "-", "cs", " ", "-", RowBox[{"rb", RowBox[{"(", RowBox[{ RowBox[{"\[Alpha]", " ", "pb"}], "+", "cs", "+", "f"}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"\[Alpha]", " ", "pb"}], "+", "cs", "+", "f"}], ")"}]}], "-", "csb"}]}], "\[IndentingNewLine]", RowBox[{"Profitib", ":=", RowBox[{ RowBox[{"rb", RowBox[{"(", RowBox[{ RowBox[{"\[Alpha]", " ", "pb"}], "+", "cs", "+", "f"}], ")"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"\[Alpha]", " ", "pb"}], "+", "cs", "+", "f"}], ")"}]}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Profitvb", ":=", RowBox[{ RowBox[{"\[Alpha]", " ", "pb"}], "+", RowBox[{"eb", " ", RowBox[{"(", RowBox[{"1", "-", "\[Alpha]"}], ")"}], "pb"}], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], "\[Theta]"}], "-", RowBox[{"kb", "*", RowBox[{"(", RowBox[{"eb", "*", "eb"}], ")"}]}], "-", "cv", "-", "cvb"}]}], "\[IndentingNewLine]"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.84774692485349*^9, 3.847746972128188*^9}, { 3.847747397472211*^9, 3.8477474241671333`*^9}, {3.847747702882146*^9, 3.847747709330879*^9}, {3.847747747143688*^9, 3.8477477763814287`*^9}, { 3.847749431163137*^9, 3.8477494324598417`*^9}, {3.84793877916921*^9, 3.847938800764226*^9}, {3.8594515624979887`*^9, 3.859451643321137*^9}, { 3.8594519059088984`*^9, 3.859451921335861*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"7a7a4910-7af5-4f1b-ab75-fed66f353db2"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.8594530186979113`*^9, 3.8594530187028437`*^9}},ExpressionUUID->"99af2956-4950-4c69-851b-\ 6290fd3c1aa2"], Cell["\<\ we assume the SA , to simplfy the question, we set the \[Alpha]=1- \ \[Beta]*pb, then the payoff of SA is :\ \>", "Text", CellChangeTimes->{{3.859451927750812*^9, 3.859452050061626*^9}, { 3.859453039937043*^9, 3.859453050198208*^9}},ExpressionUUID->"a0037cfe-5c1f-4b8d-8185-\ 20198fd6ce74"], Cell[BoxData[ RowBox[{"Profitsb2", ":=", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Beta]", " ", "pb"}]}], ")"}]}], " ", "pb"}], "+", RowBox[{"eb", " ", RowBox[{"(", RowBox[{"f", "-", RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Beta]", " ", "pb"}]}], " ", ")"}]}], ")"}], "pb"}]}], ")"}]}], "-", "cs", " ", "-", RowBox[{"rb", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Beta]", " ", "pb"}]}], " ", ")"}], "pb"}], "+", "cs", "+", "f"}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], " ", ")"}], RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Beta]", " ", "pb"}]}], " ", ")"}], " ", "pb"}], "+", "cs", "+", "f"}], ")"}]}], "-", "csb"}]}]], "Input", CellChangeTimes->{{3.8594520587305403`*^9, 3.859452076544445*^9}, 3.859452108271451*^9, {3.85945305438356*^9, 3.8594531347796297`*^9}}, CellLabel->"In[5]:=",ExpressionUUID->"16974854-80e4-4192-b50f-98d08ffa64db"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"D", "[", RowBox[{"Profitsb2", ",", "pb"}], "]"}]], "Input", CellChangeTimes->{{3.859452087522633*^9, 3.8594521164600077`*^9}}, CellLabel->"In[6]:=",ExpressionUUID->"4e0089e6-b8b4-46a0-9355-22db64854da5"], Cell[BoxData[ RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", "pb", " ", "\[Beta]"}], "-", RowBox[{"2", " ", "eb", " ", "pb", " ", "\[Beta]"}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"2", " ", "pb", " ", "\[Beta]"}]}], ")"}]}], "-", RowBox[{"rb", " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"2", " ", "pb", " ", "\[Beta]"}]}], ")"}]}]}]], "Output", CellChangeTimes->{3.859452117158161*^9, 3.85945314200105*^9}, CellLabel->"Out[6]=",ExpressionUUID->"f994b317-3b4c-4445-b469-9527375ade81"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"%", "\[Equal]", "0"}], ",", "pb"}], "]"}]], "Input", CellChangeTimes->{{3.8594521211161537`*^9, 3.8594521306451263`*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"e06f9c6e-8834-4f37-8e81-841bf9938ec2"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"pb", "\[Rule]", FractionBox[ RowBox[{"eb", "+", "rb"}], RowBox[{"2", " ", "rb", " ", "\[Beta]"}]]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.8594521315202093`*^9, 3.859453145801064*^9}, CellLabel->"Out[7]=",ExpressionUUID->"01579329-5993-4936-911d-d93c936d508c"] }, Open ]], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"The", " ", RowBox[{"IC", "'"}], "s", " ", "Premium", " ", "Rate"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]"}], "Section"]], "Input", CellChangeTimes->{{3.859453179428543*^9, 3.859453203634452*^9}},ExpressionUUID->"65bf8622-a285-46f5-8238-\ 511fe9dd057a"], Cell[BoxData[{ RowBox[{"Profitib", ":=", RowBox[{ RowBox[{"rb", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Beta]", " ", "pb"}]}], " ", ")"}], " ", "pb"}], "+", "cs", "+", "f"}], ")"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Beta]", " ", "pb"}]}], " ", ")"}], " ", "pb"}], "+", "cs", "+", "f"}], ")"}]}]}]}], "\[IndentingNewLine]", RowBox[{"pb", ":=", FractionBox[ RowBox[{"eb", "+", "rb"}], RowBox[{"2", " ", "rb", " ", "\[Beta]"}]]}]}], "Input", CellChangeTimes->{{3.859453221799615*^9, 3.859453227558364*^9}, { 3.859453466575992*^9, 3.859453472410043*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"e993ae7c-d616-46e4-b138-54de54fbbfaf"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"Profitib", "\[Equal]", "0"}], ",", "rb"}], "]"}]], "Input", CellChangeTimes->{{3.859453241206469*^9, 3.85945325638657*^9}}, CellLabel->"In[3]:=",ExpressionUUID->"ac503916-d896-4e79-b126-10b60da99146"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"rb", "\[Rule]", RowBox[{"1", "-", "eb"}]}], "}"}], ",", RowBox[{"{", RowBox[{"rb", "\[Rule]", RowBox[{"-", FractionBox["eb", SqrtBox[ RowBox[{"1", "+", RowBox[{"4", " ", "cs", " ", "\[Beta]"}], "+", RowBox[{"4", " ", "f", " ", "\[Beta]"}]}]]]}]}], "}"}], ",", RowBox[{"{", RowBox[{"rb", "\[Rule]", FractionBox["eb", SqrtBox[ RowBox[{"1", "+", RowBox[{"4", " ", "cs", " ", "\[Beta]"}], "+", RowBox[{"4", " ", "f", " ", "\[Beta]"}]}]]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.859453265359044*^9, {3.859453480586802*^9, 3.859453499448471*^9}}, CellLabel->"Out[3]=",ExpressionUUID->"ac4b8d38-105d-4c63-b923-c2a6713e0001"] }, Open ]], Cell[BoxData[ StyleBox[ RowBox[{"The", " ", RowBox[{"VM", "'"}], "s", " ", "Optimal", " ", "Decision"}], "Section"]], "Input", CellChangeTimes->{{3.859453294941037*^9, 3.8594533075528297`*^9}},ExpressionUUID->"c2b5aea2-c267-4116-9a6d-\ 5018a4695103"], Cell[BoxData[{ RowBox[{"pb", ":=", FractionBox[ RowBox[{"eb", "+", "rb"}], RowBox[{"2", " ", "rb", " ", "\[Beta]"}]]}], "\[IndentingNewLine]", RowBox[{"rb", ":=", RowBox[{"1", "-", "eb"}]}], "\[IndentingNewLine]", RowBox[{"Profitvb", ":=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Beta]", " ", "pb"}]}], " ", ")"}], " ", "pb"}], "+", RowBox[{"eb", " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[Beta]", " ", "pb"}]}], " ", ")"}]}], ")"}], "pb"}], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], "\[Theta]"}], "-", RowBox[{"kb", "*", RowBox[{"(", RowBox[{"eb", "*", "eb"}], ")"}]}], "-", "cv", "-", "cvb"}]}]}], "Input",\ CellChangeTimes->{{3.8594533254312553`*^9, 3.859453345889015*^9}, { 3.8594535171293783`*^9, 3.859453520240615*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"38dc0ebb-de78-4680-8627-50cdc1b389ed"], Cell[CellGroupData[{ Cell[BoxData["Profitvb"], "Input", CellChangeTimes->{{3.859453354393592*^9, 3.859453357645629*^9}}, CellLabel->"In[4]:=",ExpressionUUID->"7a87c76f-26a3-4b51-9a51-8eb8ecbf321b"], Cell[BoxData[ RowBox[{ RowBox[{"-", "cv"}], "-", "cvb", "-", RowBox[{ SuperscriptBox["eb", "2"], " ", "kb"}], "+", FractionBox[ RowBox[{"1", "-", FractionBox["1", RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}]}]]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], " ", "\[Beta]"}]], "+", FractionBox["eb", RowBox[{"4", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], "2"], " ", "\[Beta]"}]], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], " ", "\[Theta]"}]}]], "Output", CellChangeTimes->{3.859453358360651*^9, 3.859453530365176*^9, 3.8594536020712843`*^9}, CellLabel->"Out[4]=",ExpressionUUID->"4f641119-303a-4f4e-80b9-19a91cfa7356"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"-", "cv"}], "-", "cvb", "-", RowBox[{ SuperscriptBox["eb", "2"], " ", "kb"}], "+", FractionBox[ RowBox[{"1", "-", FractionBox["1", RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}]}]]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], " ", "\[Beta]"}]], "+", FractionBox["eb", RowBox[{"4", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], "2"], " ", "\[Beta]"}]], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "eb"}], ")"}], " ", "\[Theta]"}]}], "]"}]], "Input", NumberMarks->False, CellLabel->"In[5]:=",ExpressionUUID->"ded6f8ea-78bf-4e7d-a664-ca14d0c9ac86"], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{"1", "+", RowBox[{"4", " ", "cv", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "eb"}], ")"}], " ", "\[Beta]"}], "+", RowBox[{"4", " ", "cvb", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "eb"}], ")"}], " ", "\[Beta]"}], "-", RowBox[{"4", " ", SuperscriptBox["eb", "2"], " ", "kb", " ", "\[Beta]"}], "+", RowBox[{"4", " ", SuperscriptBox["eb", "3"], " ", "kb", " ", "\[Beta]"}], "-", RowBox[{"4", " ", "\[Beta]", " ", "\[Theta]"}], "+", RowBox[{"8", " ", "eb", " ", "\[Beta]", " ", "\[Theta]"}], "-", RowBox[{"4", " ", SuperscriptBox["eb", "2"], " ", "\[Beta]", " ", "\[Theta]"}]}], RowBox[{"4", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "eb"}], ")"}], " ", "\[Beta]"}]]}]], "Output", CellChangeTimes->{3.859453533251834*^9, 3.859453608925981*^9}, CellLabel->"Out[5]=",ExpressionUUID->"58ebcbdc-e4d4-49c3-8e27-05ee5c38b524"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"D", "[", RowBox[{"%", ",", "eb"}], "]"}]], "Input", CellChangeTimes->{{3.859453365945187*^9, 3.859453382440831*^9}}, CellLabel->"In[6]:=",ExpressionUUID->"f9bc0284-d675-418f-ba6e-284634d09043"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ RowBox[{"4", " ", "cv", " ", "\[Beta]"}], "+", RowBox[{"4", " ", "cvb", " ", "\[Beta]"}], "-", RowBox[{"8", " ", "eb", " ", "kb", " ", "\[Beta]"}], "+", RowBox[{"12", " ", SuperscriptBox["eb", "2"], " ", "kb", " ", "\[Beta]"}], "+", RowBox[{"8", " ", "\[Beta]", " ", "\[Theta]"}], "-", RowBox[{"8", " ", "eb", " ", "\[Beta]", " ", "\[Theta]"}]}], RowBox[{"4", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "eb"}], ")"}], " ", "\[Beta]"}]]}], "+", FractionBox[ RowBox[{"1", "+", RowBox[{"4", " ", "cv", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "eb"}], ")"}], " ", "\[Beta]"}], "+", RowBox[{"4", " ", "cvb", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "eb"}], ")"}], " ", "\[Beta]"}], "-", RowBox[{"4", " ", SuperscriptBox["eb", "2"], " ", "kb", " ", "\[Beta]"}], "+", RowBox[{"4", " ", SuperscriptBox["eb", "3"], " ", "kb", " ", "\[Beta]"}], "-", RowBox[{"4", " ", "\[Beta]", " ", "\[Theta]"}], "+", RowBox[{"8", " ", "eb", " ", "\[Beta]", " ", "\[Theta]"}], "-", RowBox[{"4", " ", SuperscriptBox["eb", "2"], " ", "\[Beta]", " ", "\[Theta]"}]}], RowBox[{"4", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "eb"}], ")"}], "2"], " ", "\[Beta]"}]]}]], "Output", CellChangeTimes->{3.859453383539867*^9, 3.859453542913563*^9, 3.859453619813138*^9}, CellLabel->"Out[6]=",ExpressionUUID->"5f04edeb-cc52-451b-a69d-57773979429d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"%", "\[Equal]", "0"}], ",", "eb"}], "]"}]], "Input", CellChangeTimes->{{3.859453398418365*^9, 3.859453407362144*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"5fd05f78-28a4-474c-84ee-e27c6c0f2362"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"eb", "\[Rule]", RowBox[{ FractionBox[ RowBox[{ RowBox[{"4", " ", "kb", " ", "\[Beta]"}], "+", RowBox[{"\[Beta]", " ", "\[Theta]"}]}], RowBox[{"6", " ", "kb", " ", "\[Beta]"}]], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "64"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"64", " ", "kb", " ", SuperscriptBox["\[Beta]", "2"], " ", "\[Theta]"}], "-", RowBox[{"16", " ", SuperscriptBox["\[Beta]", "2"], " ", SuperscriptBox["\[Theta]", "2"]}]}], ")"}], "/", RowBox[{"(", RowBox[{"48", " ", SuperscriptBox["2", RowBox[{"2", "/", "3"}]], " ", "kb", " ", "\[Beta]", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "27"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"16", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "3"]}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "3"], " ", "\[Theta]"}], "+", RowBox[{"12", " ", "kb", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "2"]}], "-", RowBox[{"2", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "3"]}], "+", RowBox[{"3", " ", SqrtBox["3"], " ", SqrtBox[ RowBox[{ RowBox[{"27", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "4"]}], "-", RowBox[{"32", " ", SuperscriptBox["kb", "5"], " ", SuperscriptBox["\[Beta]", "5"]}], "+", RowBox[{"48", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "5"], " ", "\[Theta]"}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "3"]}]}]]}]}], ")"}], RowBox[{"1", "/", "3"}]]}], ")"}]}], "-", RowBox[{ FractionBox["1", RowBox[{"6", " ", SuperscriptBox["2", RowBox[{"1", "/", "3"}]], " ", "kb", " ", "\[Beta]"}]], SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "27"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"16", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "3"]}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "3"], " ", "\[Theta]"}], "+", RowBox[{"12", " ", "kb", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "2"]}], "-", RowBox[{"2", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "3"]}], "+", RowBox[{"3", " ", SqrtBox["3"], " ", SqrtBox[ RowBox[{ RowBox[{"27", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "4"]}], "-", RowBox[{"32", " ", SuperscriptBox["kb", "5"], " ", SuperscriptBox["\[Beta]", "5"]}], "+", RowBox[{"48", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "5"], " ", "\[Theta]"}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "3"]}]}]]}]}], ")"}], RowBox[{"1", "/", "3"}]]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"eb", "\[Rule]", RowBox[{ FractionBox[ RowBox[{ RowBox[{"4", " ", "kb", " ", "\[Beta]"}], "+", RowBox[{"\[Beta]", " ", "\[Theta]"}]}], RowBox[{"6", " ", "kb", " ", "\[Beta]"}]], "-", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "+", RowBox[{"\[ImaginaryI]", " ", SqrtBox["3"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "64"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"64", " ", "kb", " ", SuperscriptBox["\[Beta]", "2"], " ", "\[Theta]"}], "-", RowBox[{"16", " ", SuperscriptBox["\[Beta]", "2"], " ", SuperscriptBox["\[Theta]", "2"]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{"96", " ", SuperscriptBox["2", RowBox[{"2", "/", "3"}]], " ", "kb", " ", "\[Beta]", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "27"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"16", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "3"]}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "3"], " ", "\[Theta]"}], "+", RowBox[{"12", " ", "kb", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "2"]}], "-", RowBox[{"2", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "3"]}], "+", RowBox[{"3", " ", SqrtBox["3"], " ", SqrtBox[ RowBox[{ RowBox[{"27", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "4"]}], "-", RowBox[{"32", " ", SuperscriptBox["kb", "5"], " ", SuperscriptBox["\[Beta]", "5"]}], "+", RowBox[{"48", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "5"], " ", "\[Theta]"}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "3"]}]}]]}]}], ")"}], RowBox[{"1", "/", "3"}]]}], ")"}]}], "+", RowBox[{ FractionBox["1", RowBox[{"12", " ", SuperscriptBox["2", RowBox[{"1", "/", "3"}]], " ", "kb", " ", "\[Beta]"}]], RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[ImaginaryI]", " ", SqrtBox["3"]}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "27"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"16", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "3"]}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "3"], " ", "\[Theta]"}], "+", RowBox[{"12", " ", "kb", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "2"]}], "-", RowBox[{"2", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "3"]}], "+", RowBox[{"3", " ", SqrtBox["3"], " ", SqrtBox[ RowBox[{ RowBox[{"27", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "4"]}], "-", RowBox[{"32", " ", SuperscriptBox["kb", "5"], " ", SuperscriptBox["\[Beta]", "5"]}], "+", RowBox[{"48", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "5"], " ", "\[Theta]"}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "3"]}]}]]}]}], ")"}], RowBox[{"1", "/", "3"}]]}]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"eb", "\[Rule]", RowBox[{ FractionBox[ RowBox[{ RowBox[{"4", " ", "kb", " ", "\[Beta]"}], "+", RowBox[{"\[Beta]", " ", "\[Theta]"}]}], RowBox[{"6", " ", "kb", " ", "\[Beta]"}]], "-", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"\[ImaginaryI]", " ", SqrtBox["3"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "64"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"64", " ", "kb", " ", SuperscriptBox["\[Beta]", "2"], " ", "\[Theta]"}], "-", RowBox[{"16", " ", SuperscriptBox["\[Beta]", "2"], " ", SuperscriptBox["\[Theta]", "2"]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{"96", " ", SuperscriptBox["2", RowBox[{"2", "/", "3"}]], " ", "kb", " ", "\[Beta]", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "27"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"16", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "3"]}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "3"], " ", "\[Theta]"}], "+", RowBox[{"12", " ", "kb", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "2"]}], "-", RowBox[{"2", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "3"]}], "+", RowBox[{"3", " ", SqrtBox["3"], " ", SqrtBox[ RowBox[{ RowBox[{"27", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "4"]}], "-", RowBox[{"32", " ", SuperscriptBox["kb", "5"], " ", SuperscriptBox["\[Beta]", "5"]}], "+", RowBox[{"48", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "5"], " ", "\[Theta]"}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "3"]}]}]]}]}], ")"}], RowBox[{"1", "/", "3"}]]}], ")"}]}], "+", RowBox[{ FractionBox["1", RowBox[{"12", " ", SuperscriptBox["2", RowBox[{"1", "/", "3"}]], " ", "kb", " ", "\[Beta]"}]], RowBox[{ RowBox[{"(", RowBox[{"1", "+", RowBox[{"\[ImaginaryI]", " ", SqrtBox["3"]}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "27"}], " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "2"]}], "+", RowBox[{"16", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "3"]}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "3"], " ", "\[Theta]"}], "+", RowBox[{"12", " ", "kb", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "2"]}], "-", RowBox[{"2", " ", SuperscriptBox["\[Beta]", "3"], " ", SuperscriptBox["\[Theta]", "3"]}], "+", RowBox[{"3", " ", SqrtBox["3"], " ", SqrtBox[ RowBox[{ RowBox[{"27", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "4"]}], "-", RowBox[{"32", " ", SuperscriptBox["kb", "5"], " ", SuperscriptBox["\[Beta]", "5"]}], "+", RowBox[{"48", " ", SuperscriptBox["kb", "4"], " ", SuperscriptBox["\[Beta]", "5"], " ", "\[Theta]"}], "-", RowBox[{"24", " ", SuperscriptBox["kb", "3"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["kb", "2"], " ", SuperscriptBox["\[Beta]", "5"], " ", SuperscriptBox["\[Theta]", "3"]}]}]]}]}], ")"}], RowBox[{"1", "/", "3"}]]}]}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.859453625607356*^9}, CellLabel->"Out[7]=",ExpressionUUID->"770e4804-5182-4e1d-8ae8-6cbc31eceaf0"] }, Open ]] }, WindowSize->{808, 730}, WindowMargins->{{Automatic, 53}, {Automatic, 39}}, FrontEndVersion->"12.1 for Mac OS X x86 (64-bit) \ (2020\:5e743\:670813\:65e5)", StyleDefinitions->"Default.nb", ExpressionUUID->"182912a3-39dc-4894-a73a-58952deac396" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[558, 20, 407, 10, 66, "Input",ExpressionUUID->"75a74793-df95-44ed-a4ee-4109b7b80cc1"], Cell[968, 32, 1968, 56, 115, "Input",ExpressionUUID->"7a7a4910-7af5-4f1b-ab75-fed66f353db2"], Cell[2939, 90, 156, 3, 30, InheritFromParent,ExpressionUUID->"99af2956-4950-4c69-851b-6290fd3c1aa2"], Cell[3098, 95, 306, 7, 35, "Text",ExpressionUUID->"a0037cfe-5c1f-4b8d-8185-20198fd6ce74"], Cell[3407, 104, 1231, 38, 52, "Input",ExpressionUUID->"16974854-80e4-4192-b50f-98d08ffa64db"], Cell[CellGroupData[{ Cell[4663, 146, 232, 4, 44, "Input",ExpressionUUID->"4e0089e6-b8b4-46a0-9355-22db64854da5"], Cell[4898, 152, 611, 16, 34, "Output",ExpressionUUID->"f994b317-3b4c-4445-b469-9527375ade81"] }, Open ]], Cell[CellGroupData[{ Cell[5546, 173, 261, 5, 44, "Input",ExpressionUUID->"e06f9c6e-8834-4f37-8e81-841bf9938ec2"], Cell[5810, 180, 342, 8, 53, "Output",ExpressionUUID->"01579329-5993-4936-911d-d93c936d508c"] }, Open ]], Cell[6167, 191, 325, 8, 127, "Input",ExpressionUUID->"65bf8622-a285-46f5-8238-511fe9dd057a"], Cell[6495, 201, 898, 27, 72, "Input",ExpressionUUID->"e993ae7c-d616-46e4-b138-54de54fbbfaf"], Cell[CellGroupData[{ Cell[7418, 232, 263, 5, 44, "Input",ExpressionUUID->"ac503916-d896-4e79-b126-10b60da99146"], Cell[7684, 239, 816, 24, 56, "Output",ExpressionUUID->"ac4b8d38-105d-4c63-b923-c2a6713e0001"] }, Open ]], Cell[8515, 266, 264, 7, 50, "Input",ExpressionUUID->"c2b5aea2-c267-4116-9a6d-5018a4695103"], Cell[8782, 275, 989, 28, 94, "Input",ExpressionUUID->"38dc0ebb-de78-4680-8627-50cdc1b389ed"], Cell[CellGroupData[{ Cell[9796, 307, 178, 2, 30, "Input",ExpressionUUID->"7a87c76f-26a3-4b51-9a51-8eb8ecbf321b"], Cell[9977, 311, 794, 24, 64, "Output",ExpressionUUID->"4f641119-303a-4f4e-80b9-19a91cfa7356"] }, Open ]], Cell[CellGroupData[{ Cell[10808, 340, 778, 24, 73, "Input",ExpressionUUID->"ded6f8ea-78bf-4e7d-a664-ca14d0c9ac86"], Cell[11589, 366, 1018, 25, 55, "Output",ExpressionUUID->"58ebcbdc-e4d4-49c3-8e27-05ee5c38b524"] }, Open ]], Cell[CellGroupData[{ Cell[12644, 396, 222, 4, 44, "Input",ExpressionUUID->"f9bc0284-d675-418f-ba6e-284634d09043"], Cell[12869, 402, 1638, 42, 102, "Output",ExpressionUUID->"5f04edeb-cc52-451b-a69d-57773979429d"] }, Open ]], Cell[CellGroupData[{ Cell[14544, 449, 257, 5, 44, "Input",ExpressionUUID->"5fd05f78-28a4-474c-84ee-e27c6c0f2362"], Cell[14804, 456, 14507, 359, 574, "Output",ExpressionUUID->"770e4804-5182-4e1d-8ae8-6cbc31eceaf0"] }, Open ]] } ] *)