(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 29493, 921]*) (*NotebookOutlinePosition[ 30137, 943]*) (* CellTagsIndexPosition[ 30093, 939]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(\(\(\(\(\(\(\(\(This\ file\ \ considers\ the\ characteristic\ \ polynomial\ and\ \ shows\ that\[IndentingNewLine] 1\)\()\)\)\ it\ is\ positive\ at\ 0, \ \[IndentingNewLine]\ \ \ \ \ 2\)\()\)\)\ one\ root\ is\ r, \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ 3\)\()\)\)\ it\ is\ negative\ at\ r + d, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4\)\()\)\)\ \ it\ is\ negative\ at\ - d . \[IndentingNewLine]\nSuppose\ that\ K\ is\ calibrated\ s . t . \ theta = 1\ at\ the\ steady - \(\(state\)\(.\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(u[x_] = Log[x]\)], "Input"], Cell[BoxData[ \(Log[x]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(du[x_] = D[u[x], x]\)], "Input"], Cell[BoxData[ \(1\/x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ddu[x_] = D[du[x], x]\)], "Input"], Cell[BoxData[ \(\(-\(1\/x\^2\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mu[t_] = k*t^\((1 - al)\)\)], "Input"], Cell[BoxData[ \(k\ t\^\(1 - al\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(q[t_] = k*t^\((\(-al\))\)\)], "Input"], Cell[BoxData[ \(k\ t\^\(-al\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(dmu[t_] = D[mu[t], t]\)], "Input"], Cell[BoxData[ \(\((1 - al)\)\ k\ t\^\(-al\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(theta = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(w = z - K/k*\((r + d)\)\)\(\n\) \)\)], "Input"], Cell[BoxData[ \(\(-\(\(K\ \((d + r)\)\)\/k\)\) + z\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A = \((1 - al)\)/al\)], "Input"], Cell[BoxData[ \(\(1 - al\)\/al\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(m = mu[theta]/\((mu[theta] + d)\)\)], "Input"], Cell[BoxData[ \(k\/\(d + k\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c = Simplify[m*\((z - w)\) - K*theta*\((1 - m)\), {k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \(\(K\ r\)\/\(d + k\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a = Simplify[w/ c, {k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \(\(\((d + k)\)\ \((\(-\(\(K\ \((d + r)\)\)\/k\)\) + z)\)\)\/\(K\ r\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(p = a\)], "Input"], Cell[BoxData[ \(\(\((d + k)\)\ \((\(-\(\(K\ \((d + r)\)\)\/k\)\) + z)\)\)\/\(K\ r\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Y = Simplify[z* du[c]/\((r + d)\), {k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \(\(\((d + k)\)\ z\)\/\(K\ r\ \((d + r)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(X = Simplify[\((u[c] - u[b] - theta*du[w]*K/A)\)/\((r + d)\), {k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)\(\n\) \)\)], "Input"], Cell[BoxData[ \(\(\(-\(\(al\ k\ K\)\/\(\((\(-1\) + al)\)\ \((d\ K + K\ r - k\ \ z)\)\)\)\) - Log[b] + Log[\(K\ r\)\/\(d + k\)]\)\/\(d + r\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[""], "Input"], Cell[BoxData[ \(\(d\ K - al\ d\ K + al\ k\ z\)\/\(k\ r\)\)], "Output"] }, Closed]], Cell[BoxData[ \(\(\(Derive\ expressions\ for\ partials\ of\ the\ controls\ a, c, theta\ \ with\ respect\ to\ states\ m, p, X, Y\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(aX = Simplify[\(-a\)* A/\((A/\((r + d)\) + A*X + \((A*Log[a] + 1)\)/\((r + d)\))\), {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \(\(-\(\(\((1 - al)\)\ \((d + k)\)\ \((d + r)\)\ \((\(-\(\(K\ \((d + r)\)\)\/k\)\) + z)\)\)\/\(K\ r\ \((1 + \((1 - al)\)\ \((\(-\(\(al\ k\ K\)\/\(\((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\)\)\) - Log[b] + Log[\(K\ r\)\/\(d + k\)])\) - \((\(-1\) + al)\)\ Log[\(\((d + k)\)\ \((\(-\(\(K\ \((d + r)\)\)\/k\ \)\) + z)\)\)\/\(K\ r\)])\)\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(aX = Simplify[\(-\(\(\((1 - al)\)\ \((k + d)\)\ \((d + r)\)\ \((\(-\(\(K\ \((d + r)\)\)\/k\)\) + z)\)\)\/\(K\ r\ \((1 + \((1 - al)\)\ \((\(-\(\(al\ k\ K\)\/\(\((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\)\)\) - Log[b] + Log[\(K\ r\)\/\(k + d\)])\) - \((\(-1\) + al)\)\ \((\ Log[\((k + d)\)\/\(K\ r\)] + Log[b] + al/\((1 - al)\)* K/\((k\ z - \((r + d)\)\ K)\)*\((r + d + k)\)\ )\))\)\)\)\), {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \(\(\((\(-1\) + al)\)\ \((d + k)\)\ \((d + r)\)\ \((d\ K + K\ r - k\ z)\)\ \^2\)\/\(k\ K\ r\ \((\(-d\)\ K + al\ d\ K - K\ r + al\ K\ r + k\ z + \ \((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\ Log[\(d + k\)\/\(K\ r\)] + \ \((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\ Log[\(K\ r\)\/\(d + \ k\)])\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(aX = \(\((\(-1\) + al)\)\ \((k + d)\)\ \((d + r)\)\ \((d\ K + K\ r \ - k\ z)\)\^2\)\/\(k\ K\ r\ \((\(-d\)\ K + al\ d\ K - K\ r + al\ K\ r + k\ \ z)\)\)\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[BoxData[ \(\(\((\(-1\) + al)\)\ \((d + k)\)\ \((d + r)\)\ \((d\ K + K\ r - k\ z)\)\ \^2\)\/\(k\ K\ r\ \((\(-d\)\ K + al\ d\ K - K\ r + al\ K\ r + k\ z)\)\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(aY = Simplify[1/\((A/\((r + d)\) + A*X + \((A*Log[a] + 1)\)/\((r + d)\))\), {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \(\(al\ \((d + r)\)\)\/\(1 + \((1 - al)\)\ \((\(-\(\(al\ k\ \ K\)\/\(\((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\)\)\) - Log[b] + Log[\(K\ \ r\)\/\(d + k\)])\) - \((\(-1\) + al)\)\ Log[\(\((d + k)\)\ \((\(-\(\(K\ \((d \ + r)\)\)\/k\)\) + z)\)\)\/\(K\ r\)]\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(aY = \(\(al\)\(\ \)\((d + r)\)\(\ \)\)\/\(1 + \((1 - al)\)\ \ \((\(-\(\(al\ k\ K\)\/\(\((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\)\)\) - \ Log[b] + \ Log[\ \(-\(\(K\ \((d + r)\)\)\/k\)\) + z])\)\)\)\(\n\) \)\)], "Input"], Cell[BoxData[ \(\(al\ \((d + r)\)\)\/\(1 + \((1 - al)\)\ \((\(-\(\(al\ k\ \ K\)\/\(\((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\)\)\) - Log[b] + Log[\(-\(\ \(K\ \((d + r)\)\)\/k\)\) + z])\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(aY = \(\(al\)\(\ \)\((d + r)\)\(\ \)\)\/\(1 + \((1 - al)\)\ \ \((\(-\(\(al\ k\ K\)\/\(\((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\)\)\) + \ al/\((1 - al)\)*K/\((k\ z - \((r + d)\)\ K)\)*\((r + d + k)\))\)\)\)], "Input"], Cell[BoxData[ \(\(al\ \((d + r)\)\)\/\(1 + \((1 - al)\)\ \((\(-\(\(al\ k\ \ K\)\/\(\((\(-1\) + al)\)\ \((d\ K + K\ r - k\ z)\)\)\)\) + \(al\ K\ \((d + k \ + r)\)\)\/\(\((1 - al)\)\ \((\(-K\)\ \((d + r)\) + k\ z)\)\))\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(aY = \(\(al\)\(\ \)\((d + r)\)\(\ \)\)\/\(1 + \((1 - al)\)\ \((\(al\ K\ \ \((k + d + r)\) - al\ k\ K\)\/\(\((1 - al)\)\ \((\(-K\)\ \((d + r)\) + k\ \ z)\)\))\)\)\)], "Input"], Cell[BoxData[ \(\(al\ \((d + r)\)\)\/\(1 + \(\(-al\)\ k\ K + al\ K\ \((d + k + r)\)\)\/\ \(\(-K\)\ \((d + r)\) + k\ z\)\)\)], "Output"] }, Open ]], Cell[BoxData[ \(f = \({c\ \((1 + p*m)\) - m\ z + K\ theta\ \((1 - m)\), K/q[theta] - c\ \((Y - a/\((r + d)\))\)} = 0\ determines\ c\), \(\(theta\)\(.\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(fc = Simplify[{1 + p*m, a/\((r + d)\) - Y}, {k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \({\(\(-d\)\ K + k\ z\)\/\(K\ r\), \(-\(\(d + k\)\/\(k\ r\)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fa = Simplify[{0, c/\((r + d)\)}, {k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \({0, \(K\ r\)\/\(\((d + k)\)\ \((d + r)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ftheta = Simplify[{\((1 - m)\)*K, al*theta^\((al - 1)\)*K/k}, {k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \({\(d\ K\)\/\(d + k\), \(al\ K\)\/k}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fm = Simplify[{c*p - z - K*theta, 0}, {k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \({\(-\(\(K\ \((d + k + r)\)\)\/k\)\), 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fphi = {c*m, 0}\)], "Input"], Cell[BoxData[ \({\(k\ K\ r\)\/\((d + k)\)\^2, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fX = {0, 0}\)], "Input"], Cell[BoxData[ \({0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fY = {0, \(-c\)}\)], "Input"], Cell[BoxData[ \({0, \(-\(\(K\ r\)\/\(d + k\)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Mtemp = Transpose[{fc, ftheta}]\)], "Input"], Cell[BoxData[ \({{\(\(-d\)\ K + k\ z\)\/\(K\ r\), \(d\ K\)\/\(d + k\)}, {\(-\(\(d + k\)\/\(k\ r\)\)\), \(al\ K\)\/k}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Mtemp]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\(-d\)\ K + k\ z\)\/\(K\ r\)\), \(\(d\ K\)\/\(d + k\)\)}, {\(-\(\(d + k\)\/\(k\ r\)\)\), \(\(al\ K\)\/k\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Det[Mtemp], {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \(\(d\ K - al\ d\ K + al\ k\ z\)\/\(k\ r\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(con = {c, theta}\)], "Input"], Cell[BoxData[ \({\(K\ r\)\/\(d + k\), 1}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(conm = Simplify[\(-Inverse[Mtemp] . fm\), {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \({\(al\ K\^2\ r\ \((d + k + r)\)\)\/\(k\ \((d\ \((K - al\ K)\) + al\ k\ \ z)\)\), \(\((d + k)\)\ K\ \((d + k + r)\)\)\/\(k\ \((d\ \((K - al\ K)\) + al\ \ k\ z)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(conphi = Simplify[\(-Inverse[Mtemp] . fphi\), {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \({\(al\ k\ K\^2\ r\^2\)\/\(\((d + k)\)\^2\ \((\((\(-1\) + al)\)\ d\ K - \ al\ k\ z)\)\), \(k\ K\ r\)\/\(\((d + k)\)\ \((\((\(-1\) + al)\)\ d\ K - al\ k\ \ z)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(conX = Simplify[\(-Inverse[Mtemp] . \((fX + fa*aX)\)\), {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0 && k + d > 0}]\)], "Input"], Cell[BoxData[ \({\(-\(\(\((\(-1\) + al)\)\ d\ K\ r\ \((d\ K + K\ r - k\ z)\)\^2\)\/\(\((d + k)\)\ \((\((\(-1\) + al)\)\ d\ K + \((\(-1\) + al)\)\ K\ r + k\ z)\)\ \((\((\(-1\) + al)\)\ d\ K - al\ k\ z)\)\)\)\), \(-\(\(\((\(-1\) + al)\)\ \((d\ K - k\ z)\)\ \((d\ K + K\ r - k\ z)\)\^2\)\/\(K\ \((\((\(-1\) + al)\)\ d\ K + \((\(-1\) + al)\)\ K\ r + k\ z)\)\ \((\((\(-1\) + al)\)\ d\ K - al\ k\ z)\)\)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(conY = Simplify[\(-Inverse[Mtemp] . \((fY + fa*aY)\)\), {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \({\(d\ k\ K\^2\ r\^2\ \((\((\(-1\) + 2\ al)\)\ d\ K + \((\(-1\) + 2\ al)\ \)\ K\ r - \((\(-1\) + al)\)\ k\ z)\)\)\/\(\((d + k)\)\^2\ \((\((\(-1\) + al)\ \)\ d\ K + \((\(-1\) + al)\)\ K\ r + k\ z)\)\ \((\((\(-1\) + al)\)\ d\ K - al\ \ k\ z)\)\), \(k\ r\ \((\(-d\)\ K + k\ z)\)\ \((d\ \((K - 2\ al\ K)\) + K\ \ \((r - 2\ al\ r)\) + \((\(-1\) + al)\)\ k\ z)\)\)\/\(\((d + k)\)\ \ \((\((\(-1\) + al)\)\ d\ K + \((\(-1\) + al)\)\ K\ r + k\ z)\)\ \((\((\(-1\) \ + al)\)\ d\ K - al\ k\ z)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(muc = du[c]\), "\n", \(muw = du[w]\), "\n", \(mmuw = ddu[w]\), "\n", \(mmuc = ddu[c]\)}], "Input"], Cell[BoxData[ \(\(d + k\)\/\(K\ r\)\)], "Output"], Cell[BoxData[ \(1\/\(\(-\(\(K\ \((d + r)\)\)\/k\)\) + z\)\)], "Output"], Cell[BoxData[ \(\(-\(1\/\((\(-\(\(K\ \((d + r)\)\)\/k\)\) + z)\)\^2\)\)\)], "Output"], Cell[BoxData[ \(\(-\(\((d + k)\)\^2\/\(K\^2\ r\^2\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(M = Simplify[{{\(-d\) - mu[theta] + \((1 - m)\)*dmu[theta]*conm[\([2]\)], \((1 - m)\)* dmu[theta]*conphi[\([2]\)], \((1 - m)\)*dmu[theta]* conX[\([2]\)], \((1 - m)\)*dmu[theta]* conY[\([2]\)]}, \n\ \ \ \ {0, d*\((\(-1\))\), d*aX, d*aY}, \n\ \ \ {\ \((\(-muc\) + theta*K/A*mmuw*a)\)* conm[\([1]\)] + K/A*muw*conm[\([2]\)], \((\(-muc\) + theta*K/A*mmuw*a)\)* conphi[\([1]\)] + K/A*muw*conphi[\([2]\)], r + d + \((\(-muc\) + theta*K/A*mmuw*a)\)*conX[\([1]\)] + theta*K/A*mmuw*c*aX + K/A*muw*conX[\([2]\)], \((\(-muc\) + theta*K/A*mmuw*a)\)* conY[\([1]\)] + theta*K/A*mmuw*c*aY + K/A*muw*conY[\([2]\)]}, \n{\ \ \((\(-z\)*mmuc)\)* conm[\([1]\)], \((\(-z\)*mmuc)\)* conphi[\([1]\)], \((\(-z\)*mmuc)\)*conX[\([1]\)], \ r + d - z*mmuc*conY[\([1]\)]}}, {al > 0 && k > 0 && K > 0 && z > w && d > 0 && r > 0 && al > 0 && al < 1 && w > 0}]\)], "Input"], Cell[BoxData[ \({{\(al\ k\^2\ z + d\ \((\((\(-1\) + al)\)\ K\ r + al\ k\ \ z)\)\)\/\(\((\(-1\) + al)\)\ d\ K - al\ k\ z\), \(-\(\(\((\(-1\) + al)\)\ d\ k\^2\ K\ r\)\/\(\((d + k)\)\^2\ \((\((\(-1\) + al)\)\ d\ K - al\ k\ z)\)\)\)\), \(\((\(-1\) + al)\)\^2\ d\ k\ \((d\ K \ - k\ z)\)\ \((d\ K + K\ r - k\ z)\)\^2\)\/\(\((d + k)\)\ K\ \((\((\(-1\) + \ al)\)\ d\ K + \((\(-1\) + al)\)\ K\ r + k\ z)\)\ \((\((\(-1\) + al)\)\ d\ K - \ al\ k\ z)\)\), \(-\(\(\((\(-1\) + al)\)\ d\ k\^2\ r\ \((d\ K - k\ z)\)\ \((\((\(-1\) + 2\ al)\)\ d\ K + \((\(-1\) + 2\ al)\)\ K\ r - \((\(-1\) + al)\)\ k\ z)\)\)\/\(\((d + k)\)\^2\ \((\((\(-1\) + al)\)\ d\ K + \((\(-1\) + al)\)\ K\ r + k\ z)\)\ \((\((\(-1\) + al)\)\ d\ K - al\ k\ z)\)\)\)\)}, {0, \(-d\), \(\((\(-1\) + al)\)\ d\ \ \((d + k)\)\ \((d + r)\)\ \((d\ K + K\ r - k\ z)\)\^2\)\/\(k\ K\ r\ \ \((\((\(-1\) + al)\)\ d\ K + \((\(-1\) + al)\)\ K\ r + k\ z)\)\), \(al\ d\ \ \((d + r)\)\)\/\(1 - \(al\ K\ \((d + r)\)\)\/\(d\ K + K\ r - k\ z\)\)}, {\(al\ \ \((d + k)\)\ K\ \((d + k + r)\)\ \((d\ K + K\ r + k\ \((K - z)\))\)\)\/\(k\ \ \((\(-d\)\ K - K\ r + k\ z)\)\ \((d\ \((K - al\ K)\) + al\ k\ z)\)\), \ \(-\(\(al\ k\ K\ r\ \((d\ K + K\ r + k\ \((K - z)\))\)\)\/\(\((d + k)\)\ \((d\ K + K\ r - k\ z)\)\ \((\((\(-1\) + al)\)\ d\ K - al\ k\ z)\)\)\)\), \(al\^2\ K\ \((d + r)\)\ \((d + k + r)\ \)\ \((d\ K - k\ z)\) + d\ k\ z\ \((d\ K + K\ r - k\ z)\) - al\ \((d\^3\ K\^2 \ + d\^2\ K\^2\ \((k + 2\ r)\) + d\ K\ \((k + r)\)\ \((K\ r - k\ z)\) + k\ \((k \ + r)\)\ z\ \((\(-K\)\ r + k\ z)\))\)\)\/\(\((\((\(-1\) + al)\)\ d\ K + \ \((\(-1\) + al)\)\ K\ r + k\ z)\)\ \((\((\(-1\) + al)\)\ d\ K - al\ k\ \ z)\)\), \(k\ K\ r\ \((\((1 - 2\ al)\)\ d\^3\ K\^2 + al\ k\^2\ z\ \((\((\(-1\) \ + al)\)\ K\ r + k\ z)\) - d\^2\ K\ \((al\^2\ k\ K - 2\ K\ r + 4\ al\ K\ r + 2\ \ k\ z - 3\ al\ k\ z)\) + d\ \((\((K\ r - k\ z)\)\^2 + al\^2\ k\ K\ \((\(-K\)\ \ r + k\ z)\) - al\ \((2\ K\^2\ r\^2 + k\ K\ \((k - 3\ r)\)\ z + k\^2\ \ z\^2)\))\))\)\)\/\(\((d + k)\)\ \((d\ K + K\ r - 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w)\)\(/\)\((r + d)\)\(\ \)\)\)\)\)], "Input"], Cell[BoxData[ \(\(k\ \((\(-w\) + z)\)\)\/\(d + r\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(num = Simplify[al\ d\ k\^2\ \((d + r)\)\ z\ \((al\ K\ \((2\ d + r)\)\ \((d + k + r)\) - d\ \((k\ K + 2\ K\ r - 2\ k\ z)\) + \((k + r)\)\ \((\(-K\)\ r + k\ z)\))\)]\)], "Input"], Cell[BoxData[ \(al\ d\ k\^3\ z\ \((r\ \((k + r)\)\ w - al\ \((2\ d + r)\)\ \((d + k + r)\)\ \((w - z)\) + 2\ d\^2\ z + d\ \((k\ w + r\ \((2\ w + z)\))\))\)\)], "Output"] }, Open ]], Cell[BoxData[ \(The\ numerator\ is\ \(\(positive\)\(.\)\)\)], "Input"], Cell[BoxData[{ \(The\ denominator\ \ equals\ \ - \((k\ z - \((1 - al)\) \((r + d)\)\ K)\)\ \((\((1 - al)\)\ d\ K + al\ k\ z)\)\), "\[IndentingNewLine]", \(In\ ss\ 0 < kw = \((k\ z - \((r + d)\)\ K)\) < \((k\ z - \((1 - al)\) \((r + d)\)\ K)\)\ and\ hence\ the\ denominator\ is\ strictly\ \ \(\(negative\)\(.\)\)\)}], "Input"], Cell[BoxData[ \(Characteristic\ polynomial\ is\ strictly\ negative\ at\ r + d . \[IndentingNewLine]Hence\ there\ are\ two\ distinct\ positive\ \ roots, \ one\ of\ which\ is\ r\ and\ the\ other\ bigger\ than\ r + \(\(d\)\(.\ \)\)\)], "Input"], Cell[BoxData[""], "Input", CellFrame->{{0, 0}, {0, 0.5}}], Cell[BoxData[ \(Clear[K]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(w = z - 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al)\)\ \((2\ d + r)\)\ \((d + r + k)\) + k\ z\ \((k + 2\ d + r)\)\)}], "Input"], Cell[BoxData[ \(\((\(-1\) + al)\)\ K\ \((2\ d\^2 + 3\ d\ r + r\^2)\) + k\^2\ z + k\ \((2\ d + r)\)\ \((\((\(-1\) + al)\)\ K + z)\)\)], "Output"], Cell[BoxData[ \(\(-\((1 - al)\)\)\ K\ \((2\ d + r)\)\ \((d + k + r)\) + k\ \((2\ d + k + r)\)\ z\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[temp - temp2]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[BoxData[ \(If\ temp2 > 0, \ then\ characteristic\ polynomial\ is\ negative\ at\ - \(\(d\)\(.\)\)\)], \ "Input"], Cell[BoxData[ \(One\ example\ of\ when\ this\ \(\(happens\)\(:\)\)\)], "Input", CellFrame->{{0, 0}, {0, 0.5}}], Cell[BoxData[ \(Note\ that\ temp2\ decreases\ in\ al, \ because\ K\ increases\ in\ al\ toward\ K = k \((z - b)\)/\((r + d)\) . \ Consider\ the\ value\ at\ this\ \(\(bound\)\(:\)\(\ \[IndentingNewLine]\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(temp_lb = Simplify[\(-k\) \((z - b)\)/\((r + d)\)\ \((2\ d + r)\)\ \((d + r + k)\) + k\ z\ \((k + 2\ d + r)\)]\)], "Input"], Cell[BoxData[ \(\(k\ \((b\ \((2\ d + r)\)\ \((d + k + r)\) - 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