(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 116967, 2608] NotebookOptionsPosition[ 107081, 2446] NotebookOutlinePosition[ 107853, 2471] CellTagsIndexPosition[ 107810, 2468] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[" FRW Models ", "Title", FontSize->24,ExpressionUUID->"6ab56dfe-2a2d-4adc-9284-94516474d8bf"], Cell[TextData[{ "This notebook numerically integrates the modified FRW equation \n\n", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ FractionBox["1", "2"], SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{ StyleBox["d", "TI"], OverscriptBox[ StyleBox["a", "TI"], "~"]}], RowBox[{ StyleBox["d", "TI"], OverscriptBox[ StyleBox["t", "TI"], "~"]}]], ")"}], "2"], "+", SubscriptBox[ StyleBox["U", "TI"], RowBox[{ StyleBox["e", "TI"], StyleBox["f", "TI"], StyleBox["f", "TI"]}]], "\[LongEqual]", FractionBox[ SubscriptBox["\[CapitalOmega]", StyleBox["k", "TI"]], "2"]}], TraditionalForm], "errors" -> {}, "input" -> "\\frac{1}{2} \\left( \\frac{d\\tilde{a}}{d\\tilde{t}} \\right)^2 \ +{\\mathcal U}_{eff} = \\frac{\\Omega_k}{2}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "822401ad-f808-4776-8779-34559545d832"], "\n\nwith\n\n", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ StyleBox["U", "TI"], RowBox[{ StyleBox["e", "TI"], StyleBox["f", "TI"], StyleBox["f", "TI"]}]], "\[LongEqual]", "-", FractionBox["1", "2"], RowBox[{"(", RowBox[{ SubscriptBox["\[CapitalOmega]", "\[CapitalLambda]"], SuperscriptBox[ OverscriptBox[ StyleBox["a", "TI"], "~"], "2"], "+", FractionBox[ SubscriptBox["\[CapitalOmega]", StyleBox["m", "TI"]], OverscriptBox[ StyleBox["a", "TI"], "~"]], "+", FractionBox[ SubscriptBox["\[CapitalOmega]", StyleBox["r", "TI"]], SuperscriptBox[ OverscriptBox[ StyleBox["a", "TI"], "~"], "2"]]}], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "U_{eff} = - \\frac{1}{2} \\left( \\Omega_\\Lambda \ \\tilde{a}^2+\\frac{\\Omega_m}{\\tilde{a}} + \\frac{\\Omega_r}{\\tilde{a}^2} \ \\right)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "9e56f951-e277-430e-abc6-abfa563bdd36"], "\n\nto compute a specific homogeneous isotropic cosmological model for the \ inputs (1) the Hubble constant ", Cell[BoxData[ FormBox[ SubscriptBox["H", RowBox[{"0", " "}]], TraditionalForm]],ExpressionUUID-> "3639216a-3184-44a4-a337-c255b3882439"], ", and the three \[CapitalOmega]'s for radiation, matter, and cosmological \ constant. Note bene: The program assumes these parameters are such that the \ universe started with a big bang, which we know may not always be the case. \ But it appears to be for our universe." }], "Text", CellChangeTimes->{{3.734870189567423*^9, 3.73487022526422*^9}, { 3.924003241961752*^9, 3.924003274114379*^9}, {3.924003389420176*^9, 3.924003396562475*^9}, {3.924003494391067*^9, 3.924003601293056*^9}, { 3.986976005808714*^9, 3.9869760117279577`*^9}},ExpressionUUID->"b8391488-a377-44e7-b60f-\ e71e9d98e4cb"], Cell[CellGroupData[{ Cell["Clearing the variables used:", "Subsection",ExpressionUUID->"e6aedb34-b49a-400c-af4e-68999a2f63d2"], Cell["\<\ Clear all the variables that will be used in the calculation:\ \>", "Text", CellChangeTimes->{ 3.7348702385853167`*^9},ExpressionUUID->"e4806ddc-21e9-42f4-9dd9-\ 6885efc3084b"], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{ "\[CapitalOmega]r", ",", "\[CapitalOmega]m", ",", "\[CapitalOmega]\[CapitalLambda]", ",", "\[CapitalOmega]k", ",", "h", ",", "Th", ",", " ", "x", ",", "y", ",", "a0", ",", "t0", ",", "\[CapitalOmega]crit", ",", "yend", ",", "ymax"}], "]"}]], "Input", CellChangeTimes->{{3.544186480361396*^9, 3.544186501559106*^9}, { 3.734870064257477*^9, 3.7348700965798798`*^9}, {3.9240037407156*^9, 3.924003741155175*^9}}, CellLabel->"In[43]:=",ExpressionUUID->"23463587-9071-455e-9b3f-05e542a62f83"] }, Open ]], Cell[CellGroupData[{ Cell["Parameters of a FRW model:", "Subsection",ExpressionUUID->"deb930ed-d043-4f30-a67d-2d9a6b775af9"], Cell[TextData[{ "Four parameters specify a FRW cosmological model. First, there are the \ three", StyleBox[" \[CapitalOmega] '", FontSlant->"Italic"], "s :" }], "Text", CellChangeTimes->{{3.544186507848159*^9, 3.544186514689322*^9}, 3.924003658411167*^9},ExpressionUUID->"ad8bab4c-b46f-41b2-a51a-\ fea95c48d9df"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[CapitalOmega]r", "=", RowBox[{".1", "*", RowBox[{"10", "^", RowBox[{"(", RowBox[{"-", "5"}], ")"}]}]}]}]], "Input", CellChangeTimes->{{3.543849613990481*^9, 3.543849615711028*^9}, 3.544186528259637*^9, {3.92408233265228*^9, 3.924082334957397*^9}, { 3.924189709082595*^9, 3.924189709809947*^9}, {3.986975922404738*^9, 3.986975925899373*^9}}, CellLabel->"In[44]:=",ExpressionUUID->"30cc047b-3a96-4273-a0eb-8c275e05154d"], Cell[BoxData["1.0000000000000002`*^-6"], "Output", CellChangeTimes->{ 3.543849617066078*^9, 3.544185876242299*^9, 3.544186682357336*^9, 3.607337257405223*^9, 3.924004120789707*^9, 3.924004279312621*^9, 3.924077939874192*^9, {3.924078007073792*^9, 3.9240780329237843`*^9}, 3.924078087828835*^9, 3.9240781342817383`*^9, 3.924078165930067*^9, 3.9240782304957247`*^9, 3.92408215263313*^9, {3.924082336179925*^9, 3.9240823608142767`*^9}, {3.924189727281798*^9, 3.924189851712995*^9}, 3.986975961481249*^9, 3.986979671975518*^9, 3.9869797030168467`*^9}, CellLabel->"Out[44]=",ExpressionUUID->"89ac8e4d-921f-4b33-aba2-bf6339dd19ef"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[CapitalOmega]m", "=", ".3"}]], "Input", CellChangeTimes->{{3.543849620690419*^9, 3.543849622321785*^9}, 3.544186528826647*^9, {3.5441866652498627`*^9, 3.544186669765218*^9}, { 3.544186756170513*^9, 3.544186758873932*^9}, {3.924077923800068*^9, 3.924077923903597*^9}, {3.9240780260425053`*^9, 3.924078026779439*^9}, { 3.924078154653441*^9, 3.924078156173443*^9}, {3.9240823395875273`*^9, 3.92408233996104*^9}, {3.924189715282794*^9, 3.924189847854536*^9}, { 3.986975930354418*^9, 3.986975939477872*^9}}, CellLabel->"In[45]:=",ExpressionUUID->"f252d9d9-6a33-4bc7-9c28-de6b3e4e97c8"], Cell[BoxData["0.3`"], "Output", CellChangeTimes->{ 3.543849628090397*^9, 3.5441858775317907`*^9, 3.544186683355575*^9, 3.5441867598627977`*^9, 3.60733725919232*^9, 3.924004120802409*^9, 3.924004279319478*^9, 3.924077939879917*^9, {3.924078007091679*^9, 3.924078032929825*^9}, 3.924078087834594*^9, {3.9240781342875433`*^9, 3.92407816593591*^9}, 3.924078230501398*^9, 3.924082152639534*^9, 3.924082360830448*^9, {3.924189727287821*^9, 3.924189851719092*^9}, 3.9869759615044937`*^9, 3.9869796719806356`*^9, 3.98697970302071*^9}, CellLabel->"Out[45]=",ExpressionUUID->"df38846c-2aab-452b-8345-a4504613c6bd"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[CapitalOmega]\[CapitalLambda]", "=", ".7"}]], "Input", CellChangeTimes->{ 3.544186529274458*^9, {3.5441866734054537`*^9, 3.5441866742612543`*^9}, 3.73487010520903*^9, {3.924077917726976*^9, 3.9240779189820423`*^9}, { 3.924078076374337*^9, 3.924078077862118*^9}, {3.924082344566*^9, 3.924082349990781*^9}, {3.924189758273526*^9, 3.924189758881691*^9}, { 3.986976028714679*^9, 3.9869760294020348`*^9}, {3.986979695294531*^9, 3.986979696258795*^9}, {3.986979726489603*^9, 3.9869797461843767`*^9}},ExpressionUUID->"d6a4e5a2-0971-402d-a52c-\ 3d50751f6ea5"], Cell[BoxData["0.2`"], "Output", CellChangeTimes->{ 3.543849629401236*^9, 3.5441858786010838`*^9, 3.5441866844101877`*^9, 3.6073372620372543`*^9, 3.9240041208176126`*^9, 3.9240042793374777`*^9, 3.924077939884907*^9, {3.924078007097678*^9, 3.924078032946207*^9}, 3.924078087840117*^9, 3.924078134295467*^9, 3.924078165954981*^9, 3.924078230507018*^9, 3.924082152654806*^9, 3.92408236083605*^9, { 3.9241897273056726`*^9, 3.9241898517301893`*^9}, 3.986975961508101*^9, 3.986979671992787*^9, 3.986979703023638*^9}, CellLabel->"Out[46]=",ExpressionUUID->"4f82c729-2b0b-4960-8cb1-a47b4f54b8f6"] }, Open ]], Cell[TextData[{ "The last parameter is the Hubble constant ", Cell[BoxData[ FormBox[ SubscriptBox["H", "0"], TraditionalForm]],ExpressionUUID-> "3c15b498-041c-4834-966f-cb701b70f3c1"], ", or equivalently the Hubble time ", Cell[BoxData[ FormBox[ SubscriptBox["T", "0"], TraditionalForm]],ExpressionUUID-> "80fce731-0403-4e78-924e-2c867d72fd1f"], "=1", StyleBox["/", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox["H", "0"], TraditionalForm]],ExpressionUUID-> "cccb818f-66fd-47f7-afe1-52f66a0b978f"], " which we denote here by ", StyleBox["Th. ", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ SubscriptBox["T", "0"], TraditionalForm]],ExpressionUUID-> "cd8c3419-009f-4377-bf4b-144d1677be1c"], " has the dimensions of time. A billion years (Gyr) is a convenient unit \ of time for cosmology and ", Cell[BoxData[ FormBox[ SubscriptBox["T", "0"], TraditionalForm]],ExpressionUUID-> "0ba0597f-ed50-4e22-8f1e-bac2e072f811"], StyleBox["=9.788 ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["h", RowBox[{"-", "1"}]], " ", "Gyr"}], ",", " ", RowBox[{ RowBox[{"where", " ", "h"}], "=", SubscriptBox["H", "0"]}]}], TraditionalForm]],ExpressionUUID-> "b2dc386f-08be-4cea-a277-6d168c67207b"], "/[100 (km/s)/Mpc]. Thus, for ", Cell[BoxData[ FormBox[ SubscriptBox["H", "0"], TraditionalForm]],ExpressionUUID-> "ce447692-55c2-4329-982a-ae0f17cf70ad"], "=67.3", StyleBox["(km/sec)/Mpc", FontVariations->{"CompatibilityType"->0}], StyleBox[", ", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.6072723847630587`*^9, 3.607272385964484*^9}},ExpressionUUID->"3ae14679-7710-4cb7-a0ba-\ 12b2a2f3a7c3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"h", "=", ".7"}]], "Input", CellChangeTimes->{{3.607272389925571*^9, 3.607272391677503*^9}, { 3.924077997761505*^9, 3.92407802228087*^9}, {3.986975946768263*^9, 3.986975948074192*^9}}, CellLabel->"In[47]:=",ExpressionUUID->"8ee56d3f-3d2f-4baa-af9a-f9f58d70cc79"], Cell[BoxData["0.7`"], "Output", CellChangeTimes->{ 3.543849634174951*^9, 3.544185890239428*^9, 3.5441866889985228`*^9, 3.607337267016509*^9, 3.924004120823676*^9, 3.924004279347785*^9, 3.924077939889864*^9, {3.924078007102768*^9, 3.924078032952948*^9}, 3.9240780878568287`*^9, 3.9240781343108807`*^9, 3.924078165972435*^9, 3.9240782305241537`*^9, 3.9240821526602907`*^9, 3.924082360849195*^9, { 3.92418972731434*^9, 3.92418985174833*^9}, 3.9869759615110607`*^9, 3.9869796720020447`*^9, 3.986979703029059*^9}, CellLabel->"Out[47]=",ExpressionUUID->"dd55df7d-7e22-4b8c-9f4e-701a32803d6a"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Th", "=", RowBox[{"9.788", "/", "h"}]}]], "Input", CellLabel->"In[48]:=",ExpressionUUID->"c00bea6e-86a4-48c3-891f-9bfbaadb391c"], Cell[BoxData["13.982857142857144`"], "Output", CellChangeTimes->{ 3.543849642240851*^9, 3.544185892953197*^9, 3.5441866911040783`*^9, 3.607337269794415*^9, 3.924004120841691*^9, 3.924004279365653*^9, 3.924077939895309*^9, {3.92407800711061*^9, 3.924078032971128*^9}, 3.9240780878635798`*^9, 3.924078134316928*^9, 3.924078165978362*^9, 3.924078230531205*^9, 3.924082152665372*^9, 3.92408236085483*^9, { 3.9241897273287477`*^9, 3.924189851754624*^9}, 3.986975961514104*^9, 3.986979672017254*^9, 3.986979703040361*^9}, CellLabel->"Out[48]=",ExpressionUUID->"3a70c617-6fd6-4078-9e7e-7b54a16e1be8"] }, Open ]], Cell[TextData[{ "We define ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalOmega]", "k"], TraditionalForm]],ExpressionUUID-> "cfc0d154-f04e-410b-ab82-f363963aaf94"], " for curvature. It is related to the other ", StyleBox["\[CapitalOmega]", FontSlant->"Italic"], "'s by:" }], "Text", CellChangeTimes->{ 3.5441865299978027`*^9, {3.924003701353318*^9, 3.924003724316926*^9}},ExpressionUUID->"24e96611-0854-4827-b7ea-\ 4c4bf1f88fe4"], Cell[BoxData[ RowBox[{"\[CapitalOmega]k", ":=", RowBox[{ "1", "-", "\[CapitalOmega]r", "-", "\[CapitalOmega]m", "-", "\[CapitalOmega]\[CapitalLambda]"}]}]], "Input", CellChangeTimes->{{3.544186530850996*^9, 3.54418653246659*^9}, 3.734870116642231*^9, {3.924003730160471*^9, 3.924003731009303*^9}}, CellLabel->"In[49]:=",ExpressionUUID->"d13c5cb9-d8b4-45ab-aced-923155930d27"], Cell["For the parameters above ", "Text",ExpressionUUID->"eb1643e8-f0cf-42d8-a1b8-6fb73e47c416"], Cell[CellGroupData[{ Cell[BoxData["\[CapitalOmega]k"], "Input", CellChangeTimes->{ 3.544186533001226*^9, {3.924003733207151*^9, 3.924003733644986*^9}}, CellLabel->"In[50]:=",ExpressionUUID->"2896bc8b-dfd3-4241-9fe7-f9285d66cba8"], Cell[BoxData["0.49999899999999997`"], "Output", CellChangeTimes->{ 3.543849661006405*^9, 3.544186706406258*^9, 3.5441867669958267`*^9, 3.6073372766462793`*^9, 3.924004120860054*^9, 3.924004279375374*^9, 3.924077939915461*^9, {3.9240780071285963`*^9, 3.924078032990347*^9}, 3.92407808788342*^9, 3.924078134328587*^9, 3.9240781659984417`*^9, 3.924078230539895*^9, 3.924082152687841*^9, 3.924082360863113*^9, { 3.924189727338809*^9, 3.9241898517779713`*^9}, 3.986975961537969*^9, 3.986979672022389*^9, 3.9869797030560102`*^9}, CellLabel->"Out[50]=",ExpressionUUID->"0713fa54-e78a-4d80-8bca-1943608e6ee7"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Integrating the FRW equation in dimensionless variables:", "Subsection",ExpressionUUID->"9200126f-1431-4dca-b993-724b62b14593"], Cell[TextData[{ "Its convenient to rewrite the FRW equation in terms of dimensionless \ variables as above. Here it is convenient to write ", StyleBox["x", FontSlant->"Italic"], " for ", Cell[BoxData[ FormBox[ OverscriptBox["t", "~"], TraditionalForm]],ExpressionUUID-> "422466bb-5ed6-4435-81c3-61c46ca237ec"], ", and", StyleBox[" y ", FontSlant->"Italic"], "for ", Cell[BoxData[ FormBox[ OverscriptBox["a", "~"], TraditionalForm]],ExpressionUUID-> "86ccecdb-c6a9-4b14-a44a-e584e9753bcd"], ". The FRW equation is then the same as that of a non-relativistic particle \ moving in a potential U", StyleBox["(y) ", FontSlant->"Italic"], "where ", StyleBox["y ", FontSlant->"Italic"], "is the particle's position and ", StyleBox["x ", FontSlant->"Italic"], "is the time. 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" }], "Text", CellChangeTimes->{{3.924003823638357*^9, 3.9240038332179527`*^9}, { 3.9240041873162117`*^9, 3.924004193656588*^9}},ExpressionUUID->"63ef8e17-9bfd-48f6-b1e0-\ b89c5c223bba"], Cell[TextData[{ "We plot this ", StyleBox["U(y) ", FontSlant->"Italic"], "out to a value ", StyleBox["y=yend, ", FontSlant->"Italic"], "also showing a horizontal line at the value of ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalOmega]", "k"], TraditionalForm]],ExpressionUUID-> "46217160-df75-47c3-a61b-5577058261e6"], "/2. (You might have to change ", StyleBox["yend ", FontWeight->"Bold"], StyleBox["and edit ", FontVariations->{"CompatibilityType"->0}], " ", StyleBox["PlotRange ", FontWeight->"Bold"], StyleBox["in the ", FontVariations->{"CompatibilityType"->0}], StyleBox["Show ", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox["statement below to get an informative plot.) 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", "Text", CellChangeTimes->{ 3.9240039033526573`*^9},ExpressionUUID->"ba90b5ad-34e6-4a6f-89b5-\ e9771c1e559e"], Cell[TextData[{ "For values of ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalOmega]", "k"], TraditionalForm]],ExpressionUUID-> "3b29b22d-045f-453d-82cf-4b9db6b85d81"], " above the maximum of the potential the expansion is unbounded in time; \ for values that are below the expansion is bounded and the universe \ recollapses. The value just at the maximum divides these two behaviors. We \ denote that value by ", "\[CapitalOmega]", StyleBox["crit.", FontWeight->"Bold"] }], "Text", CellChangeTimes->{ 3.54418655588161*^9, {3.924003909726821*^9, 3.924003909986376*^9}},ExpressionUUID->"8476146b-7e78-4979-9dda-\ 1d08c432c551"], Cell[BoxData[ RowBox[{"crit", ":=", " ", RowBox[{"FindMinimum", "[", RowBox[{ RowBox[{ RowBox[{"-", "2"}], RowBox[{"U", "[", "yy", "]"}]}], ",", RowBox[{"{", RowBox[{"yy", ",", " ", ".5"}], "}"}]}], "]"}]}]], "Input", CellLabel->"In[57]:=",ExpressionUUID->"3740c1c5-696a-4d99-9c68-fb1efb9a576e"], Cell[BoxData[ RowBox[{"\[CapitalOmega]crit", ":=", RowBox[{"If", "[", RowBox[{ RowBox[{"\[CapitalOmega]\[CapitalLambda]", ">", ".000001"}], ",", RowBox[{"-", RowBox[{"crit", "[", RowBox[{"[", "1", "]"}], "]"}]}], ",", " ", "0"}], "]"}]}]], "Input", CellChangeTimes->{{3.5441865570495*^9, 3.54418655809352*^9}, { 3.92400415963719*^9, 3.924004170138816*^9}}, CellLabel->"In[58]:=",ExpressionUUID->"583820f6-dafd-465b-9dd4-97e8c8612159"], Cell[CellGroupData[{ Cell[BoxData["\[CapitalOmega]crit"], "Input", CellChangeTimes->{3.544186559145056*^9}, CellLabel->"In[59]:=",ExpressionUUID->"01570fca-4386-4bbb-9d3e-58ed4c4aa626"], Cell[BoxData[ RowBox[{"-", "0.4952902987449596`"}]], "Output", CellChangeTimes->{ 3.543849722480421*^9, 3.544186218600436*^9, 3.5441868050825443`*^9, 3.607337302007023*^9, 3.924004121010762*^9, 3.924004279505962*^9, 3.9240779401231413`*^9, {3.924078007263979*^9, 3.9240780331557283`*^9}, 3.924078088000018*^9, 3.924078134438815*^9, 3.9240781661155376`*^9, 3.924078230671218*^9, 3.924082152812028*^9, 3.924082361007127*^9, { 3.92418972748875*^9, 3.9241898519230556`*^9}, 3.9869759616688967`*^9, 3.986979672415781*^9, 3.9869797031318293`*^9}, CellLabel->"Out[59]=",ExpressionUUID->"3c964217-7dca-444c-a7bf-3e6608ed9230"] }, Open ]], Cell[TextData[{ "For bounded cases the integration in (*) can extend only to the first \ turning point where the denominator vanishes. This represents the expansion \ of a universe up to its maximum scale factor ", StyleBox["ymax. 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This is relevant only for closed models where ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalOmega]", "k"], TraditionalForm]],ExpressionUUID-> "94db163c-58bb-4555-b616-abafd38076e9"], "/2 < ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["U", "max"], ".", " "}], TraditionalForm]],ExpressionUUID-> "45c365c5-9b88-4dab-8d20-c970521a3ce5"], " (If the program fails to identify the correct turning point, its possible \ that you might have to insert a statement ", StyleBox["ymax= ", FontWeight->"Bold"], StyleBox["to give it the correct value and then run the program again.) 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