(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 96800, 3098]*) (*NotebookOutlinePosition[ 111772, 3642]*) (* CellTagsIndexPosition[ 111728, 3638]*) (*WindowFrame->Normal*) Notebook[{ Cell["Linear spaces and operators", "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This notebook has been written in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by \n\n", StyleBox["Mark J. McCready\nProfessor and Chair of Chemical Engineering\n\ University of Notre Dame\nNotre Dame IN 46556\nUSA", FontSize->14], "\n\nMark.J.McCready.1@nd.edu\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy, this notice remain visible to other users. \nThere is no charge for \ copying and dissemination \n\nVersion: 9/98\nMore recent versions of this \ notebook should be available at the web site:\n", Cell[BoxData[ FormBox[ ButtonBox[\(\(\(http\)\(:\)\) // \(www . nd . edu/\(\(~\)\(mjm\)\)\)/ linear . spaces . nb\), ButtonData:>{ URL[ "http://www.nd.edu/~mjm/linear.spaces.nb"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text"], Cell["\<\ If our mutlidimensional vectors \"live\" in some appropriate \ multidimensional space (See Greenberg) we need some properties \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["vector norms", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A natural question about a vector is, how \"long\" is it? We might \ want to know this for a physical reason, say mean velocity, or for an \ abstract comparision. In Cartesian space if we want the length of a vector we might define it \ as\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(norm[v_] := \@\(\[Sum]\+\(i = 1\)\%\(Length[v]\)v\[LeftDoubleBracket]i \[RightDoubleBracket]\^2\)\)], "Input", CellLabel->"In[1]:=", AspectRatioFixed->True], Cell["which is the Euclidean norm.", "Text"], Cell[BoxData[ \(\(v = {v1, v2, v3}; \)\)], "Input", CellLabel->"In[2]:=", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(norm[v]\)], "Input", CellLabel->"In[3]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\@\(v1\^2 + v2\^2 + v3\^2\)\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[BoxData[ \(\(yy = {y1, y2, y3, y4, y5}; \)\)], "Input", CellLabel->"In[2]:=", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(norm[yy]\)], "Input", CellLabel->"In[5]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\@\(y1\^2 + y2\^2 + y3\^2 + y4\^2 + y5\^2\)\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[BoxData[ \(\(zz = {1, 3, 2, 6, 2}; \)\)], "Input", CellLabel->"In[6]:=", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(\ norm[zz]\)\)], "Input", CellLabel->"In[7]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`3\ \@6\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell["\<\ However, we might like a different measure of length for \ applications where the landscape is different. For example, the Taxi Cab \ norm or the infinity norm (take the biggest element). Here is one definition that has no individual weightings, but does allow us \ to change how we count large versus small elements. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(generalnorm[v_, n_] := \((\[Sum]\+\(i = 1\)\%\(Length[v]\)Abs[ v\[LeftDoubleBracket]i\[RightDoubleBracket]]\^n)\)\^\(1/n \)\)], "Input", CellLabel->"In[8]:=", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(N[generalnorm[zz, 2]]\)], "Input", CellLabel->"In[9]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`7.34846922834953364`\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[generalnorm[zz, 10]]\)], "Input", CellLabel->"In[10]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`6.00060599403250627`\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell["Here we can plot the norm as a function of n.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[generalnorm[zz, j], {j, 1, 5}, PlotRange \[Rule] All]\)], "Input", CellLabel->"In[11]:=", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.214286 0.238095 -0.432799 0.0740084 [ [.2619 -0.00125 -3 -9 ] [.2619 -0.00125 3 0 ] [.5 -0.00125 -3 -9 ] [.5 -0.00125 3 0 ] [.7381 -0.00125 -3 -9 ] [.7381 -0.00125 3 0 ] [.97619 -0.00125 -3 -9 ] [.97619 -0.00125 3 0 ] [.01131 .15927 -6 -4.5 ] [.01131 .15927 0 4.5 ] [.01131 .30729 -12 -4.5 ] [.01131 .30729 0 4.5 ] [.01131 .4553 -12 -4.5 ] [.01131 .4553 0 4.5 ] [.01131 .60332 -12 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .2619 .01125 m .2619 .0175 L s [(2)] .2619 -0.00125 0 1 Mshowa .5 .01125 m .5 .0175 L s [(3)] .5 -0.00125 0 1 Mshowa .7381 .01125 m .7381 .0175 L s [(4)] .7381 -0.00125 0 1 Mshowa .97619 .01125 m .97619 .0175 L s [(5)] .97619 -0.00125 0 1 Mshowa .125 Mabswid .07143 .01125 m .07143 .015 L s .11905 .01125 m .11905 .015 L s .16667 .01125 m .16667 .015 L s .21429 .01125 m .21429 .015 L s .30952 .01125 m .30952 .015 L s .35714 .01125 m .35714 .015 L s .40476 .01125 m .40476 .015 L s .45238 .01125 m .45238 .015 L s .54762 .01125 m .54762 .015 L s .59524 .01125 m .59524 .015 L s .64286 .01125 m .64286 .015 L s .69048 .01125 m .69048 .015 L s .78571 .01125 m .78571 .015 L s .83333 .01125 m .83333 .015 L s .88095 .01125 m .88095 .015 L s .92857 .01125 m .92857 .015 L s .25 Mabswid 0 .01125 m 1 .01125 L s .02381 .15927 m .03006 .15927 L s [(8)] .01131 .15927 1 0 Mshowa .02381 .30729 m .03006 .30729 L s [(10)] .01131 .30729 1 0 Mshowa .02381 .4553 m .03006 .4553 L s [(12)] .01131 .4553 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(14)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .04826 m .02756 .04826 L s .02381 .08526 m .02756 .08526 L s .02381 .12226 m .02756 .12226 L s .02381 .19627 m .02756 .19627 L s .02381 .23328 m .02756 .23328 L s .02381 .27028 m .02756 .27028 L s .02381 .34429 m .02756 .34429 L s .02381 .38129 m .02756 .38129 L s .02381 .4183 m .02756 .4183 L s .02381 .49231 m .02756 .49231 L s .02381 .52931 m .02756 .52931 L s .02381 .56631 m .02756 .56631 L s .25 Mabswid .02381 0 m .02381 .61803 L s .5 Mabswid .02381 .60332 m .04262 .50069 L .06244 .41801 L .08255 .3529 L .10458 .29701 L .12579 .25427 L .14509 .22247 L .18408 .17304 L .22401 .13666 L .26243 .11074 L .30329 .08992 L .34264 .07457 L .38048 .063 L .42077 .05326 L .45954 .0458 L .50076 .03946 L .54047 .03455 L .57866 .03071 L .61931 .02737 L .65844 .02473 L .70002 .02244 L .74009 .02063 L .77864 .01918 L .81964 .0179 L .85913 .01688 L .8971 .01605 L .93752 .01531 L .97619 .01472 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", CellLabel->"From In[11]:=", Evaluatable->False, AspectRatioFixed->True, ImageSize->{348.938, 215}, ImageMargins->{{34, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOGG>O003oLimM Lil0065cW`@004]cW`<004acW`<004]cW`<000acW`00HW>O00<007>OLil0CW>O00<007>OLil0BW>O 00<007>OLil0CG>O00<007>OLil02G>O001SLil00`00LimcW`1=Lil00`00LimcW`18Lil4001>Lil0 0`00LimcW`09Lil006AcW`03001cWg>O04YcW`8004]cW`03001cW`0004acW`<000acW`00HG>O00@0 07>OLil004icW`03001cWg>O04QcW`03001cW`0004acW`03001cWg>O00acW`00HW>O0P00C7>O0`00 C7>O0P00C7>O10002g>O003oLimMLil00?mcWeecW`0057>O00<007>OLil0og>OAW>O000DLil00`00 LimcW`3oLim6Lil001AcW`03001cWg>O0?mcWdIcW`0057>O00<007>OLil0og>OAW>O000O00ecW`03001cWg>O00acW`03001cWg>O00ecW`03001cWg>O00ecW`03 001cWg>O00ecW`03001cWg>O00ecW`03001cWg>O00acW`03001cWg>O00ecW`03001cWg>O00ecW`03 001cWg>O00ecW`03001cWg>O00acW`03001cWg>O00ecW`03001cWg>O00ecW`03001cWg>O00ecW`03 001cWg>O00acW`03001cWg>O00ecW`03001cWg>O00ecW`03001cWg>O00acWb4000acW`0057>O00<0 07>OLil0C7>O00<007>OLil0C7>O00<007>OLil0Bg>O00<007>OLil01G>O9`0087>O00<007>OLil0 2W>O000DLil00`00LimcW`3FLilK001DLil001AcW`03001cWg>O0O00<007>O Lil0_7>O3@00O7>O000DLil00`00LimcW`2_Lil=0029Lil001AcW`03001cWg>O0:EcW`X009IcW`00 57>O00<007>OLil0WW>O1`00X7>O000DLil00`00LimcW`2HLil6002WLil001AcW`03001cWg>O095c W`L00:ecW`0057>O00<007>OLil0Rg>O1P00]7>O000DLil20027Lil5002jLil001AcW`03001cWg>O 089cW`@00;mcW`0057>O00<007>OLil0OW>O1000`g>O000DLil00`00LimcW`1iLil50037Lil001Ac W`03001cWg>O07EcW`@00O00<007>OLil0LW>O0`00d7>O000DLil00`00LimcW`1_Lil3 003CLil001AcW`03001cWg>O06acW`<00=IcW`0057>O00<007>OLil0JG>O0`00fG>O000DLil00`00 LimcW`1VLil3003LLil001AcW`03001cWg>O06=cW`<00=mcW`0057>O00<007>OLil0H7>O0`00hW>O 000DLil2001NLil3003ULil001AcW`03001cWg>O05YcW`<00>QcW`0057>O00<007>OLil0F7>O0P00 jg>O000DLil00`00LimcW`1FLil2003]Lil001AcW`03001cWg>O05AcW`800>mcW`0057>O00<007>O Lil0DW>O0P00lG>O000DLil00`00LimcW`1ALil00`00LimcW`3aLil001AcW`03001cWg>O04mcW`80 0?AcW`0057>O00<007>OLil0CG>O0P00mW>O000DLil00`00LimcW`1;Lil2003hLil001AcW`03001c Wg>O04YcW`03001cWg>O0?QcW`0057>O00<007>OLil0B7>O0P00ng>O000DLil20017Lil2003mLil0 01AcW`03001cWg>O04EcW`03001cWg>O0?ecW`0057>O00<007>OLil0@g>O0P00og>O0G>O000DLil0 0`00LimcW`11Lil2003oLil3Lil001AcW`03001cWg>O041cW`03001cWg>O0?mcW`=cW`0057>O00<0 07>OLil0?W>O0P00og>O1W>O000DLil00`00LimcW`0mLil00`00LimcW`3oLil6Lil001AcW`03001c Wg>O03acW`03001cWg>O0?mcW`McW`0057>O00<007>OLil0>g>O00<007>OLil0og>O27>O000DLil0 0`00LimcW`0jLil00`00LimcW`3oLil9Lil001AcW`03001cWg>O03UcW`03001cWg>O0?mcW`YcW`00 37>O0P001W>O00<007>OLil0>7>O00<007>OLil0og>O2g>O000;Lil01000LimcW`001G>O00<007>O Lil0=g>O00<007>OLil0og>O37>O000;Lil01000LimcW`001G>O0`00=W>O00<007>OLil0og>O3G>O 000Lil000]cW`04001cWg>O0005Lil00`00 LimcW`0dLil00`00LimcW`3oLil?Lil000acW`8000IcW`03001cWg>O03=cW`03001cWg>O0?mcWa1c W`0057>O00<007>OLil0O00<007>OLil0og>O4G>O000DLil00`00LimcW`0bLil00`00LimcW`3o LilALil001AcW`03001cWg>O035cW`03001cWg>O0?mcWa9cW`0057>O00<007>OLil0<7>O00<007>O Lil0og>O4g>O000DLil00`00LimcW`0_Lil00`00LimcW`3oLilDLil001AcW`03001cWg>O02mcW`03 001cWg>O0?mcWaAcW`0057>O00<007>OLil0;W>O00<007>OLil0og>O5G>O000DLil00`00LimcW`0] Lil00`00LimcW`3oLilFLil001AcW`8002ecW`03001cWg>O0?mcWaMcW`0057>O00<007>OLil0:g>O 00<007>OLil0og>O67>O000DLil00`00LimcW`0[Lil00`00LimcW`3oLilHLil001AcW`03001cWg>O 02YcW`03001cWg>O0?mcWaUcW`0057>O00<007>OLil0:G>O00<007>OLil0og>O6W>O000DLil00`00 LimcW`0XLil00`00LimcW`3oLilKLil001AcW`03001cWg>O02QcW`03001cWg>O0?mcWa]cW`0057>O 00<007>OLil09g>O00<007>OLil0og>O77>O000DLil00`00LimcW`0VLil00`00LimcW`3oLilMLil0 01AcW`03001cWg>O02EcW`03001cWg>O0?mcWaicW`0057>O00<007>OLil09G>O00<007>OLil0og>O 7W>O000DLil00`00LimcW`0TLil00`00LimcW`3oLilOLil001AcW`8002AcW`03001cWg>O0?mcWb1c W`0057>O00<007>OLil08g>O00<007>OLil0og>O87>O000DLil00`00LimcW`0RLil00`00LimcW`3o LilQLil001AcW`03001cWg>O029cW`03001cWg>O0?mcWb5cW`0057>O00<007>OLil08G>O00<007>O Lil0og>O8W>O000DLil00`00LimcW`0PLil00`00LimcW`3oLilSLil001AcW`03001cWg>O021cW`03 001cWg>O0?mcWb=cW`0057>O00<007>OLil07g>O00<007>OLil0og>O97>O000DLil00`00LimcW`0O Lil00`00LimcW`3oLilTLil001AcW`03001cWg>O01icW`03001cWg>O0?mcWbEcW`0057>O00<007>O Lil07W>O00<007>OLil0og>O9G>O000DLil00`00LimcW`0MLil00`00LimcW`3oLilVLil001AcW`80 01icW`03001cWg>O0?mcWbIcW`0057>O00<007>OLil077>O00<007>OLil0og>O9g>O000DLil00`00 LimcW`0LLil00`00LimcW`3oLilWLil001AcW`03001cWg>O01]cW`03001cWg>O0?mcWbQcW`0057>O 00<007>OLil06g>O00<007>OLil0og>O:7>O000DLil00`00LimcW`0JLil00`00LimcW`3oLilYLil0 01AcW`03001cWg>O01YcW`03001cWg>O0?mcWbUcW`0057>O00<007>OLil06G>O00<007>OLil0og>O :W>O000DLil00`00LimcW`0ILil00`00LimcW`3oLilZLil001AcW`03001cWg>O01QcW`03001cWg>O 0?mcWb]cW`0057>O00<007>OLil067>O00<007>OLil0og>O:g>O0005Lil30004Lil20006Lil00`00 LimcW`0GLil00`00LimcW`3oLil/Lil000IcW`03001cWg>O009cW`04001cWg>O0005Lil00`00Limc W`0GLil00`00LimcW`3oLil/Lil000IcW`03001cWg>O009cW`04001cWg>O0005Lil3000GLil00`00 LimcW`3oLil/Lil000IcW`03001cWg>O009cW`04001cWg>O0005Lil00`00LimcW`0FLil00`00Limc W`3oLil]Lil000IcW`03001cWg>O009cW`04001cWg>O0005Lil00`00LimcW`0FLil00`00LimcW`3o Lil]Lil000EcW`8000EcW`8000IcW`03001cWg>O01EcW`03001cWg>O0?mcWbicW`0057>O00<007>O Lil05G>O00<007>OLil0og>O;W>O000DLil00`00LimcW`0DLil00`00LimcW`3oLil_Lil001AcW`03 001cWg>O01AcW`03001cWg>O0?mcWbmcW`0057>O00<007>OLil057>O00<007>OLil0og>O;g>O000D Lil00`00LimcW`0CLil00`00LimcW`3oLil`Lil001AcW`03001cWg>O01=cW`03001cWg>O0?mcWc1c W`0057>O00<007>OLil04W>O00<007>OLil0og>OO000DLil00`00LimcW`0BLil00`00LimcW`3o LilaLil001AcW`80019cW`03001cWg>O0?mcWc9cW`0057>O00<007>OLil04G>O00<007>OLil0og>O O000DLil00`00LimcW`0ALil00`00LimcW`3oLilbLil001AcW`03001cWg>O011cW`03001cWg>O 0?mcWc=cW`0057>O00<007>OLil047>O00<007>OLil0og>OO000DLil00`00LimcW`0@Lil00`00 LimcW`3oLilcLil001AcW`03001cWg>O00mcW`03001cWg>O0?mcWcAcW`0057>O00<007>OLil03g>O 00<007>OLil0og>O=7>O000DLil00`00LimcW`0?Lil00`00LimcW`3oLildLil001AcW`03001cWg>O 00icW`03001cWg>O0?mcWcEcW`0057>O00<007>OLil03W>O00<007>OLil0og>O=G>O000DLil00`00 LimcW`0>Lil00`00LimcW`3oLileLil001AcW`8000mcW`03001cWg>O0?mcWcEcW`0057>O00<007>O Lil03G>O00<007>OLil0og>O=W>O000DLil00`00LimcW`0=Lil00`00LimcW`3oLilfLil001AcW`03 001cWg>O00ecW`03001cWg>O0?mcWcIcW`0057>O00<007>OLil037>O00<007>OLil0og>O=g>O000D Lil00`00LimcW`0O00acW`03001cWg>O0?mcWcMc W`0057>O00<007>OLil037>O00<007>OLil0og>O=g>O000DLil00`00LimcW`0;Lil00`00LimcW`3o LilhLil001AcW`03001cWg>O00]cW`03001cWg>O0?mcWcQcW`0057>O00<007>OLil02g>O00<007>O Lil0og>O>7>O000DLil00`00LimcW`0:Lil00`00LimcW`3oLiliLil001AcW`8000]cW`03001cWg>O 0?mcWcUcW`0057>O00<007>OLil02W>O00<007>OLil0og>O>G>O000DLil00`00LimcW`0:Lil00`00 LimcW`3oLiliLil001AcW`03001cWg>O00UcW`03001cWg>O0?mcWcYcW`0057>O00<007>OLil02G>O 00<007>OLil0og>O>W>O000DLil00`00LimcW`09Lil00`00LimcW`3oLiljLil001AcW`03001cWg>O 00UcW`03001cWg>O0?mcWcYcW`0057>O00<007>OLil027>O00<007>OLil0og>O>g>O000DLil00`00 LimcW`08Lil00`00LimcW`3oLilkLil001AcW`03001cWg>O00QcW`03001cWg>O0?mcWc]cW`0057>O 00<007>OLil027>O00<007>OLil0og>O>g>O0005Lil30003Lil40005Lil00`00LimcW`07Lil00`00 LimcW`3oLillLil000IcW`03001cWg>O00=cW`03001cWg>O00EcW`03001cWg>O00McW`03001cWg>O 0?mcWcacW`001W>O00<007>OLil017>O00<007>OLil017>O0`001g>O00<007>OLil0og>O?7>O0006 Lil00`00LimcW`05Lil00`00LimcW`03Lil00`00LimcW`07Lil00`00LimcW`3oLillLil000IcW`03 001cWg>O009cW`04001cWg>O0005Lil00`00LimcW`06Lil00`00LimcW`3oLilmLil000EcW`8000Ec W`8000IcW`03001cWg>O00IcW`03001cWg>O0?mcWcecW`0057>O00<007>OLil01W>O00<007>OLil0 og>O?G>O000DLil00`00LimcW`06Lil00`00LimcW`3oLilmLil001AcW`03001cWg>O00EcW`03001c Wg>O0?mcWcicW`0057>O00<007>OLil01G>O00<007>OLil0og>O?W>O000DLil00`00LimcW`05Lil0 0`00LimcW`3oLilnLil001AcW`03001cWg>O00EcW`03001cWg>O0?mcWcicW`0057>O00<007>OLil0 17>O00<007>OLil0og>O?g>O000DLil00`00LimcW`04Lil00`00LimcW`3oLiloLil001AcW`8000Ec W`03001cWg>O0?mcWcmcW`0057>O00<007>OLil017>O00<007>OLil0og>O?g>O000DLil00`00Limc W`03Lil00`00LimcW`3oLim0Lil001AcW`03001cWg>O00=cW`03001cWg>O0?mcWd1cW`0057>O00<0 07>OLil00g>O00<007>OLil0og>O@7>O000DLil00`00LimcW`03Lil00`00LimcW`3oLim0Lil001Ac W`03001cWg>O00=cW`03001cWg>O0?mcWd1cW`0057>O00<007>OLil00W>O00<007>OLil0og>O@G>O 000DLil00`00LimcW`02Lil00`00LimcW`3oLim1Lil001AcW`03001cWg>O009cW`03001cWg>O0?mc Wd5cW`0057>O00<007>OLil00W>O00<007>OLil0og>O@G>O000DLil00`00LimcW`02Lil00`00Limc W`3oLim1Lil001AcW`80009cW`03001cWg>O0?mcWd9cW`0057>O00D007>OLimcW`000?mcWdAcW`00 57>O00D007>OLimcW`000?mcWdAcW`0057>O00D007>OLimcW`000?mcWdAcW`0057>O00D007>OLimc W`000?mcWdAcW`0057>O00D007>OLimcW`000?mcWdAcW`0057>O00@007>OLil00?mcWdEcW`0057>O 00@007>OLil00?mcWdEcW`0057>O00@007>OLil00?mcWdEcW`0057>O00@007>OLil00?mcWdEcW`00 57>O00@007>OLil00?mcWdEcW`0057>O00@007>OLil00?mcWdEcW`0057>O0`00og>OAW>O000DLil0 0`00Lil0003oLim6Lil001AcW`03001cW`000?mcWdIcW`0057>O00<007>O0000og>OAW>O000DLil0 0`00Lil0003oLim6Lil001AcW`800?mcWdMcW`0057>O0P00og>OAg>O000DLil2003oLim7Lil001Ac W`800?mcWdMcW`0057>O0P00og>OAg>O000DLil2003oLim7Lil000EcW`<000AcW`<000EcW`03001c Wg>O0?mcWdIcW`001W>O00<007>OLil017>O00<007>OLil017>O00<007>OLil0og>OAW>O0006Lil0 0`00LimcW`02Lil40005Lil3003oLim6Lil000IcW`03001cWg>O009cW`03001cW`0000IcW`03001c Wg>O0?mcWdIcW`001W>O00<007>OLil00W>O00<007>O00001W>O00<007>OLil0og>OAW>O0005Lil2 0005Lil20006Lil00`00LimcW`3oLim6Lil001AcW`03001cWg>O0?mcWdIcW`00\ \>"], ImageRangeCache->{{{0, 347.938}, {214, 0}} -> {0.740343, 5.46367, 0.0126878, 0.0408185}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[TextData[ "We see that this \"generalized\" norm allows us to change the weighting from \ a simple sum as the length to a measure on the largest element. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "If the elements are about the same size there can a be some significant % \ variation as n changes. However if there is only one very large element then \ there is not much change"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(zmax = {1, 1, 1, 2, 1000, 1, 1, 1, 3}; \)\)], "Input", CellLabel->"In[12]:=", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[generalnorm[zmax, j], {j, 2, 5}, PlotRange \[Rule] All]\)], "Input",\ CellLabel->"In[13]:=", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.611111 0.31746 -61958.6 61.9586 [ [.18254 .00222 -9 -9 ] [.18254 .00222 9 0 ] [.34127 .00222 -3 -9 ] [.34127 .00222 3 0 ] [.5 .00222 -9 -9 ] [.5 .00222 9 0 ] [.65873 .00222 -3 -9 ] [.65873 .00222 3 0 ] [.81746 .00222 -9 -9 ] [.81746 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.01131 .13863 -24 -4.5 ] [.01131 .13863 0 4.5 ] [.01131 .26255 -24 -4.5 ] [.01131 .26255 0 4.5 ] [.01131 .38647 -42 -4.5 ] [.01131 .38647 0 4.5 ] [.01131 .51038 -42 -4.5 ] [.01131 .51038 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .18254 .01472 m .18254 .02097 L s [(2.5)] .18254 .00222 0 1 Mshowa .34127 .01472 m .34127 .02097 L s [(3)] .34127 .00222 0 1 Mshowa .5 .01472 m .5 .02097 L s [(3.5)] .5 .00222 0 1 Mshowa .65873 .01472 m .65873 .02097 L s [(4)] .65873 .00222 0 1 Mshowa .81746 .01472 m .81746 .02097 L s [(4.5)] .81746 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(5)] .97619 .00222 0 1 Mshowa .125 Mabswid .05556 .01472 m .05556 .01847 L s .0873 .01472 m .0873 .01847 L s .11905 .01472 m .11905 .01847 L s .15079 .01472 m .15079 .01847 L s .21429 .01472 m .21429 .01847 L s .24603 .01472 m .24603 .01847 L s .27778 .01472 m .27778 .01847 L s .30952 .01472 m .30952 .01847 L s .37302 .01472 m .37302 .01847 L s .40476 .01472 m .40476 .01847 L s .43651 .01472 m .43651 .01847 L s .46825 .01472 m .46825 .01847 L s .53175 .01472 m .53175 .01847 L s .56349 .01472 m .56349 .01847 L s .59524 .01472 m .59524 .01847 L s .62698 .01472 m .62698 .01847 L s .69048 .01472 m .69048 .01847 L s .72222 .01472 m .72222 .01847 L s .75397 .01472 m .75397 .01847 L s .78571 .01472 m .78571 .01847 L s .84921 .01472 m .84921 .01847 L s .88095 .01472 m .88095 .01847 L s .9127 .01472 m .9127 .01847 L s .94444 .01472 m .94444 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .13863 m .03006 .13863 L s [(1000)] .01131 .13863 1 0 Mshowa .02381 .26255 m .03006 .26255 L s [(1000)] .01131 .26255 1 0 Mshowa .02381 .38647 m .03006 .38647 L s [(1000.01)] .01131 .38647 1 0 Mshowa .02381 .51038 m .03006 .51038 L s [(1000.01)] .01131 .51038 1 0 Mshowa .125 Mabswid .02381 .04569 m .02756 .04569 L s .02381 .07667 m .02756 .07667 L s .02381 .10765 m .02756 .10765 L s .02381 .16961 m .02756 .16961 L s .02381 .20059 m .02756 .20059 L s .02381 .23157 m .02756 .23157 L s .02381 .29353 m .02756 .29353 L s .02381 .32451 m .02756 .32451 L s .02381 .35549 m .02756 .35549 L s .02381 .41745 m .02756 .41745 L s .02381 .44843 m .02756 .44843 L s .02381 .4794 m .02756 .4794 L s .02381 .54136 m .02756 .54136 L s .02381 .57234 m .02756 .57234 L s .02381 .60332 m .02756 .60332 L s .25 Mabswid .02381 0 m .02381 .61803 L s .5 Mabswid .02381 .60332 m .03279 .50126 L .04262 .40984 L .06244 .27474 L .07298 .22303 L .08426 .17914 L .10458 .12219 L .1148 .10156 L .12589 .08366 L .13547 .07121 L .14603 .0601 L .16586 .04483 L .17672 .03879 L .18689 .03424 L .19727 .03049 L .20714 .0276 L .22566 .02353 L .23525 .02196 L .24547 .0206 L .26386 .01876 L .27414 .018 L .28527 .01734 L .29488 .01687 L .30544 .01646 L .32538 .01588 L .33627 .01565 L .34644 .01548 L .35689 .01533 L .36679 .01522 L .38535 .01506 L .395 .015 L .40526 .01495 L .42368 .01488 L .43305 .01485 L .44315 .01483 L .46143 .01479 L .471 .01478 L .48147 .01477 L .49253 .01476 L .50257 .01475 L .52218 .01474 L .53296 .01473 L .54313 .01473 L .55285 .01473 L .56315 .01473 L .5816 .01472 L .59103 .01472 L .60117 .01472 L .61949 .01472 L .62962 .01472 L Mistroke .64066 .01472 L .65121 .01472 L .66076 .01472 L .68047 .01472 L .69129 .01472 L .70146 .01472 L .71125 .01472 L .72158 .01472 L .74006 .01472 L .74957 .01472 L .75974 .01472 L .77809 .01472 L .78829 .01472 L .79937 .01472 L .80894 .01472 L .8195 .01472 L .83931 .01472 L .85016 .01472 L .86033 .01472 L .87019 .01472 L .88056 .01472 L .89907 .01472 L .90864 .01472 L .91885 .01472 L .93723 .01472 L .94686 .01472 L .95729 .01472 L .97619 .01472 L Mfstroke 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", CellLabel->"From In[13]:=", Evaluatable->False, ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgO8G>O003oLilQ Lil004acW`@000=cW`8000=cW`<001icW`<001mcW`<000AcW`8000=cW`<001mcW`<001icW`<000=c W`8000=cW`<001mcW`<000IcW`00CG>O00<007>OLil00g>O0P001W>O00<007>OLil07W>O00<007>O Lil07g>O00@007>OLimcW`8000IcW`03001cWg>O01ecW`03001cWg>O01icW`03001cWg>O009cW`80 00IcW`03001cWg>O01mcW`03001cWg>O00=cW`00CW>O00<007>OLil02W>O00<007>OLil07W>O00<0 07>OLil07g>O00<007>OLil02G>O00<007>OLil06g>O10007G>O10002g>O00<007>OLil07g>O00<0 07>OLil00g>O001?Lil00`00LimcW`06Lil3000OLil2000PLil20009Lil3000NLil00`00Lil0000N Lil00`00Lil00009Lil3000OLil30006Lil004acW`04001cWg>O0008Lil00`00LimcW`0QLil00`00 LimcW`0OLil00`00LimcW`06Lil00`00LimcW`0NLil00`00Lil0000NLil00`00Lil00009Lil00`00 LimcW`0OLil00`00LimcW`06Lil004ecW`8000UcW`@001ecW`<001mcW`<000UcW`@001icW`8001mc W`8000UcW`@001icW`@000EcW`00;G>O00<007>OLil0l7>O000]Lil00`00LimcW`3`Lil002ecW`03 001cWg>O0?1cW`00;G>O00<007>OLil0l7>O000WLioi000002ecW`03001cWg>O00AcW`03001cWg>O 00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O 00EcW`03001cW`0000X0009cW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001c Wg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O00AcW`03001cWg>O00EcW`03001c Wg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001c Wg>O00EcW`03001cWg>O00AcW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001c Wg>O00EcW`03001cWg>O00EcW`03001cWg>O00EcW`03001cWg>O00AcW`00;G>O00<007>OLil097>O 00<007>OLil01W>O2@005G>O00<007>OLil09G>O00<007>OLil097>O00<007>OLil097>O00<007>O Lil09G>O00<007>OLil017>O000]Lil00`00LimcW`0YLil40033Lil002ecW`03001cWg>O02McW`80 0O00<007>OLil097>O0`00bG>O000]Lil00`00LimcW`0SLil00`00LimcW`3:Lil002ec W`03001cWg>O025cW`800O0P0087>O0P00cg>O000]Lil00`00LimcW`0NLil00`00Limc W`3?Lil002ecW`03001cWg>O01ecW`03001cWg>O0=1cW`00;G>O00<007>OLil077>O00<007>OLil0 dG>O000]Lil00`00LimcW`0KLil00`00LimcW`3BLil002ecW`03001cWg>O01YcW`03001cWg>O0==c W`00;G>O00<007>OLil06G>O00<007>OLil0e7>O000]Lil00`00LimcW`0HLil00`00LimcW`3ELil0 02ecW`03001cWg>O01QcW`03001cWg>O0=EcW`00;G>O0P0067>O00<007>OLil0eW>O000]Lil00`00 LimcW`0GLil00`00LimcW`3FLil002ecW`03001cWg>O01IcW`03001cWg>O0=McW`00;G>O00<007>O Lil05W>O00<007>OLil0eg>O000]Lil00`00LimcW`0ELil00`00LimcW`3HLil002ecW`03001cWg>O 01AcW`03001cWg>O0=UcW`00;G>O00<007>OLil057>O00<007>OLil0fG>O000]Lil00`00LimcW`0C Lil00`00LimcW`3JLil002ecW`8001AcW`03001cWg>O0=YcW`00;G>O00<007>OLil04W>O00<007>O Lil0fg>O000]Lil00`00LimcW`0BLil00`00LimcW`3KLil002ecW`03001cWg>O019cW`03001cWg>O 0=]cW`00;G>O00<007>OLil04G>O00<007>OLil0g7>O000]Lil00`00LimcW`0ALil00`00LimcW`3L Lil002ecW`03001cWg>O015cW`03001cWg>O0=acW`004g>O0`0017>O0P0017>O0P0017>O0P001G>O 00<007>OLil047>O00<007>OLil0gG>O000DLil00`00LimcW`02Lil01000LimcW`000W>O00@007>O Lil0009cW`04001cWg>O0004Lil00`00LimcW`0@Lil00`00LimcW`3MLil001AcW`03001cWg>O009c W`04001cWg>O0002Lil01000LimcW`000W>O00@007>OLil000AcW`80015cW`03001cWg>O0=ecW`00 57>O00<007>OLil00W>O00@007>OLil0009cW`04001cWg>O0002Lil01000LimcW`0017>O00<007>O Lil03g>O00<007>OLil0gW>O000DLil00`00LimcW`02Lil01000LimcW`000W>O00@007>OLil0009c W`04001cWg>O0004Lil00`00LimcW`0?Lil00`00LimcW`3NLil001=cW`8000EcW`8000AcW`8000Ac W`8000EcW`03001cWg>O00mcW`03001cWg>O0=icW`00;G>O00<007>OLil03W>O00<007>OLil0gg>O 000]Lil00`00LimcW`0>Lil00`00LimcW`3OLil002ecW`03001cWg>O00icW`03001cWg>O0=mcW`00 ;G>O00<007>OLil03G>O00<007>OLil0h7>O000]Lil2000>Lil00`00LimcW`3PLil002ecW`03001c Wg>O00ecW`03001cWg>O0>1cW`00;G>O00<007>OLil037>O00<007>OLil0hG>O000]Lil00`00Limc W`0O00acW`03001cWg>O0>5cW`00;G>O00<007>OLil0 37>O00<007>OLil0hG>O000]Lil00`00LimcW`0;Lil00`00LimcW`3RLil002ecW`03001cWg>O00]c W`03001cWg>O0>9cW`00;G>O00<007>OLil02g>O00<007>OLil0hW>O000]Lil2000O00YcW`03001cWg>O0>=cW`00;G>O00<007>OLil02W>O00<007>OLil0 hg>O000]Lil00`00LimcW`0:Lil00`00LimcW`3SLil002ecW`03001cWg>O00YcW`03001cWg>O0>=c W`00;G>O00<007>OLil02G>O00<007>OLil0i7>O000]Lil00`00LimcW`09Lil00`00LimcW`3TLil0 02ecW`03001cWg>O00UcW`03001cWg>O0>AcW`00;G>O0P002W>O00<007>OLil0i7>O000]Lil00`00 LimcW`08Lil00`00LimcW`3ULil002ecW`03001cWg>O00QcW`03001cWg>O0>EcW`00;G>O00<007>O Lil027>O00<007>OLil0iG>O000]Lil00`00LimcW`08Lil00`00LimcW`3ULil002ecW`03001cWg>O 00QcW`03001cWg>O0>EcW`00;G>O00<007>OLil01g>O00<007>OLil0iW>O000CLil30004Lil20004 Lil20004Lil20005Lil00`00LimcW`07Lil00`00LimcW`3VLil001AcW`03001cWg>O009cW`04001c Wg>O0002Lil01000LimcW`000W>O00@007>OLil000AcW`03001cWg>O00McW`03001cWg>O0>IcW`00 57>O00<007>OLil00W>O00@007>OLil0009cW`04001cWg>O0002Lil01000LimcW`0017>O0P0027>O 00<007>OLil0iW>O000DLil00`00LimcW`02Lil01000LimcW`000W>O00@007>OLil0009cW`04001c Wg>O0004Lil00`00LimcW`07Lil00`00LimcW`3VLil001AcW`03001cWg>O009cW`04001cWg>O0002 Lil01000LimcW`000W>O00@007>OLil000AcW`03001cWg>O00IcW`03001cWg>O0>McW`004g>O0P00 1G>O0P0017>O0P0017>O0P001G>O00<007>OLil01W>O00<007>OLil0ig>O000]Lil00`00LimcW`06 Lil00`00LimcW`3WLil002ecW`03001cWg>O00IcW`03001cWg>O0>McW`00;G>O00<007>OLil01W>O 00<007>OLil0ig>O000]Lil00`00LimcW`06Lil00`00LimcW`3WLil002ecW`8000IcW`03001cWg>O 0>QcW`00;G>O00<007>OLil01G>O00<007>OLil0j7>O000]Lil00`00LimcW`05Lil00`00LimcW`3X Lil002ecW`03001cWg>O00EcW`03001cWg>O0>QcW`00;G>O00<007>OLil01G>O00<007>OLil0j7>O 000]Lil00`00LimcW`05Lil00`00LimcW`3XLil002ecW`03001cWg>O00EcW`03001cWg>O0>QcW`00 ;G>O00<007>OLil017>O00<007>OLil0jG>O000]Lil20005Lil00`00LimcW`3YLil002ecW`03001c Wg>O00AcW`03001cWg>O0>UcW`00;G>O00<007>OLil017>O00<007>OLil0jG>O000]Lil00`00Limc W`04Lil00`00LimcW`3YLil002ecW`03001cWg>O00AcW`03001cWg>O0>UcW`00;G>O00<007>OLil0 17>O00<007>OLil0jG>O000]Lil00`00LimcW`03Lil00`00LimcW`3ZLil002ecW`03001cWg>O00=c W`03001cWg>O0>YcW`00;G>O00<007>OLil00g>O00<007>OLil0jW>O000]Lil20004Lil00`00Limc W`3ZLil002ecW`03001cWg>O00=cW`03001cWg>O0>YcW`00;G>O00<007>OLil00g>O00<007>OLil0 jW>O000]Lil00`00LimcW`03Lil00`00LimcW`3ZLil002ecW`03001cWg>O00=cW`03001cWg>O0>Yc W`00;G>O00<007>OLil00W>O00<007>OLil0jg>O000017>O0000000017>O0P0017>O0P0017>O0P00 17>O0P0017>O0P000g>O0`001G>O00<007>OLil00W>O00<007>OLil0jg>O0002Lil00`00LimcW`02 Lil01000LimcW`000W>O00@007>OLil0009cW`04001cWg>O0003Lil20003Lil01000LimcW`000g>O 00<007>OLil017>O00<007>OLil00W>O00<007>OLil0jg>O0002Lil00`00LimcW`02Lil01000Limc W`000W>O00@007>OLil0009cW`04001cWg>O0008Lil01000LimcW`000g>O00<007>OLil017>O0P00 0g>O00<007>OLil0jg>O0002Lil00`00LimcW`02Lil01000LimcW`000W>O00@007>OLil0009cW`04 001cWg>O0008Lil01000LimcW`000g>O00<007>OLil017>O00<007>OLil00W>O00<007>OLil0jg>O 0002Lil00`00LimcW`02Lil01000LimcW`000W>O00@007>OLil0009cW`04001cWg>O0008Lil01000 LimcW`000g>O00<007>OLil017>O00<007>OLil00W>O00<007>OLil0jg>O00000g>O00000005Lil2 0004Lil20004Lil2000:Lil20003Lil20006Lil00`00LimcW`02Lil00`00LimcW`3[Lil002ecW`05 001cWg>OLil0003^Lil002ecW`05001cWg>OLil0003^Lil002ecW`05001cWg>OLil0003^Lil002ec W`05001cWg>OLil0003^Lil002ecW`05001cWg>OLil0003^Lil002ecW`80009cW`03001cWg>O0>ac W`00;G>O00D007>OLimcW`000>icW`00;G>O00D007>OLimcW`000>icW`00;G>O00D007>OLimcW`00 0>icW`00;G>O00D007>OLimcW`000>icW`00;G>O00@007>OLil00>mcW`00;G>O00@007>OLil00>mc W`00;G>O00@007>OLil00>mcW`00;G>O0P0000=cW`00Lil0kW>O000]Lil01000LimcW`00kg>O000] Lil01000LimcW`00kg>O000]Lil01000LimcW`00kg>O000]Lil01000LimcW`00kg>O000]Lil01000 LimcW`00kg>O000]Lil01000LimcW`00kg>O000]Lil01000LimcW`00kg>O000]Lil01000LimcW`00 kg>O000]Lil200000g>O001cW`3^Lil002ecW`03001cW`000?1cW`00;G>O00<007>O0000l7>O000] Lil00`00Lil0003`Lil002ecW`03001cW`000?1cW`00;G>O00<007>O0000l7>O000017>O00000000 17>O0P0017>O0P0017>O0P0017>O0P0017>O0P000g>O0`001G>O00<007>O0000l7>O0002Lil00`00 LimcW`02Lil01000LimcW`000W>O00@007>OLil0009cW`04001cWg>O0003Lil20003Lil01000Limc W`000g>O00<007>OLil017>O00<007>O0000l7>O0002Lil00`00LimcW`02Lil01000LimcW`000W>O 00@007>OLil0009cW`04001cWg>O0008Lil01000LimcW`000g>O00<007>OLil017>O0`00l7>O0002 Lil00`00LimcW`02Lil01000LimcW`000W>O00@007>OLil0009cW`04001cWg>O0008Lil01000Limc W`000g>O00<007>OLil017>O00<007>O0000l7>O0002Lil00`00LimcW`02Lil01000LimcW`000W>O 00@007>OLil0009cW`04001cWg>O0008Lil01000LimcW`000g>O00<007>OLil017>O00<007>O0000 l7>O00000g>O00000005Lil20004Lil20004Lil2000:Lil20003Lil20006Lil00`00Lil0003`Lil0 02ecW`03001cW`000?1cW`00;G>O00<007>O0000l7>O000]Lil2003aLil002ecW`800?5cW`00;G>O 0P00lG>O000]Lil2003aLil002ecW`800?5cW`00;G>O0P00lG>O000]Lil2003aLil002ecW`800?5c W`00;G>O0P00lG>O000]Lil2003aLil002ecW`800?5cW`00;G>O0P00lG>O000]Lil2003aLil002ec W`03001cWg>O0?1cW`00;G>O00<007>OLil0l7>O000]Lil00`00LimcW`3`Lil002ecW`03001cWg>O 0?1cW`00;G>O00<007>OLil0l7>O000]Lil00`00LimcW`3`Lil002ecW`800?5cW`00;G>O00<007>O Lil0l7>O000]Lil00`00LimcW`3`Lil002ecW`03001cWg>O0?1cW`00;G>O00<007>OLil0l7>O0000 \ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {1.42671, 1000, 0.0127119, 5.91837*^-5}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[13]="] }, Open ]], Cell[BoxData[ \(\(zmax2 = {1, 1020, 1, 2, 1000, 1, 1, 1, 3}; \)\)], "Input", CellLabel->"In[14]:=", AspectRatioFixed->True], Cell[TextData["With two large elements the difference can be large"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[generalnorm[zmax2, j], {j, 2, 100}, PlotRange \[Rule] All]\)], "Input", CellLabel->"In[15]:=", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.00437314 0.00971817 -1.46191 0.00144581 [ [.19874 .11597 -6 -9 ] [.19874 .11597 6 0 ] [.3931 .11597 -6 -9 ] [.3931 .11597 6 0 ] [.58746 .11597 -6 -9 ] [.58746 .11597 6 0 ] [.78183 .11597 -6 -9 ] [.78183 .11597 6 0 ] [.97619 .11597 -9 -9 ] [.97619 .11597 9 0 ] [-0.00813 .27305 -24 -4.5 ] [-0.00813 .27305 0 4.5 ] [-0.00813 .41763 -24 -4.5 ] [-0.00813 .41763 0 4.5 ] [-0.00813 .56221 -24 -4.5 ] [-0.00813 .56221 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .19874 .12847 m .19874 .13472 L s [(20)] .19874 .11597 0 1 Mshowa .3931 .12847 m .3931 .13472 L s [(40)] .3931 .11597 0 1 Mshowa .58746 .12847 m .58746 .13472 L s [(60)] .58746 .11597 0 1 Mshowa .78183 .12847 m .78183 .13472 L s [(80)] .78183 .11597 0 1 Mshowa .97619 .12847 m .97619 .13472 L s [(100)] .97619 .11597 0 1 Mshowa .125 Mabswid .05296 .12847 m .05296 .13222 L s .10155 .12847 m .10155 .13222 L s .15015 .12847 m .15015 .13222 L s .24733 .12847 m .24733 .13222 L s .29592 .12847 m .29592 .13222 L s .34451 .12847 m .34451 .13222 L s .44169 .12847 m .44169 .13222 L s .49028 .12847 m .49028 .13222 L s .53887 .12847 m .53887 .13222 L s .63605 .12847 m .63605 .13222 L s .68465 .12847 m .68465 .13222 L s .73324 .12847 m .73324 .13222 L s .83042 .12847 m .83042 .13222 L s .87901 .12847 m .87901 .13222 L s .9276 .12847 m .9276 .13222 L s .25 Mabswid 0 .12847 m 1 .12847 L s .00437 .27305 m .01062 .27305 L s [(1200)] -0.00813 .27305 1 0 Mshowa .00437 .41763 m .01062 .41763 L s [(1300)] -0.00813 .41763 1 0 Mshowa .00437 .56221 m .01062 .56221 L s [(1400)] -0.00813 .56221 1 0 Mshowa .125 Mabswid .00437 .15739 m .00812 .15739 L s .00437 .1863 m .00812 .1863 L s .00437 .21522 m .00812 .21522 L s .00437 .24414 m .00812 .24414 L s .00437 .30197 m .00812 .30197 L s .00437 .33088 m .00812 .33088 L s .00437 .3598 m .00812 .3598 L s .00437 .38872 m .00812 .38872 L s .00437 .44655 m .00812 .44655 L s .00437 .47546 m .00812 .47546 L s .00437 .50438 m .00812 .50438 L s .00437 .5333 m .00812 .5333 L s .00437 .09956 m .00812 .09956 L s .00437 .07064 m .00812 .07064 L s .00437 .04172 m .00812 .04172 L s .00437 .01281 m .00812 .01281 L s .00437 .59113 m .00812 .59113 L s .25 Mabswid .00437 0 m .00437 .61803 L s .5 Mabswid .02381 .60332 m .02605 .53053 L .02846 .46977 L .03279 .38921 L .03543 .35222 L .03793 .32315 L .04262 .27984 L .04756 .24512 L .05212 .2199 L .05745 .1963 L .06244 .17835 L .07218 .15134 L .0829 .1297 L .09334 .11383 L .10458 .1006 L .11468 .09108 L .12571 .08257 L .1458 .0706 L .16548 .06186 L .18645 .05469 L .20642 .04931 L .22803 .04461 L .2681 .03802 L .3091 .03318 L .34859 .02967 L .38656 .02704 L .42699 .02481 L .46589 .02308 L .50725 .02157 L .5471 .02038 L .58543 .01941 L .62621 .01854 L .66548 .01783 L .70323 .01724 L .74343 .0167 L .78212 .01625 L .82326 .01583 L .86288 .01549 L .90099 .01519 L .94155 .01492 L .97619 .01472 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", CellLabel->"From In[15]:=", Evaluatable->False, ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgO00<007>OLil0 og>O0g>O000KLil00`00LimcW`3oLil3Lil001]cW`03001cWg>O0?mcW`=cW`006g>O0P00og>O17>O 000KLil00`00LimcW`2/Lim=0009Lil001]cW`03001cWg>O08AcWbP005IcW`006g>O00<007>OLil0 Kg>O5@00OW>O000KLil00`00LimcW`1KLilD002CLil001]cW`03001cWg>O055cW`X00:McW`006g>O 00<007>OLil0AW>O2`00/G>O000KLil00`00LimcW`0nLil8002lLil001]cW`03001cWg>O03UcW`D0 0O0P00=W>O1000bG>O000KLil00`00LimcW`0bLil3003=Lil001]cW`03001cWg>O02mc W`<00=1cW`006g>O00<007>OLil0:g>O1000dg>O000KLil00`00LimcW`0XLil3003GLil001]cW`03 001cWg>O02EcW`<00=YcW`006g>O00<007>OLil08g>O0P00gG>O000KLil00`00LimcW`0RLil00`00 LimcW`3MLil001]cW`80025cW`800>1cW`006g>O00<007>OLil07W>O0P00hW>O000KLil00`00Limc W`0MLil00`00LimcW`3RLil001]cW`03001cWg>O01acW`03001cWg>O0>=cW`006g>O00<007>OLil0 6g>O00<007>OLil0i7>O000KLil00`00LimcW`0JLil00`00LimcW`3ULil001]cW`03001cWg>O01Uc W`03001cWg>O00icW`@000=cW`8002YcW`<000=cW`8002YcW`8000AcW`8002YcW`8000AcW`8002Mc W`<000AcW`8000AcW`8000=cW`006g>O00<007>OLil067>O00<007>OLil047>O00<007>OLil00W>O 00@007>OLil002YcW`05001cWg>OLil00002Lil00`00LimcW`0VLil01000LimcW`000W>O00@007>O Lil002QcW`04001cWg>O0002Lil01000LimcW`009g>O00<007>OLil00W>O00@007>OLil0009cW`04 001cWg>O0002Lil001]cW`8001QcW`03001cWg>O019cW`05001cWg>OLil00002Lil00`00LimcW`0V Lil40002Lil01000LimcW`00:7>O00@007>OLil0009cW`04001cWg>O000XLil01000LimcW`000W>O 00@007>OLil002McW`03001cWg>O009cW`04001cWg>O0002Lil01000LimcW`000W>O000KLil00`00 LimcW`0GLil00`00LimcW`0CLil01000LimcW`000W>O00<007>OLil09W>O00<007>O00000g>O00@0 07>OLil002QcW`<000=cW`04001cWg>O000YLil20003Lil01000LimcW`009g>O00<007>OLil00W>O 00@007>OLil0009cW`04001cWg>O0002Lil001]cW`03001cWg>O01IcW`03001cWg>O015cW`04001c Wg>O0002Lil01000LimcW`00:7>O00<007>O00000g>O00@007>OLil002QcW`03001cWg>O00=cW`04 001cWg>O000XLil01000LimcW`000W>O00@007>OLil002McW`03001cWg>O009cW`04001cWg>O0002 Lil01000LimcW`000W>O000KLil00`00LimcW`0ELil00`00LimcW`0CLil20004Lil2000ZLil20004 Lil2000ZLil30003Lil2000ZLil20004Lil2000WLil20005Lil20004Lil20003Lil001]cW`03001c Wg>O01AcW`03001cWg>O0>]cW`006g>O00<007>OLil057>O00<007>OLil0jg>O000KLil00`00Limc W`0CLil00`00LimcW`3/Lil001]cW`03001cWg>O019cW`03001cWg>O0>ecW`006W>Oo`0010000g>O 000KLil00`00LimcW`09Lil00`00LimcW`05Lil00`00LimcW`02Lil00`00LimcW`09Lil00`00Limc W`0:Lil00`00LimcW`09Lil00`00LimcW`0:Lil00`00LimcW`09Lil00`00LimcW`0:Lil00`00Limc W`0:Lil00`00LimcW`09Lil00`00LimcW`0:Lil00`00LimcW`09Lil00`00LimcW`0:Lil00`00Limc W`09Lil00`00LimcW`0:Lil00`00LimcW`09Lil00`00LimcW`0:Lil00`00LimcW`09Lil00`00Limc W`0:Lil00`00LimcW`0:Lil00`00LimcW`07Lil001]cW`03001cWg>O015cW`03001cWg>O01]cW`03 001cWg>O02mcW`03001cWg>O02mcW`03001cWg>O02mcW`03001cWg>O031cW`03001cWg>O00McW`00 6g>O00<007>OLil047>O00<007>OLil0kg>O000KLil00`00LimcW`0@Lil00`00LimcW`3_Lil001]c W`03001cWg>O00mcW`03001cWg>O0?1cW`006g>O00<007>OLil03g>O00<007>OLil0l7>O000KLil0 0`00LimcW`0>Lil00`00LimcW`3aLil001]cW`03001cWg>O00icW`03001cWg>O0?5cW`006g>O0P00 3g>O00<007>OLil0lG>O000KLil00`00LimcW`0=Lil00`00LimcW`3bLil001]cW`03001cWg>O00ec W`03001cWg>O0?9cW`006g>O00<007>OLil03G>O00<007>OLil0lW>O000KLil00`00LimcW`0=Lil0 0`00LimcW`3bLil001]cW`03001cWg>O00acW`03001cWg>O0?=cW`006g>O00<007>OLil037>O00<0 07>OLil0lg>O000KLil00`00LimcW`0O0?Ac W`006g>O00<007>OLil02g>O00<007>OLil0m7>O000KLil00`00LimcW`0;Lil00`00LimcW`3dLil0 01]cW`03001cWg>O00YcW`03001cWg>O0?EcW`006g>O00<007>OLil02W>O00<007>OLil0mG>O000K Lil00`00LimcW`0:Lil00`00LimcW`3eLil001]cW`03001cWg>O00YcW`03001cWg>O0?EcW`006g>O 00<007>OLil02G>O00<007>OLil0mW>O000KLil2000:Lil00`00LimcW`3fLil001]cW`03001cWg>O 00UcW`03001cWg>O0?IcW`006g>O00<007>OLil02G>O00<007>OLil0mW>O000KLil00`00LimcW`09 Lil00`00LimcW`3fLil001]cW`03001cWg>O00UcW`03001cWg>O0?IcW`006g>O00<007>OLil02G>O 00<007>OLil0mW>O000KLil00`00LimcW`08Lil00`00LimcW`3gLil001]cW`03001cWg>O00QcW`03 001cWg>O0?McW`006g>O0P002G>O00<007>OLil0mg>O000KLil00`00LimcW`08Lil00`00LimcW`3g Lil001]cW`03001cWg>O00QcW`03001cWg>O0?McW`006g>O00<007>OLil027>O00<007>OLil0mg>O 000KLil00`00LimcW`08Lil00`00LimcW`3gLil001]cW`03001cWg>O00QcW`03001cWg>O0?McW`00 6g>O00<007>OLil027>O00<007>OLil0mg>O000017>O000000000g>O10000g>O0P0017>O0P001G>O 00<007>OLil01g>O00<007>OLil0n7>O0002Lil00`00LimcW`03Lil00`00LimcW`02Lil01000Limc W`000W>O00@007>OLil000AcW`03001cWg>O00McW`03001cWg>O0?QcW`000W>O00<007>OLil017>O 00D007>OLimcW`00009cW`04001cWg>O0002Lil00`00LimcW`02Lil20008Lil00`00LimcW`3hLil0 009cW`03001cWg>O00EcW`04001cWg>O0002Lil01000LimcW`000W>O00<007>OLil00W>O00<007>O Lil01g>O00<007>OLil0n7>O0002Lil00`00LimcW`02Lil01000LimcW`000W>O00@007>OLil0009c W`04001cWg>O0004Lil00`00LimcW`07Lil00`00LimcW`3hLil00003Lil0000000EcW`8000AcW`80 00AcW`8000EcW`03001cWg>O00McW`03001cWg>O0?QcW`006g>O00<007>OLil01g>O00<007>OLil0 n7>O000KLil00`00LimcW`07Lil00`00LimcW`3hLil001]cW`03001cWg>O00IcW`03001cWg>O0?Uc W`006g>O00<007>OLil01W>O00<007>OLil0nG>O000KLil20007Lil00`00LimcW`3iLil001]cW`03 001cWg>O00IcW`03001cWg>O0?UcW`006g>O00<007>OLil01W>O00<007>OLil0nG>O000KLil00`00 LimcW`06Lil00`00LimcW`3iLil001]cW`03001cWg>O00EcW`03001cWg>O0?YcW`006g>O00<007>O Lil01G>O00<007>OLil0nW>O000KLil00`00LimcW`05Lil00`00LimcW`3jLil001]cW`03001cWg>O 00EcW`03001cWg>O0?YcW`006g>O0P001W>O00<007>OLil0nW>O000KLil00`00LimcW`05Lil00`00 LimcW`3jLil001]cW`03001cWg>O00EcW`03001cWg>O0?YcW`006g>O00<007>OLil01G>O00<007>O Lil0nW>O000KLil00`00LimcW`05Lil00`00LimcW`3jLil001]cW`03001cWg>O00EcW`03001cWg>O 0?YcW`006g>O00<007>OLil01G>O00<007>OLil0nW>O000KLil00`00LimcW`05Lil00`00LimcW`3j Lil001]cW`03001cWg>O00EcW`03001cWg>O0?YcW`006g>O0P001W>O00<007>OLil0nW>O000KLil0 0`00LimcW`05Lil00`00LimcW`3jLil001]cW`03001cWg>O00EcW`03001cWg>O0?YcW`006g>O00<0 07>OLil017>O00<007>OLil0ng>O000KLil00`00LimcW`04Lil00`00LimcW`3kLil001]cW`03001c Wg>O00AcW`03001cWg>O0?]cW`006g>O00<007>OLil017>O00<007>OLil0ng>O000KLil00`00Limc W`04Lil00`00LimcW`3kLil001]cW`8000EcW`03001cWg>O0?]cW`006g>O00<007>OLil017>O00<0 07>OLil0ng>O000KLil00`00LimcW`04Lil00`00LimcW`3kLil001]cW`03001cWg>O00AcW`03001c Wg>O0?]cW`006g>O00<007>OLil017>O00<007>OLil0ng>O000KLil00`00LimcW`04Lil00`00Limc W`3kLil00004Lil000000003Lil30004Lil20004Lil20005Lil00`00LimcW`04Lil00`00LimcW`3k Lil0009cW`03001cWg>O00EcW`04001cWg>O0002Lil01000LimcW`000W>O00<007>OLil00W>O00<0 07>OLil017>O00<007>OLil0ng>O0002Lil00`00LimcW`05Lil01000LimcW`000W>O00@007>OLil0 009cW`03001cWg>O009cW`8000EcW`03001cWg>O0?]cW`000W>O00<007>OLil00g>O0P000g>O00@0 07>OLil0009cW`04001cWg>O0004Lil00`00LimcW`04Lil00`00LimcW`3kLil0009cW`03001cWg>O 00EcW`04001cWg>O0002Lil01000LimcW`000W>O00<007>OLil00W>O00<007>OLil017>O00<007>O Lil0ng>O00000g>O00000004Lil30004Lil20004Lil20005Lil00`00LimcW`04Lil00`00LimcW`3k Lil001]cW`03001cWg>O00=cW`03001cWg>O0?acW`006g>O00<007>OLil00g>O00<007>OLil0o7>O 000KLil00`00LimcW`03Lil00`00LimcW`3lLil001]cW`03001cWg>O00=cW`03001cWg>O0?acW`00 6g>O0P0017>O00<007>OLil0o7>O000KLil00`00LimcW`03Lil00`00LimcW`3lLil001]cW`03001c Wg>O00=cW`03001cWg>O0?acW`006g>O00<007>OLil00g>O00<007>OLil0o7>O000KLil00`00Limc W`03Lil00`00LimcW`3lLil001]cW`03001cWg>O00=cW`03001cWg>O0?acW`006g>O00<007>OLil0 0g>O00<007>OLil0o7>O000KLil00`00LimcW`03Lil00`00LimcW`3lLil001]cW`03001cWg>O00=c W`03001cWg>O0?acW`006g>O0P0017>O00<007>OLil0o7>O000KLil00`00LimcW`03Lil00`00Limc W`3lLil001]cW`03001cWg>O00=cW`03001cWg>O0?acW`006g>O00<007>OLil00g>O00<007>OLil0 o7>O000KLil00`00LimcW`03Lil00`00LimcW`3lLil001]cW`03001cWg>O00=cW`03001cWg>O0?ac W`006g>O00<007>OLil00g>O00<007>OLil0o7>O000KLil00`00LimcW`02Lil00`00LimcW`3mLil0 01]cW`8000=cW`03001cWg>O0?ecW`006g>O00<007>OLil00W>O00<007>OLil0oG>O000KLil00`00 LimcW`02Lil00`00LimcW`3mLil001]cW`03001cWg>O009cW`03001cWg>O0?ecW`006g>O00<007>O Lil00W>O00<007>OLil0oG>O000KLil00`00LimcW`02Lil00`00LimcW`3mLil001]cW`03001cWg>O 009cW`03001cWg>O0?ecW`006g>O00<007>OLil00W>O00<007>OLil0oG>O000KLil20003Lil00`00 LimcW`3mLil001]cW`03001cWg>O009cW`03001cWg>O0?ecW`006g>O00<007>OLil00W>O00<007>O Lil0oG>O000KLil00`00LimcW`02Lil00`00LimcW`3mLil001]cW`03001cWg>O009cW`03001cWg>O 0?ecW`006g>O00<007>OLil00W>O00<007>OLil0oG>O000KLil00`00LimcW`02Lil00`00LimcW`3m Lil00004Lil000000004Lil30003Lil20004Lil20005Lil00`00LimcW`02Lil00`00LimcW`3mLil0 009cW`03001cWg>O00AcW`05001cWg>OLil00002Lil01000LimcW`000W>O00<007>OLil00W>O00<0 07>OLil00W>O00<007>OLil0oG>O0002Lil00`00LimcW`02Lil40002Lil01000LimcW`000W>O00@0 07>OLil000AcW`8000=cW`03001cWg>O0?ecW`000W>O00<007>OLil00W>O00<007>O00000g>O00@0 07>OLil0009cW`04001cWg>O0004Lil00`00LimcW`02Lil00`00LimcW`3mLil0009cW`03001cWg>O 009cW`03001cW`0000=cW`04001cWg>O0002Lil01000LimcW`0017>O00<007>OLil00W>O00<007>O Lil0oG>O00000g>O00000005Lil20004Lil20004Lil20005Lil00`00LimcW`02Lil00`00LimcW`3m Lil001]cW`03001cWg>O009cW`03001cWg>O0?ecW`006g>O00<007>OLil00W>O00<007>OLil0oG>O 000KLil00`00LimcW`02Lil00`00LimcW`3mLil001]cW`03001cWg>O009cW`03001cWg>O0?ecW`00 6g>O0P000g>O00<007>OLil0oG>O000KLil00`00LimcW`02Lil00`00LimcW`3mLil001]cW`03001c Wg>O009cW`03001cWg>O0?ecW`006g>O00<007>OLil00W>O00<007>OLil0oG>O000KLil00`00Limc W`3oLil3Lil001]cW`03001cWg>O0?mcW`=cW`006g>O00<007>OLil0og>O0g>O000KLil00`00Limc W`3oLil3Lil00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-10.8576, 1011.13, 0.39877, 2.41592}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[15]="] }, Open ]], Cell["\<\ In any case, since the norm is roughly a sum, the largest elements \ always contribute the most. For computationally intense problems often times \ a \"max\" norm is used where the norm is the just the magnitude of the \ largest element. This corresponds to n--> infinity. Of course, if you are \ using a max norm, do not calculate the entire sum. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Preserving our notion of length. ", "Subsubsection"], Cell["\<\ If a norm is a length, then how do vectors behave relative to one \ another as we change the norm? \ \>", "Text"], Cell["\<\ In the first case, the relative difference does not change \ much.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(zz1 = {2, 5, 7, 9, 11, 0}\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(TraditionalForm\`{2, 5, 7, 9, 11, 0}\)], "Output", CellLabel->"Out[1]="], Cell[CellGroupData[{ Cell[BoxData[ \(zz2 = {1, 6, 0, 5, 3, 2}\)], "Input", CellLabel->"In[48]:="], Cell[BoxData[ \(TraditionalForm\`{1, 6, 0, 5, 3, 2}\)], "Output", CellLabel->"Out[48]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{generalnorm[zz1, j], generalnorm[zz2, j]}, {j, 1, 10}, PlotRange \[Rule] All, PlotStyle -> {Dashing[{1, 0}], Dashing[{ .1, .03}]}]\)], "Input", CellLabel->"In[50]:=", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0820106 0.10582 -0.113746 0.0210901 [ [.34127 .06357 -3 -9 ] [.34127 .06357 3 0 ] [.55291 .06357 -3 -9 ] [.55291 .06357 3 0 ] [.76455 .06357 -3 -9 ] [.76455 .06357 3 0 ] [.97619 .06357 -6 -9 ] [.97619 .06357 6 0 ] [.11713 .09716 -12 -4.5 ] [.11713 .09716 0 4.5 ] [.11713 .20261 -12 -4.5 ] [.11713 .20261 0 4.5 ] [.11713 .30806 -12 -4.5 ] [.11713 .30806 0 4.5 ] [.11713 .41351 -12 -4.5 ] [.11713 .41351 0 4.5 ] [.11713 .51896 -12 -4.5 ] [.11713 .51896 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .34127 .07607 m .34127 .08232 L s [(4)] .34127 .06357 0 1 Mshowa .55291 .07607 m .55291 .08232 L s [(6)] .55291 .06357 0 1 Mshowa .76455 .07607 m .76455 .08232 L s [(8)] .76455 .06357 0 1 Mshowa .97619 .07607 m .97619 .08232 L s [(10)] .97619 .06357 0 1 Mshowa .125 Mabswid .18254 .07607 m .18254 .07982 L s .23545 .07607 m .23545 .07982 L s .28836 .07607 m .28836 .07982 L s .39418 .07607 m .39418 .07982 L s .44709 .07607 m .44709 .07982 L s .5 .07607 m .5 .07982 L s .60582 .07607 m .60582 .07982 L s .65873 .07607 m .65873 .07982 L s .71164 .07607 m .71164 .07982 L s .81746 .07607 m .81746 .07982 L s .87037 .07607 m .87037 .07982 L s .92328 .07607 m .92328 .07982 L s .07672 .07607 m .07672 .07982 L s .02381 .07607 m .02381 .07982 L s .25 Mabswid 0 .07607 m 1 .07607 L s .12963 .09716 m .13588 .09716 L s [(10)] .11713 .09716 1 0 Mshowa .12963 .20261 m .13588 .20261 L s [(15)] .11713 .20261 1 0 Mshowa .12963 .30806 m .13588 .30806 L s [(20)] .11713 .30806 1 0 Mshowa .12963 .41351 m .13588 .41351 L s [(25)] .11713 .41351 1 0 Mshowa .12963 .51896 m .13588 .51896 L s [(30)] .11713 .51896 1 0 Mshowa .125 Mabswid .12963 .11825 m .13338 .11825 L s .12963 .13934 m .13338 .13934 L s .12963 .16043 m .13338 .16043 L s .12963 .18152 m .13338 .18152 L s .12963 .2237 m .13338 .2237 L s .12963 .24479 m .13338 .24479 L s .12963 .26588 m .13338 .26588 L s .12963 .28697 m .13338 .28697 L s .12963 .32915 m .13338 .32915 L s .12963 .35024 m .13338 .35024 L s .12963 .37133 m .13338 .37133 L s .12963 .39242 m .13338 .39242 L s .12963 .4346 m .13338 .4346 L s .12963 .45569 m .13338 .45569 L s .12963 .47678 m .13338 .47678 L s .12963 .49787 m .13338 .49787 L s .12963 .05498 m .13338 .05498 L s .12963 .03389 m .13338 .03389 L s .12963 .0128 m .13338 .0128 L s .12963 .54005 m .13338 .54005 L s .12963 .56114 m .13338 .56114 L s .12963 .58223 m .13338 .58223 L s .12963 .60332 m .13338 .60332 L s .25 Mabswid .12963 0 m .12963 .61803 L s .5 Mabswid [ 1 0 ] 0 setdash .02381 .60332 m .03279 .5248 L .04262 .4602 L .05288 .40892 L .06244 .37149 L .0828 .31359 L .09421 .28986 L .10458 .272 L .12373 .24584 L .14451 .22452 L .16435 .20884 L .18538 .19573 L .22321 .17829 L .24262 .17148 L .2635 .16532 L .30472 .15578 L .34443 .14894 L .38262 .14386 L .42327 .1396 L .4624 .13633 L .50398 .13353 L .54405 .13133 L .5826 .12958 L .6236 .12802 L .66309 .12676 L .70106 .12573 L .74149 .12479 L .78039 .12402 L .82176 .12331 L .8616 .12272 L .89993 .12223 L .94071 .12177 L .97619 .12142 L s [ .1 .03 ] 0 setdash .02381 .24479 m .03279 .20657 L .04262 .17519 L .05288 .15033 L .06244 .13223 L .0828 .10435 L .09421 .09298 L .10458 .08446 L .12373 .07206 L .14451 .06202 L .16435 .0547 L .18538 .04862 L .22321 .04062 L .24262 .03753 L .2635 .03475 L .30472 .03047 L .34443 .02741 L .38262 .02515 L .42327 .02325 L .4624 .02178 L .50398 .02052 L .54405 .01952 L .5826 .01871 L .6236 .01798 L .66309 .01738 L .70106 .01689 L .74149 .01643 L .78039 .01605 L .82176 .0157 L .8616 .0154 L .89993 .01514 L .94071 .0149 L .97619 .01472 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", CellLabel->"From In[50]:=", ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgO00<007>OLil0 n7>O000ULil00`00LimcW`3hLil002EcW`03001cWg>O0?QcW`009G>O0P00nG>O000ULil00`00Limc W`2NLilM0008LilN0007Lil90007Lil002EcW`03001cWg>O069cW`l000McWah0069cW`009G>O00<0 07>OLil0Dg>O3`00UW>O000ULil00`00LimcW`0eLilG002/Lil002EcW`03001cWg>O02icW`L00<=c W`009G>O0P009G>O0P00dW>O000ULil00`00LimcW`0KLil9003DLil002EcW`03001cWg>O01EcW`H0 01ecW`<003YcW`8003]cW`8003McW`<000AcW`8000AcW`009G>O00<007>OLil047>O1@0097>O00<0 07>OLil0>7>O00@007>OLil003UcW`04001cWg>O000gLil00`00LimcW`02Lil01000LimcW`000g>O 000ULil00`00LimcW`0G>O00@007>OLil003McW`03001c Wg>O009cW`04001cWg>O0003Lil002EcW`03001cWg>O00YcW`8002]cW`03001cW`0003YcW`<003]c W`8003QcW`03001cWg>O009cW`04001cWg>O0003Lil002EcW`8000YcW`03001cWg>O02]cW`03001c W`0003YcW`03001cWg>O03YcW`04001cWg>O000gLil00`00LimcW`02Lil01000LimcW`000g>O000U Lil00`00LimcW`0hLil2000kLil3000jLil2000gLil20005Lil20004Lil002EcW`03001cWg>O0080 0?IcW`009G>O0`00n7>O000TLil2003jLil0029cW`800003Lil007>O0?UcW`00o`008@000006Lil0 0`00LimcW`0=Lil00`00LimcW`07Lil00`00LimcW`02Lil00`00LimcW`0O00=cW`03001cWg>O03UcW`03001cWg>O03YcW`03001cWg>O03YcW`03001cWg>O 03YcW`03001cWg>O00EcW`007G>O0P001W>O00<007>OLil0n7>O000FLil30003Lil30006Lil00`00 LimcW`3hLil001McW`03001cWg>O00<0009cW`03001cWg>O00=cW`03001cWg>O0?QcW`005g>O00<0 07>O00000W>O00@007>OLil000EcW`<00?QcW`005g>O0P000g>O00@007>OLil000EcW`03001cWg>O 0?QcW`005g>O00<007>OLil00W>O00@007>OLil000EcW`03001cWg>O0?QcW`005W>O0P001G>O0P00 1W>O00<007>OLil0n7>O000FLil00`00LimcW`0O00ec W`03001cWg>O0?QcW`005G>O00<007>OLil03G>O0P00nG>O000DLil00`00LimcW`0>Lil00`00Limc W`3PLilA0007Lil001=cW`03001cWg>O00mcW`03001cWg>O0:AcWc`001QcW`009G>O00<007>OLil0 QG>O7`00E7>O000ULil00`00LimcW`1^LilG001cLil002EcW`03001cWg>O069cW``008YcW`009G>O 0P00F7>O2`00UW>O000ULil00`00LimcW`1;Lil<002QLil002EcW`03001cWg>O041cW`/00:ecW`00 3g>O00<007>OLil04g>O00<007>OLil0=g>O2@00^7>O000?Lil00`00LimcW`0CLil00`00LimcW`0b Lil50031Lil000mcW`03001cWg>O01=cW`03001cWg>O02ecW`D00O00<007>OLil057>O 0P00:W>O1000bg>O000>Lil00`00LimcW`0DLil00`00LimcW`0ULil4003?Lil000ecW`03001cWg>O 01EcW`03001cWg>O029cW`<00==cW`003G>O00<007>OLil05G>O00<007>OLil07g>O0`00eW>O000= Lil00`00LimcW`0ELil00`00LimcW`0LLil3003ILil000acW`03001cWg>O01IcW`03001cWg>O01Uc W`<00=acW`0037>O00<007>OLil05W>O00<007>OLil05g>O0P00gg>O000;Lil00`00LimcW`0GLil2 000FLil2003QLil000]cW`03001cWg>O01McW`03001cWg>O019cW`<00>=cW`002g>O00<007>OLil0 5g>O00<007>OLil047>O0P00iW>O000:Lil00`00LimcW`0HLil00`00LimcW`0>Lil2003XLil000Yc W`03001cWg>O00UcW`<000=cW`<000IcW`03001cWg>O00acW`800>YcW`002W>O00<007>OLil02W>O 00<007>OLil01G>O00<007>OLil00g>O00<007>OLil02W>O0P00k7>O0009Lil00`00LimcW`0;Lil0 0`00LimcW`05Lil00`00LimcW`03Lil30008Lil2003^Lil000UcW`03001cWg>O00]cW`03001cWg>O 009cW`<000IcW`03001cWg>O00IcW`800?1cW`002G>O00<007>OLil02g>O00<007>OLil00W>O00<0 07>OLil01W>O00<007>OLil01G>O00<007>OLil0l7>O0009Lil00`00LimcW`0:Lil20004Lil40005 Lil00`00LimcW`04Lil00`00LimcW`3aLil000QcW`03001cWg>O01YcW`03001cWg>O00=cW`03001c Wg>O0?9cW`0027>O00<007>OLil06W>O00<007>OLil00W>O00<007>OLil0lg>O0008Lil00`00Limc W`0JLil20002Lil00`00LimcW`3dLil000McW`03001cWg>O01]cW`04001cWg>O003gLil000McW`03 001cWg>O01]cW`03001cW`000?QcW`001g>O00<007>OLil06g>O0P00nG>O0007Lil00`00LimcW`0K Lil00`00LimcW`3hLil000IcW`03001cWg>O01]cW`800?YcW`001W>O00<007>OLil06W>O00@007>O 00000?UcW`008g>O00<007>O0000nW>O000RLil01000LimcW`00nW>O000QLil01@00LimcWg>O0000 nW>O000PLil00`00LimcW`02Lil00`00LimcW`3hLil0021cW`03001cWg>O009cW`03001cWg>O0?Qc W`007g>O00<007>OLil00g>O0P00nG>O000NLil00`00LimcW`04Lil00`00LimcW`3hLil001icW`03 001cWg>O00AcW`03001cWg>O0?QcW`007G>O00<007>OLil01G>O00<007>OLil0n7>O000MLil00`00 LimcW`05Lil00`00LimcW`3hLil001acW`03001cWg>O00IcW`03001cWg>O0?QcW`0077>O00<007>O Lil01W>O0P00nG>O000KLil00`00LimcW`07Lil00`00LimcW`3hLil001]cW`03001cWg>O00McW`03 001cWg>O0?QcW`006W>O00<007>OLil027>O00<007>OLil0n7>O000FLil40003Lil20006Lil00`00 LimcW`3hLil001McW`03001cW`00009cW`04001cWg>O0005Lil00`00LimcW`3hLil001QcW`05001c Wg>OLil00002Lil00`00LimcW`03Lil3003hLil001McW`03001cW`00009cW`04001cWg>O0005Lil0 0`00LimcW`3hLil001IcW`800005Lil007>OLil00002Lil00`00LimcW`03Lil00`00LimcW`3hLil0 01IcW`<000AcW`8000IcW`03001cWg>O0?QcW`005W>O00<007>OLil037>O00<007>OLil0n7>O000F Lil00`00LimcW`0O00ecW`800?UcW`005G>O00<007>O Lil03G>O00<007>OLil0n7>O000ELil00`00LimcW`0=Lil00`00LimcW`3hLil001AcW`03001cWg>O 00icW`03001cWg>O0?QcW`0057>O00<007>OLil03W>O00<007>OLil0n7>O000DLil00`00LimcW`0> Lil00`00LimcW`3hLil001=cW`03001cWg>O00mcW`800?UcW`004g>O00<007>OLil03g>O00<007>O Lil0n7>O000CLil00`00LimcW`0?Lil00`00LimcW`3hLil0019cW`03001cWg>O011cW`03001cWg>O 0?QcW`004W>O00<007>OLil047>O00<007>OLil0n7>O000BLil00`00LimcW`0@Lil00`00LimcW`3h Lil0015cW`03001cWg>O015cW`800?UcW`004G>O00<007>OLil04G>O00<007>OLil0n7>O000ALil0 0`00LimcW`0ALil00`00LimcW`3hLil0015cW`03001cWg>O015cW`03001cWg>O0?QcW`0047>O00<0 07>OLil04W>O00<007>OLil0n7>O000@Lil00`00LimcW`0BLil00`00LimcW`3hLil0011cW`03001c Wg>O019cW`800?UcW`0047>O00<007>OLil04W>O00<007>OLil0n7>O000@Lil00`00LimcW`0BLil0 0`00LimcW`3hLil000mcW`03001cWg>O01=cW`03001cWg>O0?QcW`003g>O00<007>OLil017>O1000 0W>O0`001W>O00<007>OLil0n7>O000?Lil00`00LimcW`05Lil00`00LimcW`05Lil00`00LimcW`03 Lil00`00LimcW`3hLil000mcW`03001cWg>O00IcW`03001cWg>O00AcW`03001cWg>O00=cW`<00?Qc W`003g>O00<007>OLil01g>O00<007>OLil00`001W>O00<007>OLil0n7>O000>Lil00`00LimcW`05 Lil01000LimcW`000W>O00<007>OLil01W>O00<007>OLil0n7>O000>Lil00`00LimcW`06Lil20003 Lil40005Lil00`00LimcW`3hLil000icW`03001cWg>O01AcW`03001cWg>O0?QcW`003W>O00<007>O Lil057>O00<007>OLil0n7>O000>Lil00`00LimcW`0DLil2003iLil000ecW`03001cWg>O01EcW`03 001cWg>O0?QcW`003G>O00<007>OLil05G>O00<007>OLil0n7>O000=Lil00`00LimcW`0ELil00`00 LimcW`3hLil000ecW`03001cWg>O01EcW`03001cWg>O0?QcW`003G>O00<007>OLil05G>O00<007>O Lil0n7>O000O01IcW`03001cWg>O0?QcW`00 37>O00<007>OLil05W>O00<007>OLil0n7>O000O01IcW`03001cWg>O0?QcW`002g>O00<007>OLil05g>O00<007>OLil0n7>O000;Lil0 0`00LimcW`0GLil2003iLil000]cW`03001cWg>O01McW`03001cWg>O0?QcW`002g>O00<007>OLil0 5g>O00<007>OLil0n7>O000;Lil00`00LimcW`0GLil00`00LimcW`3hLil000]cW`03001cWg>O01Mc W`03001cWg>O0?QcW`002W>O00<007>OLil067>O00<007>OLil0n7>O000:Lil00`00LimcW`0HLil2 003iLil000YcW`03001cWg>O01QcW`03001cWg>O0?QcW`002W>O00<007>OLil067>O00<007>OLil0 n7>O000:Lil00`00LimcW`0HLil00`00LimcW`3hLil000YcW`03001cWg>O00UcW`<000AcW`8000Ic W`03001cWg>O0?QcW`002W>O00<007>OLil037>O00@007>OLil0009cW`03001cWg>O00=cW`03001c Wg>O0?QcW`002G>O00<007>OLil03G>O00@007>OLil0009cW`03001cWg>O00=cW`<00?QcW`002G>O 00<007>OLil02g>O0P000g>O00@007>OLil000EcW`03001cWg>O0?QcW`002G>O00<007>OLil03G>O 00@007>OLil0009cW`03001cWg>O00=cW`03001cWg>O0?QcW`002G>O00<007>OLil02W>O0`0017>O 0P001W>O00<007>OLil0n7>O0009Lil00`00LimcW`0ILil00`00LimcW`3hLil000UcW`03001cWg>O 01UcW`03001cWg>O0?QcW`002G>O00<007>OLil06G>O0P00nG>O0008Lil00`00LimcW`0JLil00`00 LimcW`3hLil000QcW`03001cWg>O01YcW`03001cWg>O0?QcW`0027>O00<007>OLil06W>O00<007>O Lil0n7>O0008Lil00`00LimcW`0JLil00`00LimcW`3hLil000QcW`03001cWg>O01YcW`03001cWg>O 0?QcW`0027>O00<007>OLil06W>O0P00nG>O0008Lil00`00LimcW`0JLil00`00LimcW`3hLil000Mc W`03001cWg>O01]cW`03001cWg>O0?QcW`001g>O00<007>OLil06g>O00<007>OLil0n7>O0007Lil0 0`00LimcW`0KLil00`00LimcW`3hLil000McW`03001cWg>O01]cW`03001cWg>O0?QcW`001g>O00<0 07>OLil06g>O0P00nG>O0007Lil00`00LimcW`0KLil00`00LimcW`3hLil000McW`03001cWg>O01]c W`03001cWg>O0?QcW`001W>O00<007>OLil077>O00<007>OLil0n7>O0006Lil00`00LimcW`0LLil0 0`00LimcW`3hLil000IcW`03001cWg>O01acW`03001cWg>O0?QcW`001W>O00<007>OLil077>O0P00 nG>O000ULil00`00LimcW`3hLil002EcW`03001cWg>O0?QcW`009G>O00<007>OLil0n7>O000ULil0 0`00LimcW`3hLil00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {0.774954, 5.39319, 0.0329272, 0.165621}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[50]="] }, Open ]] }, Open ]], Cell["\<\ In the second case, the norms become identical at large values of \ j.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(zz1 = {2, 5, 7, 9, 11, 0}\)], "Input", CellLabel->"In[51]:="], Cell[BoxData[ \(TraditionalForm\`{2, 5, 7, 9, 11, 0}\)], "Output", CellLabel->"Out[51]="], Cell[CellGroupData[{ Cell[BoxData[ \(zz2 = {1, 11, 0, 5, 3, 2}\)], "Input", CellLabel->"In[52]:="], Cell[BoxData[ \(TraditionalForm\`{1, 11, 0, 5, 3, 2}\)], "Output", CellLabel->"Out[52]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{generalnorm[zz1, j], generalnorm[zz2, j]}, {j, 1, 10}, PlotRange \[Rule] All, PlotStyle -> {Dashing[{1, 0}], Dashing[{ .1, .03}]}]\)], "Input", CellLabel->"In[53]:=", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0820106 0.10582 -0.266807 0.025592 [ [.34127 .10457 -3 -9 ] [.34127 .10457 3 0 ] [.55291 .10457 -3 -9 ] [.55291 .10457 3 0 ] [.76455 .10457 -3 -9 ] [.76455 .10457 3 0 ] [.97619 .10457 -6 -9 ] [.97619 .10457 6 0 ] [.11713 .24503 -12 -4.5 ] [.11713 .24503 0 4.5 ] [.11713 .37299 -12 -4.5 ] [.11713 .37299 0 4.5 ] [.11713 .50095 -12 -4.5 ] [.11713 .50095 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .34127 .11707 m .34127 .12332 L s [(4)] .34127 .10457 0 1 Mshowa .55291 .11707 m .55291 .12332 L s [(6)] .55291 .10457 0 1 Mshowa .76455 .11707 m .76455 .12332 L s [(8)] .76455 .10457 0 1 Mshowa .97619 .11707 m .97619 .12332 L s [(10)] .97619 .10457 0 1 Mshowa .125 Mabswid .18254 .11707 m .18254 .12082 L s .23545 .11707 m .23545 .12082 L s .28836 .11707 m .28836 .12082 L s .39418 .11707 m .39418 .12082 L s .44709 .11707 m .44709 .12082 L s .5 .11707 m .5 .12082 L s .60582 .11707 m .60582 .12082 L s .65873 .11707 m .65873 .12082 L s .71164 .11707 m .71164 .12082 L s .81746 .11707 m .81746 .12082 L s .87037 .11707 m .87037 .12082 L s .92328 .11707 m .92328 .12082 L s .07672 .11707 m .07672 .12082 L s .02381 .11707 m .02381 .12082 L s .25 Mabswid 0 .11707 m 1 .11707 L s .12963 .24503 m .13588 .24503 L s [(20)] .11713 .24503 1 0 Mshowa .12963 .37299 m .13588 .37299 L s [(25)] .11713 .37299 1 0 Mshowa .12963 .50095 m .13588 .50095 L s [(30)] .11713 .50095 1 0 Mshowa .125 Mabswid .12963 .14266 m .13338 .14266 L s .12963 .16826 m .13338 .16826 L s .12963 .19385 m .13338 .19385 L s .12963 .21944 m .13338 .21944 L s .12963 .27062 m .13338 .27062 L s .12963 .29622 m .13338 .29622 L s .12963 .32181 m .13338 .32181 L s .12963 .3474 m .13338 .3474 L s .12963 .39858 m .13338 .39858 L s .12963 .42418 m .13338 .42418 L s .12963 .44977 m .13338 .44977 L s .12963 .47536 m .13338 .47536 L s .12963 .09148 m .13338 .09148 L s .12963 .06589 m .13338 .06589 L s .12963 .0403 m .13338 .0403 L s .12963 .0147 m .13338 .0147 L s .12963 .52654 m .13338 .52654 L s .12963 .55214 m .13338 .55214 L s .12963 .57773 m .13338 .57773 L s .12963 .60332 m .13338 .60332 L s .25 Mabswid .12963 0 m .12963 .61803 L s .5 Mabswid [ 1 0 ] 0 setdash .02381 .60332 m .03279 .50804 L .04262 .42965 L .05288 .36742 L .06244 .32201 L .0828 .25175 L .09421 .22295 L .10458 .20127 L .12373 .16954 L .14451 .14366 L .16435 .12463 L .18538 .10873 L .22321 .08756 L .24262 .0793 L .2635 .07183 L .30472 .06026 L .34443 .05195 L .38262 .04578 L .42327 .04061 L .4624 .03665 L .50398 .03325 L .54405 .03058 L .5826 .02845 L .6236 .02656 L .66309 .02503 L .70106 .02378 L .74149 .02264 L .78039 .0217 L .82176 .02085 L .8616 .02013 L .89993 .01954 L .94071 .01898 L .97619 .01856 L s [ .1 .03 ] 0 setdash .02381 .29622 m .03279 .24211 L .04262 .19805 L .05288 .16354 L .06244 .13874 L .0828 .10141 L .09421 .08666 L .10458 .07585 L .12373 .06065 L .14451 .04898 L .16435 .04096 L .18538 .03474 L .20491 .03043 L .22321 .02732 L .24346 .02466 L .26198 .02276 L .28097 .02122 L .30169 .0199 L .32136 .01891 L .34234 .01808 L .36231 .01745 L .38392 .01691 L .42399 .01618 L .44392 .01592 L .46499 .0157 L .50296 .0154 L .52247 .01528 L .54339 .01518 L .5632 .0151 L .58475 .01503 L .62459 .01493 L .64437 .01489 L .66537 .01486 L .70312 .01482 L .72247 .0148 L .74333 .01478 L .76298 .01477 L .78447 .01476 L .82409 .01474 L .8437 .01474 L .86465 .01473 L .88456 .01473 L .90615 .01472 L .94613 .01472 L .97619 .01472 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", CellLabel->"From In[53]:=", ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgO00<007>OLil0 n7>O000ULil00`00LimcW`3hLil002EcW`03001cWg>O0?QcW`009G>O00<007>OLil0n7>O000ULil2 0014LilM0008LilN0007LilN0008LilM0008LilI0007Lil002EcW`03001cWg>O02]cWa0008icWbP0 00McW`009G>O00<007>OLil08G>O2P00LG>O;@00;g>O000ULil00`00LimcW`0NLil3001TLilG001L Lil002EcW`03001cWg>O015cW`D005QcWaL007=cW`009G>O00<007>OLil03W>O0`00DG>O3000RW>O 000ULil00`00LimcW`0:Lil40019Lil;002FLil002EcW`8000McW`@004AcW`T00:5cW`009G>O00<0 07>OLil017>O0P00@W>O1P00ZW>O000ULil00`00LimcW`02Lil2000lLil8002`Lil002EcW`03001c Wg>O008003IcW`P00;QcW`009G>O0`00=7>O1000`7>O000TLil2000cLil30034Lil002=cW`03001c W`0002mcW`@00O00<007>OLil00P00:W>O1000bg>O000ULil00`00LimcW`0ULil4003? Lil002EcW`03001cWg>O029cW`<00==cW`009G>O00<007>OLil07g>O0`00eW>O000ULil00`00Limc W`0LLil3000ILil3000jLil2000kLil2000gLil30004Lil20004Lil001acW`03001cWg>O00IcW`03 001cWg>O01YcW`8001ecW`03001cWg>O03QcW`04001cWg>O000iLil01000LimcW`00=g>O00<007>O Lil00W>O00@007>OLil000=cW`006g>O00<007>OLil01g>O00<007>OLil06G>O00<007>OLil06g>O 1000>G>O00@007>OLil003UcW`04001cWg>O000gLil00`00LimcW`02Lil01000LimcW`000g>O000K Lil00`00LimcW`07Lil00`00LimcW`0GLil2000NLil00`00Lil0000jLil3000kLil2000hLil00`00 LimcW`02Lil01000LimcW`000g>O000JLil00`00LimcW`08Lil2000FLil2000PLil00`00Lil0000j Lil00`00LimcW`0jLil01000LimcW`00=g>O00<007>OLil00W>O00@007>OLil000=cW`006G>O00<0 07>OLil02G>O00<007>OLil04g>O0P008g>O0P00>g>O0`00>W>O0P00=g>O0P001G>O0P0017>O000H Lil00`00LimcW`0:Lil00`00LimcW`0BLil00`00LimcW`3SLil001McW`03001cWg>O00]cW`03001c Wg>O011cW`800>IcW`005g>O00<007>OLil02g>O00<007>OLil03W>O0P00j7>O000FLil00`00Limc W`0O00ecW`03001cWg>O00]cW`03001cWg>O 0>YcW`00o`008@000006Lil00`00LimcW`0;Lil00`00Lil0000>Lil00`00LimcW`08Lil01@00Limc Wg>O00003W>O00<007>OLil037>O00<007>OLil037>O00<007>OLil03G>O00<007>OLil037>O00<0 07>OLil037>O00<007>OLil037>O00<007>OLil037>O00<007>OLil03G>O00<007>OLil037>O00<0 07>OLil037>O00<007>OLil037>O00<007>OLil037>O00<007>OLil037>O00<007>OLil03G>O00<0 07>OLil01G>O000DLil00`00LimcW`0>Lil00`00LimcW`07Lil00`00LimcW`0_Lil00`00LimcW`0j Lil00`00LimcW`0jLil00`00LimcW`0jLil00`00LimcW`05Lil001=cW`03001cWg>O00mcW`03001c Wg>O00IcW`03001cWg>O0>mcW`004W>O00<007>OLil047>O00<007>OLil01G>O00<007>OLil0l7>O 000BLil00`00LimcW`0@Lil00`00LimcW`04Lil00`00LimcW`3aLil0015cW`03001cWg>O015cW`03 001cWg>O00=cW`03001cWg>O0?9cW`004G>O00<007>OLil04G>O0P000g>O00<007>OLil0lg>O000A Lil00`00LimcW`0ALil01@00LimcWg>O0000mW>O000@Lil00`00LimcW`0BLil01@00LimcWg>O0000 mW>O000@Lil00`00LimcW`0BLil01000LimcW`00mg>O000@Lil00`00LimcW`0BLil00`00Lil0003h Lil000mcW`03001cWg>O01=cW`800?UcW`003g>O00<007>OLil04g>O00<007>OLil0n7>O000?Lil0 0`00LimcW`0BLil2003jLil000icW`03001cWg>O019cW`04001cW`00003iLil002=cW`03001cW`00 0?YcW`008W>O00@007>OLil00?YcW`008W>O00@007>OLil00?YcW`008G>O00D007>OLimcW`000?Yc W`0087>O00<007>OLil00W>O00<007>OLil0n7>O000PLil00`00LimcW`02Lil00`00LimcW`3hLil0 01mcW`03001cWg>O00=cW`800?UcW`0037>O00<007>OLil047>O00<007>OLil00g>O00<007>OLil0 n7>O000O00mc W`03001cWg>O00AcW`03001cWg>O0?QcW`002g>O00<007>OLil03g>O00<007>OLil01G>O00<007>O Lil0n7>O000;Lil00`00LimcW`0?Lil00`00LimcW`05Lil00`00LimcW`3hLil000]cW`03001cWg>O 00icW`03001cWg>O00IcW`03001cWg>O0?QcW`002g>O00<007>OLil03W>O00<007>OLil01W>O0P00 nG>O000:Lil00`00LimcW`0>Lil00`00LimcW`07Lil00`00LimcW`3hLil000YcW`03001cWg>O00ic W`03001cWg>O00McW`03001cWg>O0?QcW`002W>O00<007>OLil03G>O00<007>OLil027>O00<007>O Lil0n7>O000:Lil00`00LimcW`0=Lil00`00LimcW`08Lil00`00LimcW`3hLil000YcW`03001cWg>O 00acW`03001cWg>O00UcW`03001cWg>O0?QcW`002G>O00<007>OLil02W>O10000g>O0P001W>O00<0 07>OLil0n7>O0009Lil00`00LimcW`0;Lil00`00Lil00002Lil01000LimcW`001G>O00<007>OLil0 n7>O0009Lil00`00LimcW`0O00000W>O00<007>OLil00g>O0`00n7>O0009Lil0 0`00LimcW`0O00<007>OLil0n7>O0008Lil00`00LimcW`0;Lil2 00001G>O001cWg>O00000W>O00<007>OLil00g>O00<007>OLil0n7>O0008Lil00`00LimcW`0O00acW`03001cWg>O00]cW`03001cWg>O 0?QcW`0027>O00<007>OLil02g>O00<007>OLil037>O00<007>OLil0n7>O0008Lil00`00LimcW`0; Lil00`00LimcW`0O00acW`03001cWg>O00acW`800?Uc W`001g>O00<007>OLil02g>O00<007>OLil03G>O00<007>OLil0n7>O0007Lil00`00LimcW`0;Lil0 0`00LimcW`0=Lil00`00LimcW`3hLil000McW`03001cWg>O00]cW`03001cWg>O00ecW`03001cWg>O 0?QcW`001g>O00<007>OLil02W>O00<007>OLil03W>O00<007>OLil0n7>O0006Lil00`00LimcW`0; Lil00`00LimcW`0>Lil00`00LimcW`3hLil000IcW`03001cWg>O00]cW`03001cWg>O00icW`03001c Wg>O0?QcW`001W>O00<007>OLil02g>O00<007>OLil03W>O0P00nG>O000CLil00`00LimcW`0?Lil0 0`00LimcW`3hLil001=cW`03001cWg>O00mcW`03001cWg>O0?QcW`004g>O00<007>OLil03g>O00<0 07>OLil0n7>O000BLil00`00LimcW`0@Lil00`00LimcW`3hLil0019cW`03001cWg>O011cW`03001c Wg>O0?QcW`004W>O00<007>OLil047>O00<007>OLil0n7>O000ALil00`00LimcW`0ALil00`00Limc W`3hLil0015cW`03001cWg>O015cW`800?UcW`004G>O00<007>OLil04G>O00<007>OLil0n7>O000A Lil00`00LimcW`0ALil00`00LimcW`3hLil0015cW`03001cWg>O015cW`03001cWg>O0?QcW`0047>O 00<007>OLil04W>O00<007>OLil0n7>O000@Lil00`00LimcW`0BLil00`00LimcW`3hLil0011cW`03 001cWg>O019cW`03001cWg>O0?QcW`0047>O00<007>OLil04W>O0P00nG>O000@Lil00`00LimcW`0B Lil00`00LimcW`3hLil0011cW`03001cWg>O019cW`03001cWg>O0?QcW`0047>O00<007>OLil04W>O 00<007>OLil0n7>O000?Lil00`00LimcW`0CLil00`00LimcW`3hLil000mcW`03001cWg>O00AcW`@0 009cW`<000IcW`03001cWg>O0?QcW`003g>O00<007>OLil01G>O00<007>OLil01G>O00<007>OLil0 0g>O00<007>OLil0n7>O000?Lil00`00LimcW`06Lil00`00LimcW`04Lil00`00LimcW`03Lil3003h Lil000mcW`03001cWg>O00McW`03001cWg>O00<000IcW`03001cWg>O0?QcW`003g>O00<007>OLil0 17>O00@007>OLil0009cW`03001cWg>O00IcW`03001cWg>O0?QcW`003W>O00<007>OLil01W>O0P00 0g>O10001G>O00<007>OLil0n7>O000>Lil00`00LimcW`0DLil00`00LimcW`3hLil000icW`03001c Wg>O01AcW`03001cWg>O0?QcW`003W>O00<007>OLil057>O00<007>OLil0n7>O000>Lil00`00Limc W`0DLil00`00LimcW`3hLil000icW`03001cWg>O01AcW`800?UcW`003G>O00<007>OLil05G>O00<0 07>OLil0n7>O000=Lil00`00LimcW`0ELil00`00LimcW`3hLil000ecW`03001cWg>O01EcW`03001c Wg>O0?QcW`003G>O00<007>OLil05G>O00<007>OLil0n7>O000=Lil00`00LimcW`0ELil00`00Limc W`3hLil000ecW`03001cWg>O01EcW`03001cWg>O0?QcW`0037>O00<007>OLil05W>O0P00nG>O000< Lil00`00LimcW`0FLil00`00LimcW`3hLil000acW`03001cWg>O01IcW`03001cWg>O0?QcW`0037>O 00<007>OLil05W>O00<007>OLil0n7>O000O01IcW`03001cWg>O0?QcW`0037>O00<007>OLil05W>O00<007>OLil0n7>O000;Lil00`00 LimcW`0GLil2003iLil000]cW`03001cWg>O01McW`03001cWg>O0?QcW`002g>O00<007>OLil05g>O 00<007>OLil0n7>O000;Lil00`00LimcW`0GLil00`00LimcW`3hLil000]cW`03001cWg>O01McW`03 001cWg>O0?QcW`002g>O00<007>OLil05g>O00<007>OLil0n7>O000;Lil00`00LimcW`0GLil00`00 LimcW`3hLil000YcW`03001cWg>O01QcW`03001cWg>O0?QcW`002W>O00<007>OLil067>O0P00nG>O 000:Lil00`00LimcW`0HLil00`00LimcW`3hLil000YcW`03001cWg>O01QcW`03001cWg>O0?QcW`00 2W>O00<007>OLil067>O00<007>OLil0n7>O000:Lil00`00LimcW`0HLil00`00LimcW`3hLil000Yc W`03001cWg>O00UcW`<000AcW`8000IcW`03001cWg>O0?QcW`002G>O00<007>OLil03G>O00@007>O Lil0009cW`03001cWg>O00=cW`03001cWg>O0?QcW`002G>O00<007>OLil03G>O00@007>OLil0009c W`03001cWg>O00=cW`<00?QcW`002G>O00<007>OLil02g>O0P000g>O00@007>OLil000EcW`03001c Wg>O0?QcW`002G>O00<007>OLil03G>O00@007>OLil0009cW`03001cWg>O00=cW`03001cWg>O0?Qc W`002G>O00<007>OLil02W>O0`0017>O0P001W>O00<007>OLil0n7>O0009Lil00`00LimcW`0ILil0 0`00LimcW`3hLil000UcW`03001cWg>O01UcW`03001cWg>O0?QcW`002G>O00<007>OLil06G>O00<0 07>OLil0n7>O0008Lil00`00LimcW`0JLil2003iLil000QcW`03001cWg>O01YcW`03001cWg>O0?Qc W`0027>O00<007>OLil06W>O00<007>OLil0n7>O0008Lil00`00LimcW`0JLil00`00LimcW`3hLil0 00QcW`03001cWg>O01YcW`03001cWg>O0?QcW`0027>O00<007>OLil06W>O00<007>OLil0n7>O0008 Lil00`00LimcW`0JLil00`00LimcW`3hLil000QcW`03001cWg>O01YcW`03001cWg>O0?QcW`0027>O 00<007>OLil06W>O0P00nG>O0007Lil00`00LimcW`0KLil00`00LimcW`3hLil000McW`03001cWg>O 01]cW`03001cWg>O0?QcW`001g>O00<007>OLil06g>O00<007>OLil0n7>O0007Lil00`00LimcW`0K Lil00`00LimcW`3hLil000McW`03001cWg>O01]cW`03001cWg>O0?QcW`001g>O00<007>OLil06g>O 00<007>OLil0n7>O0007Lil00`00LimcW`0KLil2003iLil000McW`03001cWg>O01]cW`03001cWg>O 0?QcW`001g>O00<007>OLil06g>O00<007>OLil0n7>O0006Lil00`00LimcW`0LLil00`00LimcW`3h Lil000IcW`03001cWg>O01acW`03001cWg>O0?QcW`001W>O00<007>OLil077>O00<007>OLil0n7>O 0006Lil00`00LimcW`0LLil00`00LimcW`3hLil000IcW`03001cWg>O01acW`800?UcW`009G>O00<0 07>OLil0n7>O000ULil00`00LimcW`3hLil002EcW`03001cWg>O0?QcW`009G>O00<007>OLil0n7>O 0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {0.774954, 10.4253, 0.0329272, 0.136487}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[53]="] }, Open ]] }, Open ]], Cell["\<\ We can also have them cross. This does not cause a calculational \ problem but it does bother our sensibilities a bit. We are not used to \ having mathematics be subjective, ( It seems like we are saying player X is \ a better quarterback than player Y, even though player Y has better \ statistics!). However, it is not subjective at all. Our rules are telling \ how player Z will stack up given a straightforward comparison based on our \ defined rules. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(zz1 = {5, 5, 7, 9, 11, 13}\)], "Input", CellLabel->"In[54]:="], Cell[BoxData[ \(TraditionalForm\`{5, 5, 7, 9, 11, 13}\)], "Output", CellLabel->"Out[54]="], Cell[CellGroupData[{ Cell[BoxData[ \(zz2 = {10, 19, 1, 1, .5, 10}\)], "Input", CellLabel->"In[55]:="], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"10", ",", "19", ",", "1", ",", "1", ",", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "10"}], "}"}], TraditionalForm]], "Output", CellLabel->"Out[55]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{generalnorm[zz1, j], generalnorm[zz2, j]}, {j, 1, 6}, PlotRange \[Rule] All, PlotStyle -> {Dashing[{ .1, .03}], Dashing[{ .02, .02}]}]\)], "Input",\ CellLabel->"In[56]:=", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.166667 0.190476 -0.2124 0.0163144 [ [.21429 .01982 -3 -9 ] [.21429 .01982 3 0 ] [.40476 .01982 -3 -9 ] [.40476 .01982 3 0 ] [.59524 .01982 -3 -9 ] [.59524 .01982 3 0 ] [.78571 .01982 -3 -9 ] [.78571 .01982 3 0 ] [.97619 .01982 -3 -9 ] [.97619 .01982 3 0 ] [.01131 .11389 -12 -4.5 ] [.01131 .11389 0 4.5 ] [.01131 .19546 -12 -4.5 ] [.01131 .19546 0 4.5 ] [.01131 .27703 -12 -4.5 ] [.01131 .27703 0 4.5 ] [.01131 .3586 -12 -4.5 ] [.01131 .3586 0 4.5 ] [.01131 .44018 -12 -4.5 ] [.01131 .44018 0 4.5 ] [.01131 .52175 -12 -4.5 ] [.01131 .52175 0 4.5 ] [.01131 .60332 -12 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .03232 m .21429 .03857 L s [(2)] .21429 .01982 0 1 Mshowa .40476 .03232 m .40476 .03857 L s [(3)] .40476 .01982 0 1 Mshowa .59524 .03232 m .59524 .03857 L s [(4)] .59524 .01982 0 1 Mshowa .78571 .03232 m .78571 .03857 L s [(5)] .78571 .01982 0 1 Mshowa .97619 .03232 m .97619 .03857 L s [(6)] .97619 .01982 0 1 Mshowa .125 Mabswid .0619 .03232 m .0619 .03607 L s .1 .03232 m .1 .03607 L s .1381 .03232 m .1381 .03607 L s .17619 .03232 m .17619 .03607 L s .25238 .03232 m .25238 .03607 L s .29048 .03232 m .29048 .03607 L s .32857 .03232 m .32857 .03607 L s .36667 .03232 m .36667 .03607 L s .44286 .03232 m .44286 .03607 L s .48095 .03232 m .48095 .03607 L s .51905 .03232 m .51905 .03607 L s .55714 .03232 m .55714 .03607 L s .63333 .03232 m .63333 .03607 L s .67143 .03232 m .67143 .03607 L s .70952 .03232 m .70952 .03607 L s .74762 .03232 m .74762 .03607 L s .82381 .03232 m .82381 .03607 L s .8619 .03232 m .8619 .03607 L s .9 .03232 m .9 .03607 L s .9381 .03232 m .9381 .03607 L s .25 Mabswid 0 .03232 m 1 .03232 L s .02381 .11389 m .03006 .11389 L s [(20)] .01131 .11389 1 0 Mshowa .02381 .19546 m .03006 .19546 L s [(25)] .01131 .19546 1 0 Mshowa .02381 .27703 m .03006 .27703 L s [(30)] .01131 .27703 1 0 Mshowa .02381 .3586 m .03006 .3586 L s [(35)] .01131 .3586 1 0 Mshowa .02381 .44018 m .03006 .44018 L s [(40)] .01131 .44018 1 0 Mshowa .02381 .52175 m .03006 .52175 L s [(45)] .01131 .52175 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(50)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .04863 m .02756 .04863 L s .02381 .06494 m .02756 .06494 L s .02381 .08126 m .02756 .08126 L s .02381 .09757 m .02756 .09757 L s .02381 .1302 m .02756 .1302 L s .02381 .14652 m .02756 .14652 L s .02381 .16283 m .02756 .16283 L s .02381 .17915 m .02756 .17915 L s .02381 .21177 m .02756 .21177 L s .02381 .22809 m .02756 .22809 L s .02381 .2444 m .02756 .2444 L s .02381 .26072 m .02756 .26072 L s .02381 .29335 m .02756 .29335 L s .02381 .30966 m .02756 .30966 L s .02381 .32597 m .02756 .32597 L s .02381 .34229 m .02756 .34229 L s .02381 .37492 m .02756 .37492 L s .02381 .39123 m .02756 .39123 L s .02381 .40755 m .02756 .40755 L s .02381 .42386 m .02756 .42386 L s .02381 .45649 m .02756 .45649 L s .02381 .4728 m .02756 .4728 L s .02381 .48912 m .02756 .48912 L s .02381 .50543 m .02756 .50543 L s .02381 .53806 m .02756 .53806 L s .02381 .55438 m .02756 .55438 L s .02381 .57069 m .02756 .57069 L s .02381 .587 m .02756 .587 L s .02381 .016 m .02756 .016 L s .25 Mabswid .02381 0 m .02381 .61803 L s .5 Mabswid [ .1 .03 ] 0 setdash .02381 .60332 m .04262 .48638 L .06244 .3984 L .08255 .3328 L .10458 .2789 L .12415 .24182 L .14509 .21016 L .16496 .18576 L .18653 .16396 L .22646 .13283 L .26733 .10946 L .30668 .09237 L .34452 .07944 L .3848 .06841 L .42358 .05978 L .4648 .05223 L .50451 .04618 L .5427 .04126 L .58335 .0368 L .62248 .03311 L .66406 .02973 L .70413 .0269 L .74268 .02451 L .78368 .02227 L .82317 .02037 L .86114 .01873 L .90156 .01717 L .94047 .01583 L .97619 .01472 L s [ .02 .02 ] 0 setdash .02381 .46465 m .04262 .39262 L .06244 .33793 L .08255 .29683 L .10458 .26282 L .12415 .23928 L .14509 .21908 L .16496 .20344 L .18653 .18941 L .22646 .16933 L .26733 .15424 L .30668 .14325 L .34452 .13501 L .3848 .12808 L .42358 .12276 L .4648 .11822 L .50451 .11468 L .5427 .1119 L .58335 .10946 L .62248 .10753 L .66406 .10584 L .70413 .1045 L .74268 .10342 L .78368 .10247 L .82317 .1017 L .86114 .10108 L .90156 .10053 L .94047 .10008 L .97619 .09973 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", CellLabel->"From In[56]:=", ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgO8G>O003oLilQ Lil0049cW`@0035cW`<003=cW`<0035cW`<003=cW`8000McW`003g>O00<007>OLil0O00<007>O Lil0=7>O00<007>OLil0O00<007>OLil0O00<007>OLil0;g>O00@007>OLil000IcW`003g>O 00<007>OLil0O00<007>OLil0O00<007>OLil0;g>O1000=7>O00<007>OLil0;g>O00@007>O Lil000IcW`003g>O00<007>OLil0O00<007>OLil0<7>O0P00O00<007>O0000O0`00O 0`001g>O000?Lil00`00LimcW`0`Lil01000LimcW`00=7>O00<007>OLil0;g>O00<007>O0000O 00<007>OLil0O00<007>OLil01g>O000?Lil2000bLil2000bLil3000cLil2000bLil4000TLil9 0005Lil30006Lil000mcW`03001cWg>O0>=cWa@001McW`003g>O00<007>OLil0bg>O4@00O000? Lil00`00LimcW`2oLil<0013Lil000mcW`03001cWg>O0:UcW`l005IcW`0027>Oo`006@00000?Lil0 0`00LimcW`08Lil00`00LimcW`07Lil00`00LimcW`08Lil00`00LimcW`07Lil00`00LimcW`08Lil0 0`00LimcW`08Lil00`00LimcW`07Lil00`00LimcW`08Lil00`00LimcW`07Lil00`00LimcW`08Lil0 0`00LimcW`08Lil00`00LimcW`07Lil00`00LimcW`08Lil00`00LimcW`07Lil00`00LimcW`07Lil4 0008Lil00`00LimcW`07Lil00`00LimcW`08Lil00`00LimcW`07Lil00`00LimcW`08Lil00`00Limc W`07Lil00`00LimcW`08Lil00`00LimcW`08Lil00`00LimcW`07Lil00`00LimcW`08Lil00`00Limc W`05Lil000mcW`03001cWg>O039cW`03001cWg>O039cW`03001cWg>O02IcW`@000QcW`03001cWg>O 039cW`03001cWg>O039cW`03001cWg>O00EcW`003g>O00<007>OLil0R7>O2000OW>O000?Lil00`00 LimcW`20Lil80026Lil000mcW`8007acW`D008icW`003g>O00<007>OLil0Mg>O1000Tg>O000?Lil0 0`00LimcW`1^Lil2002NLil000mcW`03001cWg>O06UcW`D00:1cW`003g>O0P00IG>O1@00YG>O000? Lil00`00LimcW`1OLil5002ZLil000mcW`03001cWg>O05]cW`@00:mcW`003g>O00<007>OLil0F7>O 0`00/g>O000?Lil00`00LimcW`1ELil3002fLil000mcW`8005EcW`03001cWg>O0;McW`003g>O00<0 07>OLil0og>O3g>O000?Lil00`00LimcW`3oLil?Lil000mcW`03001cWg>O04]cW`800<5cW`003g>O 0P00BW>O0P00`g>O000?Lil00`00LimcW`16Lil3002@Lil40004Lil70005Lil60005Lil60005Lil4 0007Lil000mcW`03001cWg>O04AcW`8006mcW`L000AcW`L000AcW`L000AcW`<003EcW`0000=cW`00 00000P000g>O0P001G>O00<007>OLil0@W>O0P00Fg>O1P001G>O1P00GW>O0002Lil00`00LimcW`02 Lil01000LimcW`0017>O00<007>OLil0@7>O0P00BG>O10001G>O1P00M7>O0003Lil01@00LimcWg>O 00000W>O00<007>OLil00W>O0`00?W>O0P00@7>O10001G>O0P00Pg>O0004Lil01000LimcW`000W>O 00<007>OLil00W>O00<007>OLil0?7>O0P00=g>O100017>O0`00SW>O00001G>O001cWg>O00000W>O 00@007>OLil000AcW`03001cWg>O03YcW`8002icW`@000AcW`<009UcW`000W>O0P0017>O0P001G>O 00<007>OLil0>7>O0P00;G>O0`00Y7>O000?Lil2000gLil2000WLil4002[Lil000mcW`03001cWg>O 03EcW`03001cWg>O02AcW`<00:mcW`003g>O00<007>OLil0=G>O00<007>OLil077>O1000]W>O000? Lil00`00LimcW`0dLil00`00LimcW`0JLil3002jLil000mcW`03001cWg>O04]cW`800<5cW`003g>O 0P00BG>O0`00`g>O000?Lil00`00LimcW`17Lil00`00LimcW`34Lil000mcW`03001cWg>O041cW`80 0O00<007>OLil0;7>O0P003g>O0`00cW>O000?Lil00`00LimcW`0[Lil00`00LimcW`0> Lil00`00LimcW`3?Lil000mcW`8002]cW`03001cWg>O0>5cW`003g>O00<007>OLil0:G>O00<007>O Lil027>O1000eW>O000?Lil00`00LimcW`0XLil00`00LimcW`08Lil00`00LimcW`3HLil000mcW`03 001cWg>O02McW`03001cWg>O00QcW`03001cWg>O0=UcW`003g>O0P009g>O00<007>OLil0iG>O000? Lil00`00LimcW`0ULil00`00LimcW`3VLil000mcW`03001cWg>O02AcW`03001cWg>O00EcW`800>1c W`0000=cW`0000000P000W>O0`001G>O00<007>OLil097>O00<007>OLil00g>O0P00hW>O0002Lil0 0`00LimcW`05Lil00`00LimcW`02Lil00`00LimcW`0SLil00`00LimcW`03Lil00`00LimcW`3RLil0 00=cW`03001cWg>O00AcW`03001cWg>O009cW`<0029cW`03001cWg>O00=cW`03001cWg>O0>=cW`00 17>O00<007>OLil00`001G>O00<007>OLil08G>O00<007>OLil0jW>O00001G>O001cWg>O00000W>O 00<007>OLil01G>O00<007>OLil08G>O00<007>OLil0jW>O0002Lil20003Lil40004Lil00`00Limc W`0PLil01000Lil00000jW>O000?Lil2000PLil00`00Lil0003/Lil000mcW`03001cWg>O01mcW`80 0>ecW`003g>O00<007>OLil07W>O0P00kW>O000?Lil00`00LimcW`0MLil00`00LimcW`3^Lil000mc W`03001cWg>O01ecW`03001cWg>O0>icW`003g>O0P007G>O00<007>OLil0kg>O000?Lil00`00Limc W`0KLil00`00LimcW`3`Lil000mcW`03001cWg>O01YcW`03001cWg>O0?5cW`003g>O00<007>OLil0 6G>O00<007>OLil0lW>O000?Lil2000ILil00`00LimcW`3cLil000mcW`03001cWg>O01McW`03001c Wg>O0?AcW`003g>O00<007>OLil0og>O3g>O000?Lil00`00LimcW`3oLil?Lil000mcW`03001cWg>O 01McW`03001cWg>O0?AcW`003g>O0P005G>O00@007>OLil00?IcW`003g>O00<007>OLil04g>O00@0 07>OLil00?McW`003g>O00<007>OLil04g>O00<007>O0000n7>O000017>O0000000017>O0P001G>O 00<007>OLil04W>O00@007>OLil00?QcW`0017>O00@007>OLil0009cW`03001cWg>O009cW`03001c Wg>O019cW`03001cW`000?UcW`0017>O00@007>OLil0009cW`03001cWg>O009cW`<0015cW`03001c W`000?YcW`000W>O0P000g>O00@007>OLil000AcW`03001cWg>O01=cW`03001cWg>O0?QcW`0017>O 00@007>OLil0009cW`03001cWg>O009cW`03001cWg>O019cW`03001cWg>O0?UcW`0000AcW`000000 00AcW`8000EcW`03001cWg>O019cW`03001cWg>O0?UcW`003g>O0P004g>O00<007>OLil0nG>O000? Lil00`00LimcW`0=Lil20002Lil00`00LimcW`3jLil000mcW`03001cWg>O00ecW`05001cWg>OLil0 003lLil000mcW`03001cWg>O00acW`05001cWg>OLil0003mLil000mcW`03001cWg>O00acW`05001c Wg>OLil0003mLil000mcW`8000ecW`05001cWg>OLil0003mLil000mcW`03001cWg>O00]cW`05001c Wg>OLil0003nLil000mcW`03001cWg>O00mcW`03001cWg>O0?acW`003g>O00<007>OLil03W>O00<0 07>OLil0oG>O000?Lil2000?Lil00`00LimcW`3mLil000mcW`03001cWg>O00icW`03001cWg>O0?ec W`003g>O00<007>OLil02G>O00D007>OLimcW`000?mcW`5cW`003g>O00<007>OLil02G>O00D007>O LimcW`000?mcW`5cW`003g>O00<007>OLil027>O00<007>OLil00W>O00<007>OLil0oW>O000?Lil2 0009Lil01@00LimcWg>O0000og>O0W>O000?Lil00`00LimcW`07Lil00`00LimcW`02Lil00`00Limc W`3oLil00004Lil000000003Lil30005Lil00`00LimcW`06Lil00`00LimcW`03Lil00`00LimcW`3o Lil000AcW`03001cWg>O00=cW`03001cWg>O009cW`03001cWg>O0?mcW`mcW`0017>O00<007>OLil0 0g>O00<007>OLil00W>O0`00og>O3g>O0002Lil20003Lil30005Lil00`00LimcW`3oLil?Lil000Ac W`04001cWg>O0007Lil00`00LimcW`3oLil?Lil00004Lil000000003Lil40004Lil00`00LimcW`05 Lil00`00LimcW`3oLil7Lil000mcW`03001cWg>O00EcW`03001cWg>O0?mcW`McW`003g>O0P001G>O 00<007>OLil0og>O27>O000?Lil00`00LimcW`04Lil00`00LimcW`02Lil00`00LimcW`3oLil3Lil0 00mcW`03001cWg>O00AcW`03001cWg>O009cW`03001cWg>O0?mcW`=cW`003g>O00<007>OLil00g>O 00<007>OLil00g>O00<007>OLil0og>O0g>O000?Lil00`00LimcW`03Lil00`00LimcW`03Lil00`00 LimcW`3oLil3Lil000mcW`8000UcW`03001cWg>O0?mcW`AcW`003g>O00<007>OLil027>O00<007>O Lil0og>O17>O000?Lil00`00LimcW`08Lil00`00LimcW`3oLil4Lil000mcW`03001cWg>O00QcW`03 001cWg>O0?mcW`AcW`003g>O0P000W>O00<007>OLil017>O00<007>OLil0og>O17>O000?Lil01@00 LimcWg>O00001G>O00<007>OLil0og>O1G>O000?Lil01@00LimcWg>O00001G>O00<007>OLil0og>O 1G>O000?Lil01000LimcW`001W>O00<007>OLil0og>O1G>O000?Lil01000LimcW`001W>O00<007>O Lil0og>O1G>O000?Lil200000g>O001cW`04Lil00`00LimcW`3oLil6Lil000mcW`03001cWg>O00Ic W`03001cWg>O0?mcW`IcW`000W>O0`000g>O0P001G>O00<007>OLil01W>O00<007>OLil0og>O1W>O 0003Lil01@00LimcWg>O00000W>O00<007>OLil00W>O00<007>OLil01W>O00<007>OLil0og>O1W>O 00000g>O000000020002Lil01000LimcW`0017>O0`001G>O00<007>OLil0og>O1g>O000017>O001c W`000g>O00@007>OLil000AcW`03001cWg>O00EcW`03001cWg>O0?mcW`McW`0000AcW`00Lil000=c W`04001cWg>O0004Lil20006Lil00`00LimcW`3oLil7Lil0009cW`8000AcW`8000EcW`8000IcW`03 001cWg>O0?mcW`McW`003g>O0P001W>O00<007>OLil0og>O1g>O000?Lil20005Lil00`00LimcW`3o Lil8Lil000mcW`03001cWg>O00AcW`03001cWg>O0?mcW`QcW`003g>O00<007>OLil017>O00<007>O Lil0og>O27>O000?Lil00`00LimcW`04Lil00`00LimcW`3oLil8Lil000mcW`8000AcW`03001cWg>O 0?mcW`UcW`003g>O00<007>OLil00g>O00<007>OLil0og>O2G>O000?Lil00`00LimcW`3oLil?Lil0 00mcW`03001cWg>O0?mcW`mcW`003g>O00<007>OLil0og>O3g>O000?Lil2003oLil@Lil000mcW`03 001cWg>O0?mcW`mcW`003g>O00<007>OLil0og>O3g>O000?Lil00`00LimcW`3oLil?Lil000mcW`05 001cWg>OLil0003oLil=Lil000mcW`80009cW`03001cWg>O0?mcW`]cW`003g>O00D007>OLimcW`00 0?mcW`ecW`000W>O0`000W>O0`001G>O00D007>OLimcW`000?mcW`ecW`000g>O00<007>OLil017>O 00<007>OLil00W>O00@007>OLil00?mcW`icW`0000=cW`0000000P001G>O00<007>OLil00W>O1000 og>O3W>O000017>O001cW`000g>O0`001G>O00@007>OLil00?mcW`icW`0000AcW`00Lil000=cW`03 001cWg>O00EcW`04001cWg>O003oLil>Lil0009cW`8000=cW`@000AcW`04001cWg>O003oLil>Lil0 00mcW`04001cWg>O003oLil>Lil000mcW`800003Lil007>O0?mcW`ecW`003g>O00<007>O0000og>O 3g>O000?Lil00`00Lil0003oLil?Lil000mcW`03001cW`000?mcW`mcW`003g>O0`00og>O3g>O000? Lil00`00Lil0003oLil?Lil000mcW`03001cW`000?mcW`mcW`003g>O00<007>O0000og>O3g>O000? Lil2003oLil@Lil000mcW`800?mcWa1cW`003g>O0P00og>O47>O000?Lil2003oLil@Lil000mcW`80 0?mcWa1cW`003g>O0P00og>O47>O000?Lil2003oLil@Lil000mcW`03001cWg>O0?mcW`mcW`0000Ac W`00000000AcW`8000EcW`03001cWg>O0?mcW`mcW`0017>O00@007>OLil0009cW`03001cWg>O009c W`03001cWg>O0?mcW`mcW`0017>O00@007>OLil0009cW`03001cWg>O009cW`<00?mcW`mcW`0000Ac W`00000000=cW`04001cWg>O0004Lil00`00LimcW`3oLil?Lil00003Lil007>O00AcW`04001cWg>O 0004Lil00`00LimcW`3oLil?Lil00003Lil00000008000=cW`8000EcW`03001cWg>O0?mcW`mcW`00 3g>O00<007>OLil0og>O3g>O0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {0.707852, 12.2643, 0.0188752, 0.218838}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[56]="] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell["Inner product", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Another natural question to ask about vectors is their relative \ direction. Or (similarly) perhaps the difference between vectors. This is \ the inner product, (x,y) = inner[x,y]. Here is a definition for the inner product that corresponds to a Euclidean \ norm. Note however that we need the conjugate of the second argument to \ avoid the possibility, when we relate this back to the norm, of negative \ lengths. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(inner[x_, y_] := \[Sum]\+\(i = 1\)\%\(Length[x]\)x\[LeftDoubleBracket]i \[RightDoubleBracket]\ Conjugate[y\[LeftDoubleBracket]i\[RightDoubleBracket]]\)], "Input", CellLabel->"In[1]:=", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ \(General::"spell1"\), \( : \ \), "\<\"Possible spelling error: new symbol name \ \\\"\\!\\(TraditionalForm\\`inner\\)\\\" is similar to existing symbol \ \\\"\\!\\(TraditionalForm\\`Inner\\)\\\".\"\>"}], TraditionalForm]], "Message"] }, Open ]], Cell[BoxData[ \(\(vv = {v1, v2, v3, v4}; \)\)], "Input", CellLabel->"In[2]:=", AspectRatioFixed->True], Cell[BoxData[ \(\(ww = {w1, w2, w3, w4}; \)\)], "Input", CellLabel->"In[3]:=", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(inner[vv, ww]\)], "Input", CellLabel->"In[4]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`v1\ \(Conjugate(w1)\) + v2\ \(Conjugate(w2)\) + v3\ \(Conjugate(w3)\) + v4\ \(Conjugate(w4)\)\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[TextData[{ "We can use the ", StyleBox["Mathematica", FontSlant->"Italic"], " definition of a general Inner Product to construct one that is the same \ as ours," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Inner[Times, vv, Conjugate[ww], Plus]\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \(TraditionalForm \`v1\ \(Conjugate(w1)\) + v2\ \(Conjugate(w2)\) + v3\ \(Conjugate(w3)\) + v4\ \(Conjugate(w4)\)\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[BoxData[{ \(\(a = {1, \(-3\), \(-4\) + I}; \)\), \(\(b = {\(-1\) - I, 3, \(-2\) - 7\ I}; \)\)}], "Input", CellLabel->"In[6]:=", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(inner[a, b]\)], "Input", CellLabel->"In[8]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\(-9\) - 29\ \[ImaginaryI]\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[TextData[ "Comparing to the \"dot\" product, we see that they are different for complex \ elements."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(a . b\)], "Input", CellLabel->"In[9]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`5 + 25\ \[ImaginaryI]\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(vv . ww\)], "Input", CellLabel->"In[10]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`v1\ w1 + v2\ w2 + v3\ w3 + v4\ w4\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(inner[vv, ww]\)], "Input", CellLabel->"In[11]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`v1\ \(Conjugate(w1)\) + v2\ \(Conjugate(w2)\) + v3\ \(Conjugate(w3)\) + v4\ \(Conjugate(w4)\)\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["\<\ We see that the inner product is not the same at a standard Dot \ product, which is defined only for real vectors. If (x,y) = 0, then x and y are orthogonal in the vector space that we are \ considering. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Weighted inner product", "Subsubsection"], Cell["\<\ We will need eventually to use inner products that have some \ weighting factors. This makes different directions more important to the \ value of the norm. For example, if we were weighting test scores for high \ school students we might count Mathematics more than Social Studies \ (particularly if we were thinking of future engineers). Thus for the inner \ product of two student vectors, students would need similar Math scores to be \ \"parallel\" and differences in their social studies scores would matter \ less. \ \>", "Text"], Cell[TextData[{ "Here is a weighted Euclidean inner product.\n\n", Cell[BoxData[ \(TraditionalForm\`\((x, y)\)\_Ew\)]], "= ", Cell[BoxData[ \(TraditionalForm\`x\_i\)]], " ", Cell[BoxData[ \(TraditionalForm\`W\_ij\)]], " ", Cell[BoxData[ \(TraditionalForm\`y\_j\)]], " = ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], "T"], TraditionalForm]]], StyleBox["W y\n\n", FontWeight->"Bold"], "The matrix, ", StyleBox["W, ", FontWeight->"Bold"], " gives the weighting coefficients. " }], "Text"], Cell[BoxData[ \(weightedinner[x_, y_, wz_] := \[Sum]\+\(i = 1\)\%\(Length[x] \)\(\[Sum]\+\(j = 1\)\%\(Length[y]\)x\[LeftDoubleBracket]i \[RightDoubleBracket]\ wz[\([i, j]\)]\ y\[LeftDoubleBracket]j\[RightDoubleBracket]\)\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ \(\(ax = {ax1, ax2, ax3, ax4, ax5}; \)\)], "Input", CellLabel->"In[34]:="], Cell[BoxData[ \(\(bx = {bx1, bx2, bx3, bx4, bx5}; \)\)], "Input", CellLabel->"In[35]:="], Cell[CellGroupData[{ Cell[BoxData[ \(wx = {{wx11, wx12, wx13, wx14, wx15}, \n \t\t{wx21, wx22, wx23, wx24, wx25}, {wx31, wx32, wx33, wx34, wx35}, \n \t\t\t{wx41, wx42, wx43, wx44, wx45}, {wx51, wx52, wx53, wx54, wx55}} \)], "Input", CellLabel->"In[36]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"wx11", "wx12", "wx13", "wx14", "wx15"}, {"wx21", "wx22", "wx23", "wx24", "wx25"}, {"wx31", "wx32", "wx33", "wx34", "wx35"}, {"wx41", "wx42", "wx43", "wx44", "wx45"}, {"wx51", "wx52", "wx53", "wx54", "wx55"} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[36]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(test = weightedinner[ax, bx, wx]\)], "Input", CellLabel->"In[37]:="], Cell[BoxData[ \(TraditionalForm \`ax1\ bx1\ wx11 + ax1\ bx2\ wx12 + ax1\ bx3\ wx13 + ax1\ bx4\ wx14 + ax1\ bx5\ wx15 + ax2\ bx1\ wx21 + ax2\ bx2\ wx22 + ax2\ bx3\ wx23 + ax2\ bx4\ wx24 + ax2\ bx5\ wx25 + ax3\ bx1\ wx31 + ax3\ bx2\ wx32 + ax3\ bx3\ wx33 + ax3\ bx4\ wx34 + ax3\ bx5\ wx35 + ax4\ bx1\ wx41 + ax4\ bx2\ wx42 + ax4\ bx3\ wx43 + ax4\ bx4\ wx44 + ax4\ bx5\ wx45 + ax5\ bx1\ wx51 + ax5\ bx2\ wx52 + ax5\ bx3\ wx53 + ax5\ bx4\ wx54 + ax5\ bx5\ wx55\)], "Output", CellLabel->"Out[37]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test, ax1\ bx4\ ]\)], "Input", CellLabel->"In[38]:="], Cell[BoxData[ \(TraditionalForm\`wx14\)], "Output", CellLabel->"Out[38]="] }, Open ]], Cell[TextData[{ "Often we jsut need 1 weighting function for each element so that ", StyleBox["W", FontWeight->"Bold"], " is of the form" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(wz = {{w1, 0, 0, 0, 0}, {0, w2, 0, 0, 0}, {0, 0, w3, 0, 0}, \n \t\t{0, 0, 0, w4, 0}, {0, 0, 0, 0, w5}}\)], "Input", CellLabel->"In[39]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"w1", "0", "0", "0", "0"}, {"0", "w2", "0", "0", "0"}, {"0", "0", "w3", "0", "0"}, {"0", "0", "0", "w4", "0"}, {"0", "0", "0", "0", "w5"} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[39]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(test2 = weightedinner[ax, bx, wz]\)], "Input", CellLabel->"In[40]:="], Cell[BoxData[ \(TraditionalForm \`ax1\ bx1\ w1 + ax2\ bx2\ w2 + ax3\ bx3\ w3 + ax4\ bx4\ w4 + ax5\ bx5\ w5\)], "Output", CellLabel->"Out[40]="] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Macintosh", ScreenRectangle->{{0, 1152}, {0, 850}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{601, 770}, WindowMargins->{{177, Automatic}, {Automatic, 8}}, PrintingCopies->1, PrintingPageRange->{1, 12}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, PrivateFontOptions->{"FontType"->"Outline"}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. Make modifications to any definition using commands in the Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], ScriptMinSize->9], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ShowCellLabel->False, ImageSize->{200, 200}, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{True, True}, PrintingOptions->{"FirstPageHeader"->False, "FacingPages"->True}, CellLabelAutoDelete->False, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellFrame->{{0, 0}, {0, 0.25}}, CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CellFrameMargins->9, LineSpacing->{0.95, 0}, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->36, Background->RGBColor[0.750011, 1, 0.750011]], Cell[StyleData["Title", "Printout"], CellMargins->{{18, 30}, {4, 0}}, CellFrameMargins->4, FontSize->30] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{18, 30}, {0, 10}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, LineSpacing->{1, 0}, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->24, FontSlant->"Italic", Background->RGBColor[0.8, 0.920012, 0.920012]], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{18, 30}, {0, 10}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SectionFirst"], CellFrame->{{0, 0}, {0, 3}}, CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CellFrameMargins->3, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold"], Cell[StyleData["SectionFirst", "Printout"], FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold", Background->RGBColor[0.920012, 0.870024, 0.770016]], Cell[StyleData["Section", "Printout"], FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontSize->14, FontWeight->"Bold", Background->RGBColor[0.970001, 0.890013, 0.850004]], Cell[StyleData["Subsection", "Printout"], FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{18, 30}, {4, 12}}, CellGroupingRules->{"SectionGrouping", 60}, PageBreakBelow->False, CounterIncrements->"Subsubsection", FontSize->12, FontWeight->"Bold", Background->RGBColor[0.870008, 1, 0.960006]], Cell[StyleData["Subsubsection", "Printout"], FontSize->10] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{18, 10}, {Inherited, 6}}, TextJustification->1, LineSpacing->{1, 2}, CounterIncrements->"Text", Background->RGBColor[0.900008, 1, 0.940002]], Cell[StyleData["Text", "Printout"], CellMargins->{{18, 30}, {Inherited, 4}}, LineSpacing->{1, 3}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Caption"], CellMargins->{{55, 50}, {5, 5}}, PageBreakAbove->False, FontSize->10], Cell[StyleData["Caption", "Printout"], CellMargins->{{55, 55}, {5, 2}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{55, 10}, {5, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Input", "Printout"], CellMargins->{{55, 55}, {0, 10}}, ShowCellLabel->False, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{55, 10}, {8, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelPositioning->Left, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output", Background->RGBColor[0.940017, 0.890013, 0.990005]], Cell[StyleData["Output", "Printout"], CellMargins->{{55, 55}, {10, 10}}, ShowCellLabel->False, FontSize->9.5] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellDingbat->"\[LongDash]", CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Message", "Printout"], CellMargins->{{55, 55}, {0, 3}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, TextAlignment->Left, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Printout"], CellMargins->{{54, 72}, {2, 10}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", StyleMenuListing->None, Background->RGBColor[0.780011, 0.920012, 1], ButtonBoxOptions->{Background->RGBColor[0.750011, 0.630014, 0.760021]}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{55, 55}, {0, 15}}, ImageSize->{0.0625, 0.0625}, ImageMargins->{{35, Inherited}, {Inherited, 0}}, FontSize->8] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], CellMargins->{{9, Inherited}, {Inherited, Inherited}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontSlant->"Oblique"], Cell[StyleData["CellLabel", "Printout"], CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Unique Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{45, Inherited}, {2, 20}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Author", "Printout"], CellMargins->{{36, Inherited}, {2, 30}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Address"], CellMargins->{{45, Inherited}, {2, 2}}, CellGroupingRules->{"TitleGrouping", 30}, PageBreakBelow->False, LineSpacing->{1, 1}, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->12, FontSlant->"Italic"], Cell[StyleData["Address", "Printout"], CellMargins->{{36, Inherited}, {2, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Abstract"], CellMargins->{{45, 75}, {Inherited, 30}}, LineSpacing->{1, 0}], Cell[StyleData["Abstract", "Printout"], CellMargins->{{36, 67}, {Inherited, 50}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Reference"], CellMargins->{{18, 40}, {2, 2}}, TextJustification->1, LineSpacing->{1, 0}], Cell[StyleData["Reference", "Printout"], CellMargins->{{18, 40}, {Inherited, 0}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, \ etc. They automatically give the cell a FrameLabel containing a reference to \ a particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CellMargins->{{55, 10}, {0, 10}}, CellFrameLabels->{{None, Cell[ TextData[ {"(", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}, DefaultFormatType->DefaultInputFormatType, CounterIncrements->"NumberedEquation", FormatTypeAutoConvert->False], Cell[StyleData["NumberedEquation", "Printout"], CellMargins->{{55, 55}, {0, 10}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellMargins->{{55, 145}, {2, 10}}, CellHorizontalScrolling->True, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}], FontWeight -> "Bold"], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False], Cell[StyleData["NumberedFigure", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{55, 145}, {2, 10}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}], FontWeight -> "Bold"], None}}, TextAlignment->Center, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{55, 10}, {2, 10}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ChemicalFormula"], CellMargins->{{55, 10}, {2, 10}}, DefaultFormatType->DefaultInputFormatType, AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{18, 10}, {Inherited, 6}}, FontFamily->"Courier"], Cell[StyleData["Program", "Printout"], CellMargins->{{18, 30}, {Inherited, 4}}, FontSize->9.5] }, Closed]] }, Closed]] }, Open ]] }], MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002mT8o?mooh<" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1717, 49, 92, 2, 76, "Title", Evaluatable->False], Cell[1812, 53, 1113, 25, 319, "Text"], Cell[2928, 80, 199, 5, 54, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[3152, 89, 79, 2, 69, "Section", Evaluatable->False], Cell[3234, 93, 313, 9, 82, "Text", Evaluatable->False], Cell[3550, 104, 201, 5, 50, "Input"], Cell[3754, 111, 44, 0, 40, "Text"], Cell[3801, 113, 105, 3, 27, "Input"], Cell[CellGroupData[{ Cell[3931, 120, 90, 3, 27, "Input"], Cell[4024, 125, 102, 2, 51, "Output"] }, Open ]], Cell[4141, 130, 114, 3, 27, "Input"], Cell[CellGroupData[{ Cell[4280, 137, 91, 3, 27, "Input"], Cell[4374, 142, 121, 3, 51, "Output"] }, Open ]], Cell[4510, 148, 109, 3, 27, "Input"], Cell[CellGroupData[{ Cell[4644, 155, 97, 3, 27, "Input"], Cell[4744, 160, 81, 2, 46, "Output"] }, Open ]], Cell[4840, 165, 396, 9, 96, "Text", Evaluatable->False], Cell[5239, 176, 239, 6, 69, "Input"], Cell[CellGroupData[{ Cell[5503, 186, 104, 3, 43, "Input"], Cell[5610, 191, 95, 2, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5742, 198, 106, 3, 43, "Input"], Cell[5851, 203, 96, 2, 43, "Output"] }, Open ]], Cell[5962, 208, 61, 0, 40, "Text"], Cell[CellGroupData[{ Cell[6048, 212, 142, 3, 43, "Input"], Cell[6193, 217, 12359, 333, 239, 2984, 211, "GraphicsData", "PostScript", \ "Graphics", Evaluatable->False], Cell[18555, 552, 192, 5, 43, "Output"] }, Open ]], Cell[18762, 560, 219, 4, 54, "Text", Evaluatable->False], Cell[18984, 566, 255, 5, 54, "Text", Evaluatable->False], Cell[19242, 573, 127, 3, 43, "Input"], Cell[CellGroupData[{ Cell[19394, 580, 146, 4, 43, "Input"], Cell[19543, 586, 13221, 422, 201, 4425, 307, "GraphicsData", "PostScript", \ "Graphics", Evaluatable->False], Cell[32767, 1010, 192, 5, 43, "Output"] }, Open ]], Cell[32974, 1018, 131, 3, 43, "Input"], Cell[33108, 1023, 125, 2, 40, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[33258, 1029, 150, 4, 43, "Input"], Cell[33411, 1035, 12603, 356, 201, 3388, 236, "GraphicsData", "PostScript", \ "Graphics", Evaluatable->False], Cell[46017, 1393, 192, 5, 43, "Output"] }, Open ]], Cell[46224, 1401, 425, 8, 82, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[46674, 1413, 59, 0, 45, "Subsubsection"], Cell[46736, 1415, 123, 3, 40, "Text"], Cell[46862, 1420, 90, 3, 40, "Text"], Cell[CellGroupData[{ Cell[46977, 1427, 82, 2, 43, "Input"], Cell[47062, 1431, 94, 2, 43, "Output"], Cell[CellGroupData[{ Cell[47181, 1437, 82, 2, 43, "Input"], Cell[47266, 1441, 94, 2, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47397, 1448, 235, 5, 58, "Input"], Cell[47635, 1455, 12305, 392, 201, 4077, 285, "GraphicsData", "PostScript", \ "Graphics"], Cell[59943, 1849, 192, 5, 43, "Output"] }, Open ]] }, Open ]], Cell[60162, 1858, 94, 3, 40, "Text"], Cell[CellGroupData[{ Cell[60281, 1865, 83, 2, 43, "Input"], Cell[60367, 1869, 95, 2, 43, "Output"], Cell[CellGroupData[{ Cell[60487, 1875, 83, 2, 43, "Input"], Cell[60573, 1879, 95, 2, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[60705, 1886, 235, 5, 58, "Input"], Cell[60943, 1893, 12558, 388, 201, 3941, 276, "GraphicsData", "PostScript", \ "Graphics"], Cell[73504, 2283, 192, 5, 43, "Output"] }, Open ]] }, Open ]], Cell[73723, 2292, 486, 8, 96, "Text"], Cell[CellGroupData[{ Cell[74234, 2304, 84, 2, 43, "Input"], Cell[74321, 2308, 96, 2, 43, "Output"], Cell[CellGroupData[{ Cell[74442, 2314, 87, 2, 43, "Input"], Cell[74532, 2318, 279, 8, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[74848, 2331, 242, 6, 73, "Input"], Cell[75093, 2339, 13235, 442, 201, 4675, 331, "GraphicsData", "PostScript", \ "Graphics"], Cell[88331, 2783, 192, 5, 43, "Output"] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[88574, 2794, 80, 2, 69, "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[88679, 2800, 494, 11, 110, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[89198, 2815, 262, 6, 68, "Input"], Cell[89463, 2823, 291, 6, 20, "Message"] }, Open ]], Cell[89769, 2832, 110, 3, 43, "Input"], Cell[89882, 2837, 110, 3, 43, "Input"], Cell[CellGroupData[{ Cell[90017, 2844, 96, 3, 43, "Input"], Cell[90116, 2849, 180, 4, 43, "Output"] }, Open ]], Cell[90311, 2856, 188, 6, 40, "Text"], Cell[CellGroupData[{ Cell[90524, 2866, 94, 2, 43, "Input"], Cell[90621, 2870, 180, 4, 43, "Output"] }, Open ]], Cell[90816, 2877, 169, 4, 58, "Input"], Cell[CellGroupData[{ Cell[91010, 2885, 94, 3, 43, "Input"], Cell[91107, 2890, 101, 2, 43, "Output"] }, Open ]], Cell[91223, 2895, 163, 4, 40, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[91411, 2903, 88, 3, 43, "Input"], Cell[91502, 2908, 96, 2, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[91635, 2915, 91, 3, 43, "Input"], Cell[91729, 2920, 109, 2, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[91875, 2927, 97, 3, 43, "Input"], Cell[91975, 2932, 181, 4, 43, "Output"] }, Open ]], Cell[92171, 2939, 278, 8, 68, "Text", Evaluatable->False] }, Open ]], Cell[CellGroupData[{ Cell[92486, 2952, 47, 0, 45, "Subsubsection"], Cell[92536, 2954, 548, 9, 96, "Text"], Cell[93087, 2965, 613, 25, 96, "Text"], Cell[93703, 2992, 308, 6, 77, "Input"], Cell[94014, 3000, 94, 2, 43, "Input"], Cell[94111, 3004, 94, 2, 43, "Input"], Cell[CellGroupData[{ Cell[94230, 3010, 264, 5, 73, "Input"], Cell[94497, 3017, 401, 9, 107, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[94935, 3031, 90, 2, 43, "Input"], Cell[95028, 3035, 545, 9, 103, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[95610, 3049, 87, 2, 43, "Input"], Cell[95700, 3053, 80, 2, 43, "Output"] }, Open ]], Cell[95795, 3058, 159, 5, 40, "Text"], Cell[CellGroupData[{ Cell[95979, 3067, 168, 3, 58, "Input"], Cell[96150, 3072, 331, 9, 107, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[96518, 3086, 91, 2, 43, "Input"], Cell[96612, 3090, 160, 4, 43, "Output"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)