(************** 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[ 113590, 2554]*) (*NotebookOutlinePosition[ 114282, 2578]*) (* CellTagsIndexPosition[ 114238, 2574]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[""], "Input"], Cell[BoxData[ \(Quit\)], "Input"], Cell[TextData[{ StyleBox["Kahler-Einstein metric on dP3 by constrained optimization", FontVariations->{"Underline"->True}], "\n\nCharles Doran, Chris Herzog, Josh Kantor (U. Wash), Matthew Headrick \ (Stanford), Toby Wiseman (Imperial)\n\n", StyleBox["Copyright March 2007\n\nThis notebook computes the approximation \ to the Kahler-Einstein metric using constrained optmiziation as described by \ us in section 6 of hep-th/0703057.\n\nThe calculation below computes the \ approximation for 18th order polynomial expansions and some low lying \ eigenvalues transforming in the trivial representation of the dihedral group.\ \n\nNotation differs from paper; x=", FontSize->18], Cell[BoxData[ \(TraditionalForm\`x\_1\)], FontSize->18], StyleBox[",y=", FontSize->18], Cell[BoxData[ \(TraditionalForm\`x\_2\)], FontSize->18] }], "Text"], Cell[BoxData[""], "Input"], Cell["\<\ We construct a power series solution in our symmetric variables U \ and V.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(UVrels\ \ = \ {V\ -> \ x^2\ y^2\ \((x + y)\)^2, \[IndentingNewLine]U -> \ x^2\ + \ x\ y\ + \ y^2}\)], "Input"], Cell[BoxData[ \({V \[Rule] x\^2\ y\^2\ \((x + y)\)\^2, U \[Rule] x\^2 + x\ y + y\^2}\)], "Output"] }, Open ]], Cell[BoxData[ \(gp[x_, \ y_]\ := 1/2 \((\ \((1 - x)\)\ Log[1 - x]\ + \((1 + x)\) Log[1 + x] + \ \((1 - y)\)\ Log[1 - y]\ + \((1 + y)\) Log[1 + y]\ + \ \((1 - \((x + y)\))\) Log[1 - \((x + y)\)]\ + \ \((1 + x + y)\) Log[1 + x + y])\)\)], "Input"], Cell[BoxData[ RowBox[{\(MAeq[x_, y_, f_]\), " ", ":=", RowBox[{ RowBox[{"(", " ", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["f", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}], RowBox[{ SuperscriptBox["f", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["f", TagBox[\((1, 1)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}], ")"}], "^", "2"}]}], ")"}], " ", "-", RowBox[{"Exp", "[", " ", RowBox[{ RowBox[{"2", " ", "x", " ", RowBox[{ SuperscriptBox["f", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}], " ", "+", " ", RowBox[{"2", " ", "y", " ", RowBox[{ SuperscriptBox["f", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}], " ", "-", " ", \(2 f[x, y]\)}], "]"}]}]}]], "Input"], Cell["\<\ The coefficients of the U^n series will be Ai, of the V U^n series \ Bi, and so on. We adopted a slightly different formula for A1 because it's easier to express \ the higher coefficients in terms of \[Alpha] than in terms of A1.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(-1\)/2\ Log[3]\ - Log[\[Alpha]] \[Equal] A1, \ \[Alpha]]\)], "Input"], Cell[BoxData[ \({{\[Alpha] \[Rule] \[ExponentialE]\^\(-A1\)\/\@3}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Aa = \ {\(-1\)/2\ Log[3]\ - Log[\[Alpha]], \ A1, \ A2, \ A3, \ A4, \ A5, \ A6, \ A7, \ A8, \ A9, \ A10}\)], "Input"], Cell[BoxData[ \({\(-\(Log[3]\/2\)\) - Log[\[Alpha]], A1, A2, A3, A4, A5, A6, A7, A8, A9, A10}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Bb\ = \ {B0, \ B1, \ B2, \ B3, \ B4, \ B5, \ B6, \ B7}\)], "Input"], Cell[BoxData[ \({B0, B1, B2, B3, B4, B5, B6, B7}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Cc = {C0, \ C1, \ C2, \ C3, \ C4}\)], "Input"], Cell[BoxData[ \({C0, C1, C2, C3, C4}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Dd\ = \ {D0, \ D1}\)], "Input"], Cell[BoxData[ \({D0, D1}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(hguessintermediate\ = \ \((Sum[ U^n\ Aa[\([n + 1]\)], \ {n, 0, 9}]\ + \ Sum[V\ U^n\ Bb[\([n + 1]\)], \ {n, 0, 6}] + \ Sum[V^2\ \ U^n\ Cc[\([n + 1]\)], \ {n, 0, 3}]\ + \ Sum[V^3\ U^n\ Dd[\([n + 1]\)], \ {n, 0, 0}]\ )\) /. \ UVrels\)], "Input"], Cell[BoxData[ \(B0\ x\^2\ y\^2\ \((x + y)\)\^2 + C0\ x\^4\ y\^4\ \((x + y)\)\^4 + D0\ x\^6\ y\^6\ \((x + y)\)\^6 + A1\ \((x\^2 + x\ y + y\^2)\) + B1\ x\^2\ y\^2\ \((x + y)\)\^2\ \((x\^2 + x\ y + y\^2)\) + C1\ x\^4\ y\^4\ \((x + y)\)\^4\ \((x\^2 + x\ y + y\^2)\) + A2\ \((x\^2 + x\ y + y\^2)\)\^2 + B2\ x\^2\ y\^2\ \((x + y)\)\^2\ \((x\^2 + x\ y + y\^2)\)\^2 + C2\ x\^4\ y\^4\ \((x + y)\)\^4\ \((x\^2 + x\ y + y\^2)\)\^2 + A3\ \((x\^2 + x\ y + y\^2)\)\^3 + B3\ x\^2\ y\^2\ \((x + y)\)\^2\ \((x\^2 + x\ y + y\^2)\)\^3 + C3\ x\^4\ y\^4\ \((x + y)\)\^4\ \((x\^2 + x\ y + y\^2)\)\^3 + A4\ \((x\^2 + x\ y + y\^2)\)\^4 + B4\ x\^2\ y\^2\ \((x + y)\)\^2\ \((x\^2 + x\ y + y\^2)\)\^4 + A5\ \((x\^2 + x\ y + y\^2)\)\^5 + B5\ x\^2\ y\^2\ \((x + y)\)\^2\ \((x\^2 + x\ y + y\^2)\)\^5 + A6\ \((x\^2 + x\ y + y\^2)\)\^6 + B6\ x\^2\ y\^2\ \((x + y)\)\^2\ \((x\^2 + x\ y + y\^2)\)\^6 + A7\ \((x\^2 + x\ y + y\^2)\)\^7 + A8\ \((x\^2 + x\ y + y\^2)\)\^8 + A9\ \((x\^2 + x\ y + y\^2)\)\^9 - Log[3]\/2 - Log[\[Alpha]]\)], "Output"] }, Open ]], Cell[BoxData[ \(\(hguess[x_, \ y_]\ = \ hguessintermediate;\)\)], "Input"], Cell[BoxData[ \(gguess[x_, y_]\ := \ gp[x, y]\ + \ hguess[x, y]\)], "Input"], Cell[BoxData[ \(\(myseries\ = \ CoefficientList[ Normal[Series[ MAeq[x, y, gguess]\ /. \ {x \[Rule] e\ x, \ y \[Rule] \ e\ y}, {e, 0, 16}, {y, 0, 16}]], {e, y}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(myseries[\([1]\)]\)], "Input"], Cell[BoxData[ \({\(-\((1 + A1)\)\^2\) + \((2 + 2\ A1)\)\^2 - \[ExponentialE]\^\(\(-2\)\ \ \((\(-\(Log[3]\/2\)\) - Log[\[Alpha]])\)\), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A1rels\ = \ Solve[myseries[\([1]\)] \[Equal] 0, \ Aa[\([2]\)]]\)], "Input"], Cell[BoxData[ \({{A1 \[Rule] \(-1\) - \[Alpha]}, {A1 \[Rule] \(-1\) + \[Alpha]}}\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A2rel\ = \ \(Solve[\((myseries[\([3]\)]\ /. \ A1rels[\([2]\)])\)\ \[Equal] 0, \ Aa[\([3]\)]]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({A2 \[Rule] 1\/12\ \((\(-2\) + 3\ \[Alpha]\^2)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A3rel\ = \ \(Solve[ Simplify[ myseries[\([5]\)]\ /. \ Join[A1rels[\([2]\)], \ A2rel]] \[Equal] 0, \ A3]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({A3 \[Rule] 1\/216\ \((\(-16\) - 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.35591 .50693 .36967 .52014 .39211 .51301 .37846 .49969 Mtetra .37846 .49969 .39211 .51301 .41474 .50582 .40121 .49239 Mtetra .40121 .49239 .41474 .50582 .43756 .49857 .42416 .48503 Mtetra .42416 .48503 .43756 .49857 .46057 .49126 .4473 .4776 Mtetra .4473 .4776 .46057 .49126 .48379 .48389 .47064 .47011 Mtetra .47064 .47011 .48379 .48389 .5072 .47645 .49418 .46256 Mtetra .49418 .46256 .5072 .47645 .53081 .46895 .51792 .45494 Mtetra .51792 .45494 .53081 .46895 .55463 .46139 .54187 .44725 Mtetra .54187 .44725 .55463 .46139 .57865 .45376 .56604 .43949 Mtetra .56604 .43949 .57865 .45376 .60288 .44606 .59041 .43167 Mtetra .59041 .43167 .60288 .44606 .62733 .4383 .615 .42378 Mtetra .615 .42378 .62733 .4383 .65199 .43046 .63981 .41582 Mtetra .63981 .41582 .65199 .43046 .67687 .42256 .66484 .40779 Mtetra .66484 .40779 .67687 .42256 .70197 .41459 .6901 .39968 Mtetra .6901 .39968 .70197 .41459 .7273 .40654 .71558 .3915 Mtetra .71558 .3915 .7273 .40654 .75285 .39843 .74129 .38325 Mtetra .74129 .38325 .75285 .39843 .77863 .39024 .76723 .37493 Mtetra .76723 .37493 .77863 .39024 .80465 .38199 .79342 .36654 Mtetra .663 .742 .909 r .79342 .36654 .80465 .38199 .8309 .37368 .81984 .35808 Mtetra .662 .741 .909 r .81984 .35808 .8309 .37368 .8574 .36533 .84651 .34961 Mtetra .66 .74 .91 r .84651 .34961 .8574 .36533 .88414 .35693 .87345 .34121 Mtetra .642 .735 .916 r .87345 .34121 .88414 .35693 .91089 .34645 .90078 .3339 Mtetra .663 .742 .908 r .29715 .50804 .31136 .52123 .33354 .51411 .31945 .50081 Mtetra .31945 .50081 .33354 .51411 .35591 .50693 .34193 .49352 Mtetra .34193 .49352 .35591 .50693 .37846 .49969 .3646 .48616 Mtetra .3646 .48616 .37846 .49969 .40121 .49239 .38747 .47875 Mtetra .38747 .47875 .40121 .49239 .42416 .48503 .41054 .47127 Mtetra .41054 .47127 .42416 .48503 .4473 .4776 .43381 .46372 Mtetra .43381 .46372 .4473 .4776 .47064 .47011 .45727 .45611 Mtetra .45727 .45611 .47064 .47011 .49418 .46256 .48095 .44844 Mtetra .48095 .44844 .49418 .46256 .51792 .45494 .50483 .44069 Mtetra .50483 .44069 .51792 .45494 .54187 .44725 .52892 .43288 Mtetra .52892 .43288 .54187 .44725 .56604 .43949 .55322 .425 Mtetra .55322 .425 .56604 .43949 .59041 .43167 .57774 .41705 Mtetra .57774 .41705 .59041 .43167 .615 .42378 .60247 .40903 Mtetra .60247 .40903 .615 .42378 .63981 .41582 .62743 .40093 Mtetra .62743 .40093 .63981 .41582 .66484 .40779 .65261 .39277 Mtetra .65261 .39277 .66484 .40779 .6901 .39968 .67802 .38453 Mtetra .67802 .38453 .6901 .39968 .71558 .3915 .70366 .37621 Mtetra .70366 .37621 .71558 .3915 .74129 .38325 .72954 .36782 Mtetra .72954 .36782 .74129 .38325 .76723 .37493 .75565 .35935 Mtetra .75565 .35935 .76723 .37493 .79342 .36654 .782 .35081 Mtetra .782 .35081 .79342 .36654 .81984 .35808 .80859 .34221 Mtetra .662 .741 .909 r .80859 .34221 .81984 .35808 .84651 .34961 .83544 .33358 Mtetra .659 .74 .909 r .83544 .33358 .84651 .34961 .87345 .34121 .86256 .32507 Mtetra .618 .722 .921 r .86256 .32507 .87345 .34121 .90078 .3339 .89026 .31917 Mtetra .663 .742 .908 r .28272 .49464 .29715 .50804 .31945 .50081 .30513 .4873 Mtetra .30513 .4873 .31945 .50081 .34193 .49352 .32773 .47989 Mtetra .32773 .47989 .34193 .49352 .3646 .48616 .35053 .47242 Mtetra .35053 .47242 .3646 .48616 .38747 .47875 .37352 .46489 Mtetra .37352 .46489 .38747 .47875 .41054 .47127 .39671 .45729 Mtetra .39671 .45729 .41054 .47127 .43381 .46372 .4201 .44962 Mtetra .4201 .44962 .43381 .46372 .45727 .45611 .4437 .44189 Mtetra .4437 .44189 .45727 .45611 .48095 .44844 .4675 .43409 Mtetra .4675 .43409 .48095 .44844 .50483 .44069 .49152 .42622 Mtetra .49152 .42622 .50483 .44069 .52892 .43288 .51575 .41828 Mtetra .51575 .41828 .52892 .43288 .55322 .425 .54019 .41026 Mtetra .54019 .41026 .55322 .425 .57774 .41705 .56485 .40218 Mtetra .56485 .40218 .57774 .41705 .60247 .40903 .58973 .39403 Mtetra .58973 .39403 .60247 .40903 .62743 .40093 .61484 .3858 Mtetra .61484 .3858 .62743 .40093 .65261 .39277 .64018 .3775 Mtetra .64018 .3775 .65261 .39277 .67802 .38453 .66575 .36912 Mtetra .66575 .36912 .67802 .38453 .70366 .37621 .69155 .36066 Mtetra .69155 .36066 .70366 .37621 .72954 .36782 .71758 .35213 Mtetra .71758 .35213 .72954 .36782 .75565 .35935 .74386 .34352 Mtetra .74386 .34352 .75565 .35935 .782 .35081 .77038 .33482 Mtetra .77038 .33482 .782 .35081 .80859 .34221 .79715 .32606 Mtetra .663 .741 .908 r .79715 .32606 .80859 .34221 .83544 .33358 .82417 .31723 Mtetra .661 .74 .909 r .82417 .31723 .83544 .33358 .86256 .32507 .85147 .30847 Mtetra .627 .724 .916 r .85147 .30847 .86256 .32507 .89026 .31917 .87927 .30185 Mtetra .663 .742 .908 r .26807 .48104 .28272 .49464 .30513 .4873 .29059 .47358 Mtetra .29059 .47358 .30513 .4873 .32773 .47989 .31331 .46605 Mtetra .31331 .46605 .32773 .47989 .35053 .47242 .33622 .45846 Mtetra .33622 .45846 .35053 .47242 .37352 .46489 .35934 .4508 Mtetra .35934 .4508 .37352 .46489 .39671 .45729 .38265 .44308 Mtetra .38265 .44308 .39671 .45729 .4201 .44962 .40617 .43529 Mtetra .40617 .43529 .4201 .44962 .4437 .44189 .4299 .42743 Mtetra .4299 .42743 .4437 .44189 .4675 .43409 .45384 .4195 Mtetra .45384 .4195 .4675 .43409 .49152 .42622 .47799 .4115 Mtetra .47799 .4115 .49152 .42622 .51575 .41828 .50236 .40343 Mtetra .50236 .40343 .51575 .41828 .54019 .41026 .52694 .39529 Mtetra .52694 .39529 .54019 .41026 .56485 .40218 .55175 .38707 Mtetra .55175 .38707 .56485 .40218 .58973 .39403 .57678 .37878 Mtetra .57678 .37878 .58973 .39403 .61484 .3858 .60204 .37041 Mtetra .60204 .37041 .61484 .3858 .64018 .3775 .62753 .36197 Mtetra .62753 .36197 .64018 .3775 .66575 .36912 .65326 .35345 Mtetra .65326 .35345 .66575 .36912 .69155 .36066 .67922 .34485 Mtetra .67922 .34485 .69155 .36066 .71758 .35213 .70543 .33617 Mtetra .70543 .33617 .71758 .35213 .74386 .34352 .73187 .32741 Mtetra .73187 .32741 .74386 .34352 .77038 .33482 .75857 .31856 Mtetra .75857 .31856 .77038 .33482 .79715 .32606 .78551 .30963 Mtetra .78551 .30963 .79715 .32606 .82417 .31723 .81271 .30061 Mtetra .663 .741 .908 r .81271 .30061 .82417 .31723 .85147 .30847 .84017 .29152 Mtetra .656 .735 .908 r .84017 .29152 .85147 .30847 .87927 .30185 .86795 .28285 Mtetra .663 .742 .908 r .25318 .46721 .26807 .48104 .29059 .47358 .27582 .45963 Mtetra .27582 .45963 .29059 .47358 .31331 .46605 .29865 .45199 Mtetra .29865 .45199 .31331 .46605 .33622 .45846 .32169 .44427 Mtetra .32169 .44427 .33622 .45846 .35934 .4508 .34493 .43649 Mtetra .34493 .43649 .35934 .4508 .38265 .44308 .36837 .42864 Mtetra .36837 .42864 .38265 .44308 .40617 .43529 .39202 .42073 Mtetra .39202 .42073 .40617 .43529 .4299 .42743 .41588 .41274 Mtetra .41588 .41274 .4299 .42743 .45384 .4195 .43995 .40468 Mtetra .43995 .40468 .45384 .4195 .47799 .4115 .46424 .39654 Mtetra .46424 .39654 .47799 .4115 .50236 .40343 .48875 .38834 Mtetra .48875 .38834 .50236 .40343 .52694 .39529 .51348 .38006 Mtetra .51348 .38006 .52694 .39529 .55175 .38707 .53843 .3717 Mtetra .53843 .3717 .55175 .38707 .57678 .37878 .56361 .36327 Mtetra .56361 .36327 .57678 .37878 .60204 .37041 .58903 .35476 Mtetra .58903 .35476 .60204 .37041 .62753 .36197 .61468 .34617 Mtetra .61468 .34617 .62753 .36197 .65326 .35345 .64056 .33751 Mtetra .64056 .33751 .65326 .35345 .67922 .34485 .66669 .32876 Mtetra .66669 .32876 .67922 .34485 .70543 .33617 .69306 .31993 Mtetra .69306 .31993 .70543 .33617 .73187 .32741 .71968 .31102 Mtetra .71968 .31102 .73187 .32741 .75857 .31856 .74655 .30202 Mtetra .74655 .30202 .75857 .31856 .78551 .30963 .77367 .29293 Mtetra .664 .742 .908 r .77367 .29293 .78551 .30963 .81271 .30061 .80105 .28372 Mtetra .665 .742 .908 r .80105 .28372 .81271 .30061 .84017 .29152 .82868 .27434 Mtetra .682 .747 .9 r .82868 .27434 .84017 .29152 .86795 .28285 .85644 .26359 Mtetra .663 .742 .908 r .23805 .45317 .25318 .46721 .27582 .45963 .26081 .44547 Mtetra .26081 .44547 .27582 .45963 .29865 .45199 .28376 .4377 Mtetra .28376 .4377 .29865 .45199 .32169 .44427 .30692 .42986 Mtetra .30692 .42986 .32169 .44427 .34493 .43649 .33028 .42195 Mtetra .33028 .42195 .34493 .43649 .36837 .42864 .35385 .41397 Mtetra .35385 .41397 .36837 .42864 .39202 .42073 .37763 .40592 Mtetra .37763 .40592 .39202 .42073 .41588 .41274 .40162 .3978 Mtetra .40162 .3978 .41588 .41274 .43995 .40468 .42583 .38961 Mtetra .42583 .38961 .43995 .40468 .46424 .39654 .45026 .38134 Mtetra .45026 .38134 .46424 .39654 .48875 .38834 .47491 .37299 Mtetra .47491 .37299 .48875 .38834 .51348 .38006 .49978 .36457 Mtetra .49978 .36457 .51348 .38006 .53843 .3717 .52489 .35608 Mtetra .52489 .35608 .53843 .3717 .56361 .36327 .55022 .3475 Mtetra .55022 .3475 .56361 .36327 .58903 .35476 .57579 .33885 Mtetra .57579 .33885 .58903 .35476 .61468 .34617 .6016 .33011 Mtetra .6016 .33011 .61468 .34617 .64056 .33751 .62764 .32129 Mtetra .62764 .32129 .64056 .33751 .66669 .32876 .65394 .31239 Mtetra .65394 .31239 .66669 .32876 .69306 .31993 .68048 .30341 Mtetra .68048 .30341 .69306 .31993 .71968 .31102 .70727 .29435 Mtetra .70727 .29435 .71968 .31102 .74655 .30202 .73431 .28519 Mtetra .663 .742 .909 r .73431 .28519 .74655 .30202 .77367 .29293 .76162 .27595 Mtetra .664 .742 .908 r .76162 .27595 .77367 .29293 .80105 .28372 .78918 .26659 Mtetra .665 .743 .908 r .78918 .26659 .80105 .28372 .82868 .27434 .817 .25701 Mtetra .694 .754 .898 r .817 .25701 .82868 .27434 .85644 .26359 .84487 .24525 Mtetra .663 .742 .908 r .22268 .4389 .23805 .45317 .26081 .44547 .24555 .43107 Mtetra .24555 .43107 .26081 .44547 .28376 .4377 .26863 .42317 Mtetra .26863 .42317 .28376 .4377 .30692 .42986 .29191 .4152 Mtetra .29191 .4152 .30692 .42986 .33028 .42195 .31539 .40716 Mtetra .31539 .40716 .33028 .42195 .35385 .41397 .33909 .39905 Mtetra .33909 .39905 .35385 .41397 .37763 .40592 .363 .39087 Mtetra .363 .39087 .37763 .40592 .40162 .3978 .38713 .38261 Mtetra .38713 .38261 .40162 .3978 .42583 .38961 .41147 .37428 Mtetra .41147 .37428 .42583 .38961 .45026 .38134 .43604 .36587 Mtetra .43604 .36587 .45026 .38134 .47491 .37299 .46084 .35739 Mtetra .46084 .35739 .47491 .37299 .49978 .36457 .48586 .34883 Mtetra .48586 .34883 .49978 .36457 .52489 .35608 .51111 .34018 Mtetra .51111 .34018 .52489 .35608 .55022 .3475 .5366 .33146 Mtetra .5366 .33146 .55022 .3475 .57579 .33885 .56232 .32266 Mtetra .56232 .32266 .57579 .33885 .6016 .33011 .58829 .31377 Mtetra .58829 .31377 .6016 .33011 .62764 .32129 .6145 .3048 Mtetra .6145 .3048 .62764 .32129 .65394 .31239 .64096 .29574 Mtetra .64096 .29574 .65394 .31239 .68048 .30341 .66767 .2866 Mtetra .66767 .2866 .68048 .30341 .70727 .29435 .69464 .27738 Mtetra .69464 .27738 .70727 .29435 .73431 .28519 .72187 .26807 Mtetra .663 .742 .909 r .72187 .26807 .73431 .28519 .76162 .27595 .74936 .25867 Mtetra .74936 .25867 .76162 .27595 .78918 .26659 .77711 .24919 Mtetra .664 .743 .909 r .77711 .24919 .78918 .26659 .817 .25701 .80513 .23954 Mtetra .685 .753 .903 r .80513 .23954 .817 .25701 .84487 .24525 .83327 .22829 Mtetra .663 .742 .908 r .20706 .42439 .22268 .4389 .24555 .43107 .23005 .41643 Mtetra .23005 .41643 .24555 .43107 .26863 .42317 .25324 .40841 Mtetra .25324 .40841 .26863 .42317 .29191 .4152 .27665 .40031 Mtetra .27665 .40031 .29191 .4152 .31539 .40716 .30026 .39214 Mtetra .30026 .39214 .31539 .40716 .33909 .39905 .32409 .38389 Mtetra .32409 .38389 .33909 .39905 .363 .39087 .34813 .37557 Mtetra .34813 .37557 .363 .39087 .38713 .38261 .37239 .36717 Mtetra .37239 .36717 .38713 .38261 .41147 .37428 .39688 .3587 Mtetra .39688 .3587 .41147 .37428 .43604 .36587 .42158 .35015 Mtetra .42158 .35015 .43604 .36587 .46084 .35739 .44652 .34152 Mtetra .44652 .34152 .46084 .35739 .48586 .34883 .47169 .33281 Mtetra .47169 .33281 .48586 .34883 .51111 .34018 .4971 .32402 Mtetra .4971 .32402 .51111 .34018 .5366 .33146 .52274 .31514 Mtetra .52274 .31514 .5366 .33146 .56232 .32266 .54862 .30618 Mtetra .54862 .30618 .56232 .32266 .58829 .31377 .57475 .29714 Mtetra .57475 .29714 .58829 .31377 .6145 .3048 .60113 .28802 Mtetra .60113 .28802 .6145 .3048 .64096 .29574 .62776 .2788 Mtetra .62776 .2788 .64096 .29574 .66767 .2866 .65464 .2695 Mtetra .65464 .2695 .66767 .2866 .69464 .27738 .68179 .26011 Mtetra .663 .742 .909 r .68179 .26011 .69464 .27738 .72187 .26807 .70919 .25063 Mtetra .70919 .25063 .72187 .26807 .74936 .25867 .73687 .24108 Mtetra .662 .742 .909 r .73687 .24108 .74936 .25867 .77711 .24919 .76482 .23147 Mtetra .662 .742 .91 r .76482 .23147 .77711 .24919 .80513 .23954 .79305 .22182 Mtetra .659 .743 .912 r .79305 .22182 .80513 .23954 .83327 .22829 .82158 .21225 Mtetra .663 .742 .908 r .19118 .40965 .20706 .42439 .23005 .41643 .21429 .40156 Mtetra .21429 .40156 .23005 .41643 .25324 .40841 .2376 .3934 Mtetra .2376 .3934 .25324 .40841 .27665 .40031 .26113 .38516 Mtetra .26113 .38516 .27665 .40031 .30026 .39214 .28487 .37685 Mtetra .28487 .37685 .30026 .39214 .32409 .38389 .30883 .36847 Mtetra .30883 .36847 .32409 .38389 .34813 .37557 .333 .36001 Mtetra .333 .36001 .34813 .37557 .37239 .36717 .3574 .35147 Mtetra .3574 .35147 .37239 .36717 .39688 .3587 .38203 .34285 Mtetra .38203 .34285 .39688 .3587 .42158 .35015 .40688 .33415 Mtetra .40688 .33415 .42158 .35015 .44652 .34152 .43196 .32537 Mtetra .43196 .32537 .44652 .34152 .47169 .33281 .45728 .31651 Mtetra .45728 .31651 .47169 .33281 .4971 .32402 .48284 .30757 Mtetra .48284 .30757 .4971 .32402 .52274 .31514 .50864 .29854 Mtetra .50864 .29854 .52274 .31514 .54862 .30618 .53468 .28942 Mtetra .53468 .28942 .54862 .30618 .57475 .29714 .56097 .28022 Mtetra .56097 .28022 .57475 .29714 .60113 .28802 .58752 .27094 Mtetra .58752 .27094 .60113 .28802 .62776 .2788 .61432 .26156 Mtetra .663 .742 .909 r .61432 .26156 .62776 .2788 .65464 .2695 .64138 .25209 Mtetra .64138 .25209 .65464 .2695 .68179 .26011 .6687 .24253 Mtetra .6687 .24253 .68179 .26011 .70919 .25063 .69629 .23288 Mtetra .69629 .23288 .70919 .25063 .73687 .24108 .72416 .22315 Mtetra .662 .741 .909 r .72416 .22315 .73687 .24108 .76482 .23147 .7523 .21339 Mtetra .66 .741 .91 r .7523 .21339 .76482 .23147 .79305 .22182 .78074 .2037 Mtetra .629 .729 .919 r .78074 .2037 .79305 .22182 .82158 .21225 .80966 .19594 Mtetra .663 .742 .908 r .17503 .39467 .19118 .40965 .21429 .40156 .19826 .38644 Mtetra .19826 .38644 .21429 .40156 .2376 .3934 .2217 .37814 Mtetra .2217 .37814 .2376 .3934 .26113 .38516 .24536 .36976 Mtetra .24536 .36976 .26113 .38516 .28487 .37685 .26922 .36131 Mtetra .26922 .36131 .28487 .37685 .30883 .36847 .29331 .35279 Mtetra .29331 .35279 .30883 .36847 .333 .36001 .31762 .34418 Mtetra .31762 .34418 .333 .36001 .3574 .35147 .34216 .3355 Mtetra .34216 .3355 .3574 .35147 .38203 .34285 .36692 .32673 Mtetra .36692 .32673 .38203 .34285 .40688 .33415 .39192 .31788 Mtetra .39192 .31788 .40688 .33415 .43196 .32537 .41715 .30895 Mtetra .41715 .30895 .43196 .32537 .45728 .31651 .44262 .29993 Mtetra .44262 .29993 .45728 .31651 .48284 .30757 .46833 .29083 Mtetra .46833 .29083 .48284 .30757 .50864 .29854 .49429 .28164 Mtetra .49429 .28164 .50864 .29854 .53468 .28942 .52049 .27237 Mtetra .52049 .27237 .53468 .28942 .56097 .28022 .54695 .26301 Mtetra .54695 .26301 .56097 .28022 .58752 .27094 .57367 .25355 Mtetra .663 .742 .909 r .57367 .25355 .58752 .27094 .61432 .26156 .60064 .24401 Mtetra .60064 .24401 .61432 .26156 .64138 .25209 .62788 .23438 Mtetra .62788 .23438 .64138 .25209 .6687 .24253 .65538 .22465 Mtetra .65538 .22465 .6687 .24253 .69629 .23288 .68316 .21483 Mtetra .68316 .21483 .69629 .23288 .72416 .22315 .71121 .20491 Mtetra .662 .742 .909 r .71121 .20491 .72416 .22315 .7523 .21339 .73955 .19493 Mtetra .66 .74 .909 r .73955 .19493 .7523 .21339 .78074 .2037 .76818 .18506 Mtetra .617 .722 .92 r .76818 .18506 .78074 .2037 .80966 .19594 .79737 .17789 Mtetra .664 .742 .909 r .15862 .37945 .17503 .39467 .19826 .38644 .18197 .37108 Mtetra .663 .742 .909 r .18197 .37108 .19826 .38644 .2217 .37814 .20553 .36263 Mtetra .663 .742 .908 r .20553 .36263 .2217 .37814 .24536 .36976 .22931 .35411 Mtetra .22931 .35411 .24536 .36976 .26922 .36131 .25331 .34551 Mtetra .25331 .34551 .26922 .36131 .29331 .35279 .27753 .33684 Mtetra .27753 .33684 .29331 .35279 .31762 .34418 .30198 .32808 Mtetra .30198 .32808 .31762 .34418 .34216 .3355 .32665 .31925 Mtetra .32665 .31925 .34216 .3355 .36692 .32673 .35156 .31033 Mtetra .35156 .31033 .36692 .32673 .39192 .31788 .3767 .30133 Mtetra .3767 .30133 .39192 .31788 .41715 .30895 .40208 .29224 Mtetra .40208 .29224 .41715 .30895 .44262 .29993 .4277 .28306 Mtetra .4277 .28306 .44262 .29993 .46833 .29083 .45357 .2738 Mtetra .45357 .2738 .46833 .29083 .49429 .28164 .47969 .26445 Mtetra .47969 .26445 .49429 .28164 .52049 .27237 .50605 .25501 Mtetra .50605 .25501 .52049 .27237 .54695 .26301 .53268 .24548 Mtetra .663 .742 .909 r .53268 .24548 .54695 .26301 .57367 .25355 .55956 .23586 Mtetra .55956 .23586 .57367 .25355 .60064 .24401 .58671 .22616 Mtetra .58671 .22616 .60064 .24401 .62788 .23438 .61413 .21636 Mtetra .61413 .21636 .62788 .23438 .65538 .22465 .64182 .20648 Mtetra .64182 .20648 .65538 .22465 .68316 .21483 .66979 .19649 Mtetra .663 .743 .909 r .66979 .19649 .68316 .21483 .71121 .20491 .69803 .18638 Mtetra .69803 .18638 .71121 .20491 .73955 .19493 .72656 .17617 Mtetra .662 .742 .909 r .72656 .17617 .73955 .19493 .76818 .18506 .75538 .16593 Mtetra .64 .73 .913 r .75538 .16593 .76818 .18506 .79737 .17789 .78463 .15713 Mtetra .664 .743 .909 r .14192 .36403 .15862 .37945 .18197 .37108 .1654 .35548 Mtetra .664 .742 .909 r .1654 .35548 .18197 .37108 .20553 .36263 .18909 .34686 Mtetra .663 .742 .909 r .18909 .34686 .20553 .36263 .22931 .35411 .21299 .33819 Mtetra .663 .742 .908 r .21299 .33819 .22931 .35411 .25331 .34551 .23712 .32944 Mtetra .23712 .32944 .25331 .34551 .27753 .33684 .26148 .32061 Mtetra .26148 .32061 .27753 .33684 .30198 .32808 .28606 .31171 Mtetra .28606 .31171 .30198 .32808 .32665 .31925 .31087 .30272 Mtetra .31087 .30272 .32665 .31925 .35156 .31033 .33592 .29364 Mtetra .33592 .29364 .35156 .31033 .3767 .30133 .36121 .28448 Mtetra .36121 .28448 .3767 .30133 .40208 .29224 .38674 .27523 Mtetra .38674 .27523 .40208 .29224 .4277 .28306 .41252 .26589 Mtetra .41252 .26589 .4277 .28306 .45357 .2738 .43854 .25646 Mtetra .43854 .25646 .45357 .2738 .47969 .26445 .46482 .24695 Mtetra .46482 .24695 .47969 .26445 .50605 .25501 .49135 .23733 Mtetra .49135 .23733 .50605 .25501 .53268 .24548 .51815 .22763 Mtetra .663 .742 .909 r .51815 .22763 .53268 .24548 .55956 .23586 .54521 .21785 Mtetra .54521 .21785 .55956 .23586 .58671 .22616 .57253 .20798 Mtetra .662 .742 .909 r .57253 .20798 .58671 .22616 .61413 .21636 .60013 .19804 Mtetra .60013 .19804 .61413 .21636 .64182 .20648 .62801 .18801 Mtetra .662 .743 .91 r .62801 .18801 .64182 .20648 .66979 .19649 .65617 .1779 Mtetra .662 .744 .91 r .65617 .1779 .66979 .19649 .69803 .18638 .68461 .16766 Mtetra .663 .745 .911 r .68461 .16766 .69803 .18638 .72656 .17617 .71334 .15725 Mtetra .666 .746 .91 r .71334 .15725 .72656 .17617 .75538 .16593 .74235 .14654 Mtetra .689 .752 .899 r .74235 .14654 .75538 .16593 .78463 .15713 .77151 .134 Mtetra .666 .745 .91 r .12491 .34851 .14192 .36403 .1654 .35548 .14853 .3397 Mtetra .665 .744 .909 r .14853 .3397 .1654 .35548 .18909 .34686 .17235 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.908 r .31307 .23212 .3294 .24988 .35524 .24029 .33906 .22235 Mtetra .33906 .22235 .35524 .24029 .38133 .23062 .3653 .21249 Mtetra .3653 .21249 .38133 .23062 .40767 .22085 .39181 .20255 Mtetra .663 .742 .909 r .39181 .20255 .40767 .22085 .43427 .21099 .41858 .19252 Mtetra .41858 .19252 .43427 .21099 .46114 .20103 .44562 .18239 Mtetra .44562 .18239 .46114 .20103 .48828 .19096 .47293 .17213 Mtetra .47293 .17213 .48828 .19096 .51569 .18081 .50052 .16178 Mtetra .662 .742 .909 r .50052 .16178 .51569 .18081 .54338 .17061 .52839 .15139 Mtetra .66 .741 .91 r .52839 .15139 .54338 .17061 .57135 .1604 .55654 .14103 Mtetra .656 .74 .911 r .55654 .14103 .57135 .1604 .59961 .15023 .585 .13085 Mtetra .65 .74 .915 r .585 .13085 .59961 .15023 .62817 .14014 .61377 .12099 Mtetra .641 .742 .922 r .61377 .12099 .62817 .14014 .65705 .13013 .64289 .11161 Mtetra .629 .747 .933 r .64289 .11161 .65705 .13013 .68624 .12013 .67237 .10283 Mtetra .613 .76 .949 r .67237 .10283 .68624 .12013 .71574 .10975 .70226 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.29548 .07547 .29174 Mtetra .81 .922 .911 r .07547 .29174 .09549 .29548 .12009 .28344 .10134 .27206 Mtetra .746 .831 .917 r .10134 .27206 .12009 .28344 .14472 .27264 .12664 .25732 Mtetra .699 .776 .911 r .12664 .25732 .14472 .27264 .16946 .2626 .15174 .24551 Mtetra .673 .75 .908 r .15174 .24551 .16946 .2626 .19438 .25294 .17685 .23524 Mtetra .662 .742 .909 r .17685 .23524 .19438 .25294 .21952 .24337 .20213 .22557 Mtetra .661 .742 .91 r .20213 .22557 .21952 .24337 .24491 .23373 .22764 .21591 Mtetra .663 .745 .911 r .22764 .21591 .24491 .23373 .27054 .22396 .25343 .206 Mtetra .666 .746 .91 r .25343 .206 .27054 .22396 .29644 .21406 .27948 .19582 Mtetra .667 .745 .909 r .27948 .19582 .29644 .21406 .32258 .20406 .3058 .18546 Mtetra .666 .743 .908 r .3058 .18546 .32258 .20406 .34899 .194 .33238 .17507 Mtetra .663 .74 .907 r .33238 .17507 .34899 .194 .37566 .18389 .35921 .16475 Mtetra .661 .739 .908 r .35921 .16475 .37566 .18389 .40259 .17371 .38631 .15448 Mtetra .66 .74 .909 r .38631 .15448 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0.09]^2, \ {i, 0, 10}, {j, \(-10\), 0}], \ EV > 5}, \ {EV, X1, \ X2, \ X3, \ Y1}]\)], "Input"], Cell[BoxData[ \({0.00003391761790337019`, {EV \[Rule] 6.322347634282059`, X1 \[Rule] \(-0.2451242712791892`\), X2 \[Rule] 0.006108877513251701`, X3 \[Rule] 0.006102419975037478`, Y1 \[Rule] \(-0.024410562142409976`\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NMinimize[{Sum[ tomin[i\ 0.09, j\ 0.09]^2, \ {i, 0, 10}, {j, \(-10\), 0}], \ EV > 12}, \ {EV, X1, \ X2, \ X3, \ Y1}]\)], "Input"], Cell[BoxData[ \({0.00847857191560664`, {EV \[Rule] 17.23556364291644`, X1 \[Rule] \(-0.6655036614509152`\), X2 \[Rule] 0.7034479530555982`, X3 \[Rule] 0.15983348488484153`, Y1 \[Rule] \(-1.2759580065477767`\)}}\)], "Output"] }, Open ]], Cell["And the second stored as EV -> 17.2356", "Text"], Cell["\<\ Here are the eigenvalues using the canonical metric -- a little \ lower.\ \>", "Text"], Cell[BoxData[ \(\(\(tomin[x_, \ y_]\ = \ myLap[x, y, gp\ , \ psitest]\ - \ EV\ psitest[x, y];\)\(\ 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