(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1177342, 26414]*) (*NotebookOutlinePosition[ 1178099, 26440]*) (* CellTagsIndexPosition[ 1178055, 26436]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Notebooks of the Examples", FontSize->24], "\n", StyleBox["in", FontSize->18] }], "Title", TextAlignment->Center, FontSize->30], Cell[TextData[{ StyleBox["Gy\[ODoubleDot]rgy Popper", FontSize->16, FontSlant->"Italic"], StyleBox["\nSome Concepts of Functional Analysis Using ", FontSize->18], StyleBox["Matematic", FontSize->18, FontSlant->"Italic"], Cell[BoxData[ SuperscriptBox[ StyleBox["a", FontFamily->"Times New Roman", FontSize->18, FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], "*"]]] }], "Subtitle", TextAlignment->Center, FontWeight->"Bold"], Cell[TextData[{ "Gy\[ODoubleDot]rgy Poppe", Cell[BoxData[ \(r\^1\)]], " and B\[EAcute]la Pal\[AAcute]nc", Cell[BoxData[ \(z\^2\)]] }], "Text", TextAlignment->Center], Cell[BoxData[{\(\(\^1\) Department\ of\ \ Structural\ Mechanics\), "\n", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ StyleBox[".", FontColor->GrayLevel[1], Background->GrayLevel[1]], "2"], "Department"}], " ", "of", " ", "Photogrammetry", " ", "and", " ", "Geoinformatics"}], "\n"}], "\[IndentingNewLine]", RowBox[{\(Budapest\ University\ of\ Technology\ and\ Economics, H - 1111\ Budapest, Hungary, M\[UDoubleDot]egyetem\ rkp . 3. \), "\n"}], "\[IndentingNewLine]", \(e - \(mail : palancz@epito . bme . hu\)\)}], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ Cell[BoxData[ \(\(\^*\)\)]], "The PDF version of this textbook can be downloaded from: \ http://www.me.bme.hu/ and English version (see flag), Personal uploads" }], "Text"], Cell[CellGroupData[{ Cell["1.Vector spaces, subspaces, linear manifolds", "Section", TextAlignment->Left, FontSize->16], Cell[TextData[{ StyleBox["E.1.2", "Section", FontSize->14, FontColor->RGBColor[1, 0, 0]], StyleBox[".", FontColor->RGBColor[1, 0, 0]], " The set ", Cell[BoxData[ \(C\^\((0)\)\)]], "[a, b] of all (real-valued) continuous functions on a finite interval a\ \[LessEqual] t\[LessEqual] b with addition and real number multiplication" }], "Text"], Cell["\<\ (f + g)(t) = f(t) + g(t), (\[Alpha] f)(t) = \[Alpha] f(t), \ t\[Element][a,b]\ \>", "Text", TextAlignment->Center], Cell[TextData[{ "forms a linear space.", StyleBox["The zero vector \[CapitalTheta]: f(t) = 0 for all ", FontFamily->"Times New Roman"], " t\[Element][a,b]", StyleBox[".\nNote that ", FontFamily->"Times New Roman"], Cell[BoxData[ \(C\^\((0)\)\)]], "[a, b]", StyleBox["\[Subset] ", FontFamily->"Times New Roman"], Cell[BoxData[ \(C\^\((0)\)\)]], "(a, b) is true because f\[Element]", Cell[BoxData[ \(C\^\((0)\)\)]], "[a, b]\[DoubleRightArrow]f\[Element]", Cell[BoxData[ \(C\^\((0)\)\)]], "(a, b).\nMore generally ", Cell[BoxData[ \(C\^\((k)\)\)]], "[a, b] denotes the linear space of k-times continuously differentiable \ functions on a close interval [a, b]. (This means the set of functions whose \ derivatives at least up to order k inclusive are continuous in [a, b]." }], "Text"], Cell["For example, the function", "Text"], Cell[BoxData[ RowBox[{\(f \((t)\)\), "=", RowBox[{"\[LeftBracketingBar]", GridBox[{ {\(t + t\^2\/2, if\ - 1 \[LessEqual] t \[LessEqual] 0\)}, {\(t - t\^2\/2, \ \ if\ \ \ 0\ \[LessEqual] t \[LessEqual] \ 1\)} }]}]}]], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "defined on the interval -1\[LessEqual]t\[LessEqual]1, is once continuously \ differentiable but only once, i.e. f(t)\[Element]", Cell[BoxData[ \(C\^\((1)\)\)]], "[-1, 1] but its derivative" }], "Text"], Cell[BoxData[ \(g \((t)\) = \(\(df\/dt\) \((t)\) = \(\(1\)\(-\)\) \[VerticalSeparator] t \[VerticalSeparator] \), \(-1\) \[LessEqual] t \[LessEqual] 1\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "belongs only ", Cell[BoxData[ \(C\^\((0)\)\)]], "[-1, 1], see Fig.1.1." }], "Text"], 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.73403 .51404 Metetra .554 .706 .942 r .73403 .51404 .7459 .52867 .77259 .52651 .76087 .51177 Metetra .505 .683 .946 r .76087 .51177 .77259 .52651 .79979 .52655 .78822 .5117 Metetra .443 .651 .948 r .78822 .5117 .79979 .52655 .82761 .52921 .81619 .51427 Metetra .366 .608 .942 r .81619 .51427 .82761 .52921 .85617 .53494 .8449 .51993 Metetra .272 .551 .927 r .8449 .51993 .85617 .53494 .88561 .54423 .87448 .52914 Metetra .163 .48 .898 r .87448 .52914 .88561 .54423 .91608 .5576 .9051 .54246 Metetra .042 .395 .853 r .9051 .54246 .91608 .5576 .94779 .57563 .93696 .56046 Metetra 0 .212 .699 r .31153 .51847 .32549 .53168 .34637 .5469 .33242 .53366 Metetra 0 .375 .83 r .33242 .53366 .34637 .5469 .36788 .55869 .35395 .54541 Metetra .143 .502 .909 r .35395 .54541 .36788 .55869 .38995 .56723 .37607 .5539 Metetra .294 .589 .945 r .37607 .5539 .38995 .56723 .41252 .57277 .39871 .55938 Metetra .406 .647 .956 r .39871 .55938 .41252 .57277 .43553 .57558 .42181 .56211 Metetra .488 .684 .954 r .42181 .56211 .43553 .57558 .45893 .57594 .4453 .56239 Metetra .546 .708 .947 r .4453 .56239 .45893 .57594 .48266 .57416 .46915 .56051 Metetra .587 .724 .938 r .46915 .56051 .48266 .57416 .50669 .57056 .4933 .55681 Metetra .617 .735 .93 r .4933 .55681 .50669 .57056 .53098 .56548 .51772 .55161 Metetra .637 .742 .924 r .51772 .55161 .53098 .56548 .55551 .55927 .54238 .54528 Metetra .649 .746 .92 r .54238 .54528 .55551 .55927 .58027 .55227 .56728 .53816 Metetra .655 .748 .918 r .56728 .53816 .58027 .55227 .60524 .54484 .5924 .53061 Metetra .655 .748 .918 r .5924 .53061 .60524 .54484 .63044 .53735 .61776 .52299 Metetra .649 .746 .92 r .61776 .52299 .63044 .53735 .65589 .53015 .64336 .51566 Metetra .637 .741 .924 r .64336 .51566 .65589 .53015 .6816 .52362 .66923 .509 Metetra .618 .734 .929 r .66923 .509 .6816 .52362 .70764 .51812 .69541 .50338 Metetra .591 .723 .935 r .69541 .50338 .70764 .51812 .73403 .51404 .72197 .49918 Metetra .554 .707 .942 r .72197 .49918 .73403 .51404 .76087 .51177 .74895 .49679 Metetra .505 .684 .947 r .74895 .49679 .76087 .51177 .78822 .5117 .77646 .49662 Metetra .443 .651 .948 r .77646 .49662 .78822 .5117 .81619 .51427 .80458 .49909 Metetra .366 .608 .942 r .80458 .49909 .81619 .51427 .8449 .51993 .83343 .50466 Metetra .271 .551 .927 r .83343 .50466 .8449 .51993 .87448 .52914 .86316 .51381 Metetra .162 .48 .898 r .86316 .51381 .87448 .52914 .9051 .54246 .89393 .52707 Metetra .042 .395 .852 r .89393 .52707 .9051 .54246 .93696 .56046 .92593 .54503 Metetra 0 .211 .698 r .29735 .50506 .31153 .51847 .33242 .53366 .31824 .52023 Metetra 0 .375 .83 r .31824 .52023 .33242 .53366 .35395 .54541 .33981 .53193 Metetra .142 .502 .909 r .33981 .53193 .35395 .54541 .37607 .5539 .36198 .54037 Metetra .294 .589 .945 r .36198 .54037 .37607 .5539 .39871 .55938 .38468 .54578 Metetra .406 .647 .956 r .38468 .54578 .39871 .55938 .42181 .56211 .40786 .54844 Metetra .487 .684 .954 r .40786 .54844 .42181 .56211 .4453 .56239 .43145 .54862 Metetra .546 .709 .947 r .43145 .54862 .4453 .56239 .46915 .56051 .45541 .54665 Metetra .587 .725 .938 r .45541 .54665 .46915 .56051 .4933 .55681 .47968 .54284 Metetra .617 .735 .931 r .47968 .54284 .4933 .55681 .51772 .55161 .50423 .53753 Metetra .637 .742 .924 r .50423 .53753 .51772 .55161 .54238 .54528 .52904 .53108 Metetra .649 .746 .92 r .52904 .53108 .54238 .54528 .56728 .53816 .55408 .52383 Metetra .655 .748 .918 r .55408 .52383 .56728 .53816 .5924 .53061 .57935 .51615 Metetra .655 .748 .918 r .57935 .51615 .5924 .53061 .61776 .52299 .60486 .5084 Metetra .649 .746 .92 r .60486 .5084 .61776 .52299 .64336 .51566 .63061 .50094 Metetra .637 .741 .924 r .63061 .50094 .64336 .51566 .66923 .509 .65664 .49415 Metetra .618 .734 .929 r .65664 .49415 .66923 .509 .69541 .50338 .68298 .4884 Metetra .591 .723 .936 r .68298 .4884 .69541 .50338 .72197 .49918 .70969 .48408 Metetra .554 .707 .942 r .70969 .48408 .72197 .49918 .74895 .49679 .73683 .48157 Metetra .505 .684 .947 r .73683 .48157 .74895 .49679 .77646 .49662 .76449 .48129 Metetra .443 .652 .948 r .76449 .48129 .77646 .49662 .80458 .49909 .79276 .48366 Metetra .365 .608 .942 r .79276 .48366 .80458 .49909 .83343 .50466 .82176 .48913 Metetra .271 .551 .927 r .82176 .48913 .83343 .50466 .86316 .51381 .85164 .49821 Metetra .162 .48 .897 r .85164 .49821 .86316 .51381 .89393 .52707 .88255 .51141 Metetra .041 .395 .852 r .88255 .51141 .89393 .52707 .92593 .54503 .9147 .52933 Metetra 0 .211 .698 r .28296 .49146 .29735 .50506 .31824 .52023 .30385 .5066 Metetra 0 .374 .83 r .30385 .5066 .31824 .52023 .33981 .53193 .32544 .51826 Metetra .142 .501 .909 r .32544 .51826 .33981 .53193 .36198 .54037 .34766 .52664 Metetra .294 .59 .945 r .34766 .52664 .36198 .54037 .38468 .54578 .37043 .53198 Metetra .406 .647 .956 r .37043 .53198 .38468 .54578 .40786 .54844 .39369 .53456 Metetra .487 .685 .954 r .39369 .53456 .40786 .54844 .43145 .54862 .41738 .53465 Metetra .545 .709 .947 r .41738 .53465 .43145 .54862 .45541 .54665 .44144 .53257 Metetra .587 .725 .939 r .44144 .53257 .45541 .54665 .47968 .54284 .46584 .52865 Metetra .617 .735 .931 r .46584 .52865 .47968 .54284 .50423 .53753 .49052 .52322 Metetra .637 .742 .925 r .49052 .52322 .50423 .53753 .52904 .53108 .51547 .51665 Metetra .649 .746 .92 r .51547 .51665 .52904 .53108 .55408 .52383 .54066 .50927 Metetra .655 .748 .918 r .54066 .50927 .55408 .52383 .57935 .51615 .56608 .50146 Metetra .655 .748 .918 r .56608 .50146 .57935 .51615 .60486 .5084 .59174 .49357 Metetra .649 .746 .92 r .59174 .49357 .60486 .5084 .63061 .50094 .61765 .48598 Metetra .637 .742 .924 r .61765 .48598 .63061 .50094 .65664 .49415 .64384 .47906 Metetra .618 .734 .93 r .64384 .47906 .65664 .49415 .68298 .4884 .67034 .47318 Metetra .591 .723 .936 r .67034 .47318 .68298 .4884 .70969 .48408 .6972 .46873 Metetra .554 .707 .942 r .6972 .46873 .70969 .48408 .73683 .48157 .7245 .4661 Metetra .505 .684 .947 r .7245 .4661 .73683 .48157 .76449 .48129 .75231 .4657 Metetra .443 .652 .948 r .75231 .4657 .76449 .48129 .79276 .48366 .78073 .46797 Metetra .365 .608 .943 r .78073 .46797 .79276 .48366 .82176 .48913 .80989 .47335 Metetra .271 .551 .927 r .80989 .47335 .82176 .48913 .85164 .49821 .83991 .48234 Metetra .161 .48 .897 r .83991 .48234 .85164 .49821 .88255 .51141 .87097 .49548 Metetra .041 .395 .852 r .87097 .49548 .88255 .51141 .9147 .52933 .90326 .51336 Metetra 0 .211 .697 r .26833 .47766 .28296 .49146 .30385 .5066 .28923 .49276 Metetra 0 .374 .83 r .28923 .49276 .30385 .5066 .32544 .51826 .31084 .50437 Metetra .141 .501 .908 r .31084 .50437 .32544 .51826 .34766 .52664 .3331 .5127 Metetra .293 .59 .945 r .3331 .5127 .34766 .52664 .37043 .53198 .35594 .51797 Metetra .406 .647 .956 r .35594 .51797 .37043 .53198 .39369 .53456 .37928 .52046 Metetra .487 .685 .954 r .37928 .52046 .39369 .53456 .41738 .53465 .40307 .52045 Metetra .545 .709 .947 r .40307 .52045 .41738 .53465 .44144 .53257 .42725 .51827 Metetra .587 .725 .939 r .42725 .51827 .44144 .53257 .46584 .52865 .45176 .51424 Metetra .616 .735 .931 r .45176 .51424 .46584 .52865 .49052 .52322 .47658 .50869 Metetra .637 .742 .925 r .47658 .50869 .49052 .52322 .51547 .51665 .50167 .50199 Metetra .649 .746 .921 r .50167 .50199 .51547 .51665 .54066 .50927 .52701 .49448 Metetra .655 .748 .918 r .52701 .49448 .54066 .50927 .56608 .50146 .55258 .48653 Metetra .655 .748 .918 r .55258 .48653 .56608 .50146 .59174 .49357 .5784 .4785 Metetra .649 .746 .92 r .5784 .4785 .59174 .49357 .61765 .48598 .60447 .47078 Metetra .637 .742 .924 r .60447 .47078 .61765 .48598 .64384 .47906 .63081 .46372 Metetra .618 .735 .93 r .63081 .46372 .64384 .47906 .67034 .47318 .65747 .4577 Metetra .59 .723 .936 r .65747 .4577 .67034 .47318 .6972 .46873 .6845 .45312 Metetra .553 .707 .942 r .6845 .45312 .6972 .46873 .7245 .4661 .71195 .45037 Metetra .505 .684 .947 r .71195 .45037 .7245 .4661 .75231 .4657 .73992 .44985 Metetra .443 .652 .948 r .73992 .44985 .75231 .4657 .78073 .46797 .76849 .45201 Metetra .365 .608 .943 r .76849 .45201 .78073 .46797 .80989 .47335 .7978 .45729 Metetra .271 .551 .927 r .7978 .45729 .80989 .47335 .83991 .48234 .82797 .4662 Metetra .161 .48 .897 r .82797 .4662 .83991 .48234 .87097 .49548 .85918 .47927 Metetra .04 .394 .852 r .85918 .47927 .87097 .49548 .90326 .51336 .89161 .4971 Metetra 0 .21 .697 r .25347 .46365 .26833 .47766 .28923 .49276 .27437 .47872 Metetra 0 .374 .829 r .27437 .47872 .28923 .49276 .31084 .50437 .29601 .49028 Metetra .141 .501 .908 r .29601 .49028 .31084 .50437 .3331 .5127 .31831 .49854 Metetra .293 .59 .945 r .31831 .49854 .3331 .5127 .35594 .51797 .34121 .50374 Metetra .406 .647 .956 r .34121 .50374 .35594 .51797 .37928 .52046 .36464 .50614 Metetra .487 .685 .954 r .36464 .50614 .37928 .52046 .40307 .52045 .38852 .50604 Metetra .545 .709 .947 r .38852 .50604 .40307 .52045 .42725 .51827 .41281 .50375 Metetra .587 .725 .939 r .41281 .50375 .42725 .51827 .45176 .51424 .43745 .4996 Metetra .616 .736 .931 r .43745 .4996 .45176 .51424 .47658 .50869 .46241 .49393 Metetra .637 .742 .925 r .46241 .49393 .47658 .50869 .50167 .50199 .48764 .48709 Metetra .649 .746 .921 r .48764 .48709 .50167 .50199 .52701 .49448 .51312 .47945 Metetra .655 .748 .919 r .51312 .47945 .52701 .49448 .55258 .48653 .53885 .47136 Metetra .655 .748 .919 r .53885 .47136 .55258 .48653 .5784 .4785 .56483 .46319 Metetra .649 .746 .921 r .56483 .46319 .5784 .4785 .60447 .47078 .59106 .45532 Metetra .637 .742 .924 r .59106 .45532 .60447 .47078 .63081 .46372 .61756 .44812 Metetra .618 .735 .93 r .61756 .44812 .63081 .46372 .65747 .4577 .64439 .44197 Metetra .59 .724 .936 r .64439 .44197 .65747 .4577 .6845 .45312 .67157 .43725 Metetra .553 .707 .942 r .67157 .43725 .6845 .45312 .71195 .45037 .69918 .43437 Metetra .505 .684 .947 r .69918 .43437 .71195 .45037 .73992 .44985 .7273 .43373 Metetra .442 .652 .948 r .7273 .43373 .73992 .44985 .76849 .45201 .75603 .43577 Metetra .365 .608 .943 r .75603 .43577 .76849 .45201 .7978 .45729 .78549 .44096 Metetra .27 .551 .927 r .78549 .44096 .7978 .45729 .82797 .4662 .81581 .44977 Metetra .161 .479 .897 r .81581 .44977 .82797 .4662 .85918 .47927 .84716 .46277 Metetra .04 .394 .852 r .84716 .46277 .85918 .47927 .89161 .4971 .87975 .48056 Metetra 0 .21 .697 r .23838 .44944 .25347 .46365 .27437 .47872 .25927 .46446 Metetra 0 .374 .829 r .25927 .46446 .27437 .47872 .29601 .49028 .28093 .47597 Metetra .14 .501 .908 r .28093 .47597 .29601 .49028 .31831 .49854 .30328 .48417 Metetra .293 .59 .945 r .30328 .48417 .31831 .49854 .34121 .50374 .32624 .48929 Metetra .405 .648 .956 r .32624 .48929 .34121 .50374 .36464 .50614 .34975 .4916 Metetra .487 .685 .954 r .34975 .4916 .36464 .50614 .38852 .50604 .37373 .49139 Metetra .545 .709 .947 r .37373 .49139 .38852 .50604 .41281 .50375 .39813 .48899 Metetra .587 .725 .939 r .39813 .48899 .41281 .50375 .43745 .4996 .4229 .48472 Metetra .616 .736 .931 r .4229 .48472 .43745 .4996 .46241 .49393 .44799 .47892 Metetra .636 .743 .925 r .44799 .47892 .46241 .49393 .48764 .48709 .47336 .47195 Metetra .649 .747 .921 r .47336 .47195 .48764 .48709 .51312 .47945 .499 .46417 Metetra .655 .749 .919 r .499 .46417 .51312 .47945 .53885 .47136 .52489 .45594 Metetra .655 .749 .919 r .52489 .45594 .53885 .47136 .56483 .46319 .55102 .44763 Metetra .649 .747 .921 r .55102 .44763 .56483 .46319 .59106 .45532 .57741 .43961 Metetra .637 .742 .925 r .57741 .43961 .59106 .45532 .61756 .44812 .60408 .43227 Metetra .618 .735 .93 r .60408 .43227 .61756 .44812 .64439 .44197 .63107 .42597 Metetra .59 .724 .936 r .63107 .42597 .64439 .44197 .67157 .43725 .65841 .42112 Metetra .553 .707 .943 r .65841 .42112 .67157 .43725 .69918 .43437 .68618 .4181 Metetra .505 .684 .947 r .68618 .4181 .69918 .43437 .7273 .43373 .71446 .41734 Metetra .442 .652 .948 r .71446 .41734 .7273 .43373 .75603 .43577 .74335 .41926 Metetra .364 .608 .943 r .74335 .41926 .75603 .43577 .78549 .44096 .77296 .42434 Metetra .27 .551 .927 r .77296 .42434 .78549 .44096 .81581 .44977 .80343 .43306 Metetra .16 .479 .897 r .80343 .43306 .81581 .44977 .84716 .46277 .83493 .44598 Metetra .039 .394 .852 r .83493 .44598 .84716 .46277 .87975 .48056 .86766 .46371 Metetra 0 .21 .696 r .22304 .43501 .23838 .44944 .25927 .46446 .24393 .44999 Metetra 0 .374 .829 r .24393 .44999 .25927 .46446 .28093 .47597 .26561 .46144 Metetra .14 .501 .908 r .26561 .46144 .28093 .47597 .30328 .48417 .28799 .46957 Metetra .292 .59 .945 r .28799 .46957 .30328 .48417 .32624 .48929 .31102 .4746 Metetra .405 .648 .956 r .31102 .4746 .32624 .48929 .34975 .4916 .33461 .47682 Metetra .487 .685 .955 r .33461 .47682 .34975 .4916 .37373 .49139 .35869 .47652 Metetra .545 .709 .948 r .35869 .47652 .37373 .49139 .39813 .48899 .3832 .474 Metetra .587 .725 .939 r .3832 .474 .39813 .48899 .4229 .48472 .40809 .4696 Metetra .616 .736 .931 r .40809 .4696 .4229 .48472 .44799 .47892 .43332 .46367 Metetra .636 .743 .925 r .43332 .46367 .44799 .47892 .47336 .47195 .45884 .45656 Metetra .649 .747 .921 r .45884 .45656 .47336 .47195 .499 .46417 .48463 .44864 Metetra .655 .749 .919 r .48463 .44864 .499 .46417 .52489 .45594 .51067 .44026 Metetra .655 .749 .919 r .51067 .44026 .52489 .45594 .55102 .44763 .53697 .4318 Metetra .649 .747 .921 r .53697 .4318 .55102 .44763 .57741 .43961 .56353 .42364 Metetra .637 .742 .925 r .56353 .42364 .57741 .43961 .60408 .43227 .59036 .41614 Metetra .618 .735 .93 r .59036 .41614 .60408 .43227 .63107 .42597 .61751 .4097 Metetra .59 .724 .936 r .61751 .4097 .63107 .42597 .65841 .42112 .64502 .4047 Metetra .553 .708 .943 r .64502 .4047 .65841 .42112 .68618 .4181 .67295 .40155 Metetra .505 .684 .947 r .67295 .40155 .68618 .4181 .71446 .41734 .70139 .40065 Metetra .442 .652 .949 r .70139 .40065 .71446 .41734 .74335 .41926 .73043 .40246 Metetra .364 .609 .943 r .73043 .40246 .74335 .41926 .77296 .42434 .76019 .40742 Metetra .27 .551 .927 r .76019 .40742 .77296 .42434 .80343 .43306 .79081 .41605 Metetra .16 .479 .898 r .79081 .41605 .80343 .43306 .83493 .44598 .82246 .42889 Metetra .039 .394 .852 r .82246 .42889 .83493 .44598 .86766 .46371 .85534 .44655 Metetra 0 .21 .696 r .20745 .42035 .22304 .43501 .24393 .44999 .22833 .43529 Metetra 0 .374 .829 r .22833 .43529 .24393 .44999 .26561 .46144 .25002 .44668 Metetra .139 .501 .908 r .25002 .44668 .26561 .46144 .28799 .46957 .27245 .45474 Metetra .292 .59 .945 r .27245 .45474 .28799 .46957 .31102 .4746 .29554 .45969 Metetra .405 .648 .956 r .29554 .45969 .31102 .4746 .33461 .47682 .31921 .46181 Metetra .486 .685 .955 r .31921 .46181 .33461 .47682 .35869 .47652 .34339 .4614 Metetra .545 .71 .948 r .34339 .4614 .35869 .47652 .3832 .474 .36801 .45876 Metetra .587 .726 .939 r .36801 .45876 .3832 .474 .40809 .4696 .39303 .45424 Metetra .616 .736 .932 r .39303 .45424 .40809 .4696 .43332 .46367 .41839 .44817 Metetra .636 .743 .925 r .41839 .44817 .43332 .46367 .45884 .45656 .44406 .44092 Metetra .649 .747 .921 r .44406 .44092 .45884 .45656 .48463 .44864 .47001 .43285 Metetra .655 .749 .919 r .47001 .43285 .48463 .44864 .51067 .44026 .49621 .42432 Metetra .655 .749 .919 r .49621 .42432 .51067 .44026 .53697 .4318 .52267 .41571 Metetra .649 .747 .921 r .52267 .41571 .53697 .4318 .56353 .42364 .54939 .40739 Metetra .637 .743 .925 r .54939 .40739 .56353 .42364 .59036 .41614 .57639 .39974 Metetra .618 .735 .93 r .57639 .39974 .59036 .41614 .61751 .4097 .60371 .39315 Metetra .59 .724 .937 r .60371 .39315 .61751 .4097 .64502 .4047 .63138 .38801 Metetra .553 .708 .943 r .63138 .38801 .64502 .4047 .67295 .40155 .65948 .38471 Metetra .504 .685 .948 r .65948 .38471 .67295 .40155 .70139 .40065 .68808 .38368 Metetra .442 .652 .949 r .68808 .38368 .70139 .40065 .73043 .40246 .71727 .38536 Metetra .364 .609 .943 r .71727 .38536 .73043 .40246 .76019 .40742 .74719 .3902 Metetra .269 .551 .927 r .74719 .3902 .76019 .40742 .79081 .41605 .77796 .39873 Metetra .159 .479 .898 r .77796 .39873 .79081 .41605 .82246 .42889 .80976 .41148 Metetra .038 .394 .852 r .80976 .41148 .82246 .42889 .85534 .44655 .84278 .42908 Metetra 0 .209 .695 r .19161 .40547 .20745 .42035 .22833 .43529 .21247 .42036 Metetra 0 .373 .828 r .21247 .42036 .22833 .43529 .25002 .44668 .23418 .43169 Metetra .139 .501 .908 r .23418 .43169 .25002 .44668 .27245 .45474 .25665 .43967 Metetra .292 .59 .945 r .25665 .43967 .27245 .45474 .29554 .45969 .27979 .44453 Metetra .405 .648 .957 r .27979 .44453 .29554 .45969 .31921 .46181 .30354 .44656 Metetra .486 .685 .955 r .30354 .44656 .31921 .46181 .34339 .4614 .32782 .44603 Metetra .545 .71 .948 r .32782 .44603 .34339 .4614 .36801 .45876 .35256 .44327 Metetra .587 .726 .939 r .35256 .44327 .36801 .45876 .39303 .45424 .37771 .43862 Metetra .616 .736 .932 r .37771 .43862 .39303 .45424 .41839 .44817 .40321 .43241 Metetra .636 .743 .925 r .40321 .43241 .41839 .44817 .44406 .44092 .42902 .42502 Metetra .649 .747 .921 r .42902 .42502 .44406 .44092 .47001 .43285 .45512 .41679 Metetra .655 .749 .919 r .45512 .41679 .47001 .43285 .49621 .42432 .48149 .40811 Metetra .655 .749 .919 r .48149 .40811 .49621 .42432 .52267 .41571 .50811 .39934 Metetra .649 .747 .921 r .50811 .39934 .52267 .41571 .54939 .40739 .535 .39087 Metetra .637 .743 .925 r .535 .39087 .54939 .40739 .57639 .39974 .56217 .38306 Metetra .618 .735 .93 r .56217 .38306 .57639 .39974 .60371 .39315 .58965 .37632 Metetra .59 .724 .937 r .58965 .37632 .60371 .39315 .63138 .38801 .61749 .37102 Metetra .553 .708 .943 r .61749 .37102 .63138 .38801 .65948 .38471 .64576 .36758 Metetra .504 .685 .948 r .64576 .36758 .65948 .38471 .68808 .38368 .67452 .36641 Metetra .442 .652 .949 r .67452 .36641 .68808 .38368 .71727 .38536 .70387 .36795 Metetra .364 .609 .943 r .70387 .36795 .71727 .38536 .74719 .3902 .73394 .37267 Metetra .269 .551 .927 r .73394 .37267 .74719 .3902 .77796 .39873 .76486 .38109 Metetra .159 .479 .898 r .76486 .38109 .77796 .39873 .80976 .41148 .79681 .39375 Metetra .038 .394 .852 r .79681 .39375 .80976 .41148 .84278 .42908 .82997 .41128 Metetra 0 .209 .695 r .1755 .39036 .19161 .40547 .21247 .42036 .19635 .4052 Metetra 0 .373 .828 r .19635 .4052 .21247 .42036 .23418 .43169 .21807 .41646 Metetra .139 .501 .908 r .21807 .41646 .23418 .43169 .25665 .43967 .24057 .42436 Metetra .291 .59 .945 r .24057 .42436 .25665 .43967 .27979 .44453 .26378 .42913 Metetra .404 .648 .957 r .26378 .42913 .27979 .44453 .30354 .44656 .2876 .43105 Metetra .486 .686 .955 r .2876 .43105 .30354 .44656 .32782 .44603 .31198 .43041 Metetra .545 .71 .948 r .31198 .43041 .32782 .44603 .35256 .44327 .33683 .42752 Metetra .587 .726 .94 r .33683 .42752 .35256 .44327 .37771 .43862 .36211 .42273 Metetra .616 .736 .932 r .36211 .42273 .37771 .43862 .40321 .43241 .38775 .41639 Metetra .636 .743 .926 r .38775 .41639 .40321 .43241 .42902 .42502 .41371 .40884 Metetra .649 .747 .921 r .41371 .40884 .42902 .42502 .45512 .41679 .43997 .40046 Metetra .655 .749 .919 r .43997 .40046 .45512 .41679 .48149 .40811 .4665 .39162 Metetra .655 .749 .919 r .4665 .39162 .48149 .40811 .50811 .39934 .49329 .38269 Metetra .649 .747 .921 r .49329 .38269 .50811 .39934 .535 .39087 .52035 .37406 Metetra .637 .743 .925 r .52035 .37406 .535 .39087 .56217 .38306 .54769 .36609 Metetra .618 .736 .931 r .54769 .36609 .56217 .38306 .58965 .37632 .57534 .35918 Metetra .59 .724 .937 r .57534 .35918 .58965 .37632 .61749 .37102 .60335 .35373 Metetra .553 .708 .943 r .60335 .35373 .61749 .37102 .64576 .36758 .63178 .35014 Metetra .504 .685 .948 r .63178 .35014 .64576 .36758 .67452 .36641 .6607 .34882 Metetra .442 .653 .949 r .6607 .34882 .67452 .36641 .70387 .36795 .69021 .35023 Metetra .363 .609 .943 r .69021 .35023 .70387 .36795 .73394 .37267 .72044 .35483 Metetra .269 .551 .927 r .72044 .35483 .73394 .37267 .76486 .38109 .75151 .36313 Metetra .159 .479 .898 r .75151 .36313 .76486 .38109 .79681 .39375 .7836 .37569 Metetra .038 .394 .852 r .7836 .37569 .79681 .39375 .82997 .41128 .81691 .39314 Metetra 0 .209 .695 r .15913 .37501 .1755 .39036 .19635 .4052 .17996 .38979 Metetra 0 .373 .828 r .17996 .38979 .19635 .4052 .21807 .41646 .20169 .40098 Metetra .138 .501 .908 r .20169 .40098 .21807 .41646 .24057 .42436 .22422 .4088 Metetra .291 .59 .945 r .22422 .4088 .24057 .42436 .26378 .42913 .24748 .41348 Metetra .404 .648 .957 r .24748 .41348 .26378 .42913 .2876 .43105 .27139 .41529 Metetra .486 .686 .955 r .27139 .41529 .2876 .43105 .31198 .43041 .29586 .41453 Metetra .545 .71 .948 r .29586 .41453 .31198 .43041 .33683 .42752 .32083 .41151 Metetra .587 .726 .94 r .32083 .41151 .33683 .42752 .36211 .42273 .34623 .40658 Metetra .616 .737 .932 r .34623 .40658 .36211 .42273 .38775 .41639 .37201 .40009 Metetra .636 .743 .926 r .37201 .40009 .38775 .41639 .41371 .40884 .39813 .39239 Metetra .649 .748 .922 r .39813 .39239 .41371 .40884 .43997 .40046 .42455 .38385 Metetra .655 .75 .919 r .42455 .38385 .43997 .40046 .4665 .39162 .45124 .37485 Metetra .655 .75 .919 r .45124 .37485 .4665 .39162 .49329 .38269 .4782 .36575 Metetra .649 .747 .921 r .4782 .36575 .49329 .38269 .52035 .37406 .50543 .35695 Metetra .637 .743 .925 r .50543 .35695 .52035 .37406 .54769 .36609 .53294 .34882 Metetra .618 .736 .931 r .53294 .34882 .54769 .36609 .57534 .35918 .56076 .34175 Metetra .59 .725 .937 r .56076 .34175 .57534 .35918 .60335 .35373 .58894 .33613 Metetra .553 .708 .943 r .58894 .33613 .60335 .35373 .63178 .35014 .61754 .33238 Metetra .504 .685 .948 r .61754 .33238 .63178 .35014 .6607 .34882 .64662 .33092 Metetra .441 .653 .949 r .64662 .33092 .6607 .34882 .69021 .35023 .6763 .33218 Metetra .363 .609 .943 r .6763 .33218 .69021 .35023 .72044 .35483 .70668 .33665 Metetra .268 .551 .927 r .70668 .33665 .72044 .35483 .75151 .36313 .7379 .34483 Metetra .158 .479 .898 r .7379 .34483 .75151 .36313 .7836 .37569 .77014 .35728 Metetra .037 .394 .852 r .77014 .35728 .7836 .37569 .81691 .39314 .8036 .37465 Metetra 0 .209 .694 r .14247 .35942 .15913 .37501 .17996 .38979 .16329 .37413 Metetra 0 .373 .828 r .16329 .37413 .17996 .38979 .20169 .40098 .18502 .38525 Metetra .138 .501 .908 r .18502 .38525 .20169 .40098 .22422 .4088 .20759 .39298 Metetra .291 .59 .945 r .20759 .39298 .22422 .4088 .24748 .41348 .2309 .39756 Metetra .404 .648 .957 r .2309 .39756 .24748 .41348 .27139 .41529 .25489 .39926 Metetra .486 .686 .955 r .25489 .39926 .27139 .41529 .29586 .41453 .27946 .39838 Metetra .545 .71 .948 r .27946 .39838 .29586 .41453 .32083 .41151 .30454 .39523 Metetra .587 .726 .94 r .30454 .39523 .32083 .41151 .34623 .40658 .33007 .39016 Metetra .616 .737 .932 r .33007 .39016 .34623 .40658 .37201 .40009 .356 .38351 Metetra .636 .744 .926 r .356 .38351 .37201 .40009 .39813 .39239 .38226 .37565 Metetra .649 .748 .922 r .38226 .37565 .39813 .39239 .42455 .38385 .40884 .36695 Metetra .655 .75 .92 r .40884 .36695 .42455 .38385 .45124 .37485 .4357 .35778 Metetra .655 .75 .92 r .4357 .35778 .45124 .37485 .4782 .36575 .46283 .34852 Metetra .649 .748 .922 r .46283 .34852 .4782 .36575 .50543 .35695 .49023 .33954 Metetra .637 .743 .925 r .49023 .33954 .50543 .35695 .53294 .34882 .51792 .33124 Metetra .617 .736 .931 r .51792 .33124 .53294 .34882 .56076 .34175 .54591 .324 Metetra .59 .725 .937 r .54591 .324 .56076 .34175 .58894 .33613 .57426 .31822 Metetra .553 .708 .943 r .57426 .31822 .58894 .33613 .61754 .33238 .60303 .31431 Metetra .504 .685 .948 r .60303 .31431 .61754 .33238 .64662 .33092 .63228 .31268 Metetra .441 .653 .949 r .63228 .31268 .64662 .33092 .6763 .33218 .66211 .3138 Metetra .363 .609 .944 r .66211 .3138 .6763 .33218 .70668 .33665 .69265 .31813 Metetra .268 .551 .927 r .69265 .31813 .70668 .33665 .7379 .34483 .72402 .32618 Metetra .158 .479 .898 r .72402 .32618 .7379 .34483 .77014 .35728 .75641 .33853 Metetra .037 .394 .852 r .75641 .33853 .77014 .35728 .8036 .37465 .79001 .3558 Metetra 0 .208 .694 r .12554 .34357 .14247 .35942 .16329 .37413 .14632 .35822 Metetra 0 .373 .828 r .14632 .35822 .16329 .37413 .18502 .38525 .16806 .36927 Metetra .137 .501 .908 r .16806 .36927 .18502 .38525 .20759 .39298 .19066 .37691 Metetra .29 .59 .946 r .19066 .37691 .20759 .39298 .2309 .39756 .21403 .38138 Metetra .404 .648 .957 r .21403 .38138 .2309 .39756 .25489 .39926 .23809 .38296 Metetra .486 .686 .955 r .23809 .38296 .25489 .39926 .27946 .39838 .26276 .38195 Metetra .545 .711 .948 r .26276 .38195 .27946 .39838 .30454 .39523 .28795 .37866 Metetra .587 .726 .94 r .28795 .37866 .30454 .39523 .33007 .39016 .31362 .37344 Metetra .616 .737 .932 r .31362 .37344 .33007 .39016 .356 .38351 .33968 .36664 Metetra .636 .744 .926 r .33968 .36664 .356 .38351 .38226 .37565 .3661 .35862 Metetra .649 .748 .922 r .3661 .35862 .38226 .37565 .40884 .36695 .39284 .34975 Metetra .655 .75 .92 r .39284 .34975 .40884 .36695 .4357 .35778 .41987 .34041 Metetra .655 .75 .92 r .41987 .34041 .4357 .35778 .46283 .34852 .44718 .33097 Metetra .649 .748 .922 r .44718 .33097 .46283 .34852 .49023 .33954 .47475 .32182 Metetra .637 .743 .926 r .47475 .32182 .49023 .33954 .51792 .33124 .50261 .31334 Metetra .617 .736 .931 r .50261 .31334 .51792 .33124 .54591 .324 .53078 .30593 Metetra .59 .725 .937 r .53078 .30593 .54591 .324 .57426 .31822 .55931 .29998 Metetra .553 .709 .944 r .55931 .29998 .57426 .31822 .60303 .31431 .58824 .2959 Metetra .504 .685 .948 r .58824 .2959 .60303 .31431 .63228 .31268 .61766 .29411 Metetra .441 .653 .949 r .61766 .29411 .63228 .31268 .66211 .3138 .64765 .29507 Metetra .363 .609 .944 r .64765 .29507 .66211 .3138 .69265 .31813 .67834 .29926 Metetra .268 .552 .928 r .67834 .29926 .69265 .31813 .72402 .32618 .70987 .30718 Metetra .158 .479 .898 r .70987 .30718 .72402 .32618 .75641 .33853 .74241 .31941 Metetra .036 .394 .852 r .74241 .31941 .75641 .33853 .79001 .3558 .77615 .33659 Metetra 0 .208 .694 r .10831 .32747 .12554 .34357 .14632 .35822 .12907 .34205 Metetra 0 .373 .828 r .12907 .34205 .14632 .35822 .16806 .36927 .1508 .35301 Metetra .137 .501 .908 r .1508 .35301 .16806 .36927 .19066 .37691 .17343 .36056 Metetra .29 .59 .946 r .17343 .36056 .19066 .37691 .21403 .38138 .19685 .36492 Metetra .404 .649 .957 r .19685 .36492 .21403 .38138 .23809 .38296 .22099 .36638 Metetra .486 .686 .956 r .22099 .36638 .23809 .38296 .26276 .38195 .24575 .36524 Metetra .544 .711 .949 r .24575 .36524 .26276 .38195 .28795 .37866 .27106 .36181 Metetra .587 .727 .94 r .27106 .36181 .28795 .37866 .31362 .37344 .29686 .35644 Metetra .616 .737 .932 r .29686 .35644 .31362 .37344 .33968 .36664 .32307 .34948 Metetra .636 .744 .926 r .32307 .34948 .33968 .36664 .3661 .35862 .34965 .34129 Metetra .649 .748 .922 r .34965 .34129 .3661 .35862 .39284 .34975 .37655 .33225 Metetra .655 .75 .92 r .37655 .33225 .39284 .34975 .41987 .34041 .40375 .32273 Metetra .655 .75 .92 r .40375 .32273 .41987 .34041 .44718 .33097 .43122 .31311 Metetra .649 .748 .922 r .43122 .31311 .44718 .33097 .47475 .32182 .45898 .30378 Metetra .637 .744 .926 r .45898 .30378 .47475 .32182 .50261 .31334 .48701 .29512 Metetra .617 .736 .931 r .48701 .29512 .50261 .31334 .53078 .30593 .51536 .28753 Metetra .59 .725 .937 r .51536 .28753 .53078 .30593 .55931 .29998 .54406 .2814 Metetra .553 .709 .944 r .54406 .2814 .55931 .29998 .58824 .2959 .57316 .27715 Metetra .504 .686 .948 r .57316 .27715 .58824 .2959 .61766 .29411 .60275 .27519 Metetra .441 .653 .949 r .60275 .27519 .61766 .29411 .64765 .29507 .63291 .27599 Metetra .363 .609 .944 r .63291 .27599 .64765 .29507 .67834 .29926 .66375 .28003 Metetra .268 .552 .928 r .66375 .28003 .67834 .29926 .70987 .30718 .69543 .28781 Metetra .157 .479 .898 r .69543 .28781 .70987 .30718 .74241 .31941 .72811 .29991 Metetra .036 .394 .852 r .72811 .29991 .74241 .31941 .77615 .33659 .762 .31699 Metetra 0 .208 .693 r .09079 .3111 .10831 .32747 .12907 .34205 .11151 .32561 Metetra 0 .373 .827 r .11151 .32561 .12907 .34205 .1508 .35301 .13324 .33648 Metetra .137 .501 .908 r .13324 .33648 .1508 .35301 .17343 .36056 .15589 .34393 Metetra .29 .59 .946 r .15589 .34393 .17343 .36056 .19685 .36492 .17937 .34818 Metetra .403 .649 .957 r .17937 .34818 .19685 .36492 .22099 .36638 .20358 .34952 Metetra .486 .686 .956 r .20358 .34952 .22099 .36638 .24575 .36524 .22844 .34824 Metetra .544 .711 .949 r .22844 .34824 .24575 .36524 .27106 .36181 .25386 .34466 Metetra .586 .727 .94 r .25386 .34466 .27106 .36181 .29686 .35644 .27979 .33914 Metetra .616 .737 .933 r .27979 .33914 .29686 .35644 .32307 .34948 .30614 .33201 Metetra .636 .744 .926 r .30614 .33201 .32307 .34948 .34965 .34129 .33288 .32365 Metetra .649 .748 .922 r .33288 .32365 .34965 .34129 .37655 .33225 .35994 .31443 Metetra .655 .75 .92 r .35994 .31443 .37655 .33225 .40375 .32273 .38732 .30473 Metetra .655 .75 .92 r .38732 .30473 .40375 .32273 .43122 .31311 .41497 .29493 Metetra .649 .748 .922 r .41497 .29493 .43122 .31311 .45898 .30378 .4429 .28541 Metetra .637 .744 .926 r .4429 .28541 .45898 .30378 .48701 .29512 .47112 .27656 Metetra .617 .737 .931 r .47112 .27656 .48701 .29512 .51536 .28753 .49964 .26878 Metetra .59 .725 .938 r .49964 .26878 .51536 .28753 .54406 .2814 .52852 .26247 Metetra .553 .709 .944 r .52852 .26247 .54406 .2814 .57316 .27715 .55779 .25804 Metetra .504 .686 .949 r .55779 .25804 .57316 .27715 .60275 .27519 .58755 .25592 Metetra .441 .653 .95 r .58755 .25592 .60275 .27519 .63291 .27599 .61787 .25655 Metetra .362 .609 .944 r .61787 .25655 .63291 .27599 .66375 .28003 .64887 .26042 Metetra .267 .552 .928 r .64887 .26042 .66375 .28003 .69543 .28781 .6807 .26806 Metetra .157 .479 .898 r .6807 .26806 .69543 .28781 .72811 .29991 .71353 .28003 Metetra .036 .394 .852 r .71353 .28003 .72811 .29991 .762 .31699 .74755 .29699 Metetra 0 .208 .693 r .07295 .29446 .09079 .3111 .11151 .32561 .09364 .30889 Metetra 0 .372 .827 r .09364 .30889 .11151 .32561 .13324 .33648 .11537 .31968 Metetra .136 .501 .908 r .11537 .31968 .13324 .33648 .15589 .34393 .13804 .32702 Metetra .29 .59 .946 r .13804 .32702 .15589 .34393 .17937 .34818 .16156 .33115 Metetra .403 .649 .957 r .16156 .33115 .17937 .34818 .20358 .34952 .18584 .33236 Metetra .485 .687 .956 r .18584 .33236 .20358 .34952 .22844 .34824 .2108 .33094 Metetra .544 .711 .949 r .2108 .33094 .22844 .34824 .25386 .34466 .23634 .32721 Metetra .586 .727 .94 r .23634 .32721 .25386 .34466 .27979 .33914 .2624 .32152 Metetra .616 .738 .933 r .2624 .32152 .27979 .33914 .30614 .33201 .2889 .31423 Metetra .636 .744 .926 r .2889 .31423 .30614 .33201 .33288 .32365 .31579 .30569 Metetra .649 .748 .922 r .31579 .30569 .33288 .32365 .35994 .31443 .34303 .29629 Metetra .655 .75 .92 r .34303 .29629 .35994 .31443 .38732 .30473 .37057 .2864 Metetra .655 .75 .92 r .37057 .2864 .38732 .30473 .41497 .29493 .3984 .27641 Metetra .649 .748 .922 r .3984 .27641 .41497 .29493 .4429 .28541 .42651 .2667 Metetra .637 .744 .926 r .42651 .2667 .4429 .28541 .47112 .27656 .45491 .25766 Metetra .617 .737 .931 r .45491 .25766 .47112 .27656 .49964 .26878 .48362 .24969 Metetra .59 .725 .938 r .48362 .24969 .49964 .26878 .52852 .26247 .51267 .24319 Metetra .553 .709 .944 r .51267 .24319 .52852 .26247 .55779 .25804 .54212 .23857 Metetra .504 .686 .949 r .54212 .23857 .55779 .25804 .58755 .25592 .57204 .23627 Metetra .441 .653 .95 r .57204 .23627 .58755 .25592 .61787 .25655 .60252 .23673 Metetra .362 .609 .944 r .60252 .23673 .61787 .25655 .64887 .26042 .63369 .24044 Metetra .267 .552 .928 r .63369 .24044 .64887 .26042 .6807 .26806 .66567 .24792 Metetra .157 .479 .898 r .66567 .24792 .6807 .26806 .71353 .28003 .69864 .25975 Metetra .035 .393 .852 r .69864 .25975 .71353 .28003 .74755 .29699 .7328 .27659 Metetra 0 .207 .693 r .05481 .27754 .07295 .29446 .09364 .30889 .07545 .29189 Metetra 0 .372 .827 r .07545 .29189 .09364 .30889 .11537 .31968 .09717 .30258 Metetra .136 .501 .908 r .09717 .30258 .11537 .31968 .13804 .32702 .11986 .30981 Metetra .289 .59 .946 r .11986 .30981 .13804 .32702 .16156 .33115 .14343 .31383 Metetra .403 .649 .957 r .14343 .31383 .16156 .33115 .18584 .33236 .16778 .3149 Metetra .485 .687 .956 r .16778 .3149 .18584 .33236 .2108 .33094 .19283 .31333 Metetra .544 .711 .949 r .19283 .31333 .2108 .33094 .23634 .32721 .21849 .30945 Metetra .586 .727 .941 r .21849 .30945 .23634 .32721 .2624 .32152 .24468 .30359 Metetra .616 .738 .933 r .24468 .30359 .2624 .32152 .2889 .31423 .27133 .29612 Metetra .636 .745 .927 r .27133 .29612 .2889 .31423 .31579 .30569 .29838 .2874 Metetra .649 .749 .922 r .29838 .2874 .31579 .30569 .34303 .29629 .32578 .27781 Metetra .655 .751 .92 r .32578 .27781 .34303 .29629 .37057 .2864 .3535 .26773 Metetra .655 .751 .92 r .3535 .26773 .37057 .2864 .3984 .27641 .38151 .25754 Metetra .649 .749 .922 r .38151 .25754 .3984 .27641 .42651 .2667 .40981 .24763 Metetra .637 .744 .926 r .40981 .24763 .42651 .2667 .45491 .25766 .43839 .2384 Metetra .617 .737 .932 r .43839 .2384 .45491 .25766 .48362 .24969 .46727 .23023 Metetra .59 .726 .938 r .46727 .23023 .48362 .24969 .51267 .24319 .4965 .22354 Metetra .553 .709 .944 r .4965 .22354 .51267 .24319 .54212 .23857 .52613 .21873 Metetra .504 .686 .949 r .52613 .21873 .54212 .23857 .57204 .23627 .55622 .21624 Metetra .441 .653 .95 r .55622 .21624 .57204 .23627 .60252 .23673 .58687 .21652 Metetra .362 .609 .944 r .58687 .21652 .60252 .23673 .63369 .24044 .61819 .22005 Metetra .267 .552 .928 r .61819 .22005 .63369 .24044 .66567 .24792 .65032 .22737 Metetra .157 .479 .898 r .65032 .22737 .66567 .24792 .69864 .25975 .68344 .23906 Metetra .035 .393 .851 r .68344 .23906 .69864 .25975 .7328 .27659 .71774 .25577 Metetra 0 .207 .692 r .03634 .26033 .05481 .27754 .07545 .29189 .05693 .2746 Metetra 0 .372 .827 r .05693 .2746 .07545 .29189 .09717 .30258 .07863 .28519 Metetra .135 .501 .908 r .07863 .28519 .09717 .30258 .11986 .30981 .10134 .2923 Metetra .289 .59 .946 r .10134 .2923 .11986 .30981 .14343 .31383 .12496 .29619 Metetra .403 .649 .958 r .12496 .29619 .14343 .31383 .16778 .3149 .14938 .29713 Metetra .485 .687 .956 r .14938 .29713 .16778 .3149 .19283 .31333 .17453 .29541 Metetra .544 .711 .949 r .17453 .29541 .19283 .31333 .21849 .30945 .2003 .29136 Metetra .586 .727 .941 r .2003 .29136 .21849 .30945 .24468 .30359 .22662 .28533 Metetra .616 .738 .933 r .22662 .28533 .24468 .30359 .27133 .29612 .25342 .27768 Metetra .636 .745 .927 r .25342 .27768 .27133 .29612 .29838 .2874 .28063 .26877 Metetra .649 .749 .922 r .28063 .26877 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Illustration to the use of Descartes product. \ \>", "Text", CellFrame->{{0, 0}, {2, 0}}, TextAlignment->Center, Background->RGBColor[0.776471, 0.941176, 0.992157]] }, Open ]], Cell[CellGroupData[{ Cell["2. Dimension, spanning sets and (algebraic) basis", "Section", TextAlignment->Left, FontSize->16], Cell[TextData[{ StyleBox["E.2.2", "Section", FontSize->14, FontColor->RGBColor[1, 0, 0]], ". Let ", Cell[BoxData[ \(C\_1\^\((0)\)\)]], "[-1, 1] denote that linear subspace of the vector space ", Cell[BoxData[ \(C\^\((0)\)\)]], "[-1, 1], which consists of piecewise linear polynomials on the intervals \ [-1,0] and [0, 1]. This means that an arbitrary function p(t)\[Element]", Cell[BoxData[ \(C\_1\^\((0)\)\)]], "[-1, 1], can be written in aunique way as " }], "Text"], Cell[BoxData[ RowBox[{\(p \((t)\)\), "=", RowBox[{"\[LeftBracketingBar]", GridBox[{ {\(\(p\_1\) \((t)\) \[Congruent] \(a\_1\) t\ + b\_1, \ for\ \ \ t \[Element] \([\(-1\), 0]\)\)}, {\(\(p\_2\) \((t)\) \[Congruent] \ \(a\_2\) t\ + b\_2, \ \ \ for\ \ \ t \[Element] \([0, 1]\)\)} }]}]}]], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "where in consequence of the continuity condition ", Cell[BoxData[ \(p\_1\)]], "(0) = ", Cell[BoxData[ \(p\_2\)]], "(0) follows", StyleBox[" ", FontFamily->"Times New Roman"], Cell[BoxData[ \(b\_1\)], FontFamily->"Times New Roman"], StyleBox["=", FontFamily->"Times New Roman"], Cell[BoxData[ \(b\_2\)], FontFamily->"Times New Roman"], ". 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ooooooooooooooooooooYjNW00000000oooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooofMWIO7al00000000fMWIoooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooYjNW00000000oooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooYjNW00000000oooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooooooS8b<0000oooooooohN7Q0000S8b7QhOoooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooojNWY`0000000?oooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 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oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo fMWI00000000S8b"], "Graphics", ImageSize->{533.5, 67.75}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ RowBox[{ StyleBox["Figure", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], RowBox[{ StyleBox["2.1", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[".", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Basis", FormatType->StandardForm, FontFamily->"Times New Roman"]}], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["in", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["the", 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Cell[BoxData[ RowBox[{ StyleBox[\(\(C\_3\^\((1)\)\)[\(-1\), 1]\), FontFamily->"Times New Roman"], " "}]]], "denote the linear subspace of the vector space ", Cell[BoxData[ RowBox[{ StyleBox[\(\(C\_1\^\((1)\)\)[\(-1\), 1]\), FontFamily->"Times New Roman"], " "}]]], ", which consists of piecewise cubic polynomials on the interval [-1, 0] \ and [0, 1]. Similarly to example ", StyleBox["E.2.2", FontFamily->"Arial", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontFamily->"Courier New", FontWeight->"Bold"], "determine the basis of the function space ", Cell[BoxData[ RowBox[{ StyleBox[\(\(C\_3\^\((1)\)\)[\(-1\), 1]\), FontFamily->"Times New Roman"], " "}]]], " and plot it using ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", CellFrame->{{0, 0}, {0, 2}}], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(Clear["\"]\)\(\ \ \ \ \ \ \ \ \ \ \)\( (*\ Training\ problem\ *) \)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(p\_1[t_] := \(a\_1\) t\^3 + \(b\_1\) t\^2 + \(c\_1\) t + d\_1\), "\[IndentingNewLine]", \(\(\(p\_2[t_] := \(a\_2\) t\^3 + \(b\_2\) t\^2 + \(c\_2\) t + d\_2\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\( (*\ conditions\ for\ continuity\ in\ t \[Equal] 0\ *) \)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(p\_1[0] \[Equal] p\_2[0]\)}], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \(d\_1 \[Equal] d\_2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((D[p\_1[t], t] /. t \[Rule] 0)\) \[Equal] \((D[p\_2[t], t] /. t \[Rule] 0)\)\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \(c\_1 \[Equal] c\_2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(d\_1 = \(d\_2 = d\); c\_1 = \(c\_2 = c\);\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(\(p[t_] := Piecewise[{{p\_1[t], \(-1\) \[LessEqual] t \[LessEqual] 0}, {p\_2[t], 0 \[LessEqual] t \[LessEqual] 1}}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(dp[t_] := Piecewise[{{\(p\_1'\)[t], \(-1\) \[LessEqual] t \[LessEqual] 0}, {\(p\_2'\)[t], 0 \[LessEqual] t \[LessEqual] 1}}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(s = Solve[{Map[p, {\(-1\), 0, 1}] \[Equal] Map[g, {\(-1\), 0, 1}], \[IndentingNewLine]Map[ dp, {\(-1\), 0, 1}] \[Equal] Map[dg, {\(-1\), 0, 1}]} // Flatten, {a\_1, b\_1, a\_2, b\_2, c, d}] 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CF5dJ6E]HGAYHf4PAg9QL6QYHg"], ImageRangeCache->{{{143.938, 430.938}, {289.625, 112.688}} -> {-0.676026, \ 0.309372, 0.00394728, 0.0127737}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["11. Generalized derivatives, distributions, Sobolev space", "Section", TextAlignment->Left, FontSize->16], Cell[TextData[{ StyleBox["E.11.2", "Section", FontSize->14, FontColor->RGBColor[1, 0, 0]], StyleBox[".", FontColor->RGBColor[1, 0, 0]], " In ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalR]\^1\)]], " the" }], "Text"], Cell[BoxData[ \(\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\[CurlyPhi] \((x)\ \) = \ \[ExponentialE]\^\(1/\((x\^2 - 1)\)\)\ \ \ \ \ \ if\ \ \[VerticalBar] x \[VerticalBar] \ < \ 1\ \ \[IndentingNewLine]\[IndentingNewLine] \[CurlyPhi] \((x)\) = \ \(\(0\)\(\ \ \ \ \ \ \)\(ortherwise\)\(\ \)\)\)\)\ \)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "is an example of a smooth function with compact support in \[CapitalOmega] \ =(-2, 2). But \[CurlyPhi](x) is not a test function with respect to \ \[CapitalOmega] = (-1, 1), because supp \[CurlyPhi] = {-1\[LessEqual] x \ \[LessEqual] 1} is not a proper subset of the domain \[CapitalOmega] = {-1 < \ x < 1}. \nUsing ", StyleBox["Mathematica", FontSlant->"Italic"], " we can easily plot this function: " }], "Text"], Cell[BoxData[ \(Clear[\[CurlyPhi]]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \(\[CurlyPhi][x_] := Piecewise[{{0, x \[LessEqual] \(-1\)}, {\[ExponentialE]\^\(1\/\(x\^2 - 1\)\), \ \(-1\) < x < 1}, {0, x \[GreaterEqual] 1}}]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Plot[\[CurlyPhi][x], {x, \(-2\), 2}, PlotStyle \[Rule] {Thickness[0.008]}];\)\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def 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Cell["\<\ This function has limits at the points x =-1, x =1 which are equal to zero. \ indeed, e.g. at the point x = 1 the limit from the left\ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\[CurlyPhi][x], x -> 1, Direction \[Rule] 1]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["and obviously the limit from the right", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\[CurlyPhi][x], x \[Rule] 1, Direction \[Rule] \(-1\)]\)], "Input",\ Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "(The option ", StyleBox["Direction", FontSlant->"Italic"], " \[Rule] -1 and +1 denotes the right - and left -hand side limit, \ respectively.) At the points x = -1 and x = 1 there exist all derivates of \ \[CurlyPhi](x) and they are also equal to zero. 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For \ example, if we want to solve the boundary-value problem" }], "Text"], Cell[BoxData[{ \(\(\(\(\(d\^2\) u\)\/dx\^2 = \[Delta] \((x - 1\/2)\), \ \ \ \ 0 \[LessEqual] x \[LessEqual] 1\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(u \((0)\)\ = \ \(u \((1)\)\ = 0\)\)}], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ (establish the shape of a cable under force), then this problem can be \ approximated by sequence of bounfary-value problems\ \>", "Text"], Cell[BoxData[{ \(\(\(\(\(d\^2\) u\_n\)\/dx\^2 = \(s\_n\) \((x - 1\/2)\), \ \ \ \ 0 \[LessEqual] x \[LessEqual] 1\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(u\_n\) \((0)\)\ = \ \(\(u\_n\) \((1)\)\ = 0\)\)}], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["where ", "Text"], Cell[BoxData[ \(\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(s\_n\) \((x - \[Xi])\) = \ n, \ if\ \[Xi] - 1\/\(2 n\) \[LessEqual] x \[LessEqual] \[Xi] + 1\/\(2 n\)\[IndentingNewLine] \(s\_n\) \((x - \[Xi])\) = \ 0, \ if\ \[VerticalBar] x - \[Xi] \[VerticalBar] \ > 1\/\(2 n\)\)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["Then taking the limit", "Text"], Cell[BoxData[ \(lim\ \(u\_n\) \((x)\)\ = \ u \((x)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ we can produce the correct solution for the examined boundary-value \ problem.\ \>", "Text"], Cell["\<\ In more detail: If solve the previous differential equation separately on the \ intervals,\ \>", "Text"], Cell[BoxData[ StyleBox[\(\(\(\([0, 1\/2 - 1\/\(2 n\)\)\()\)\), \([1\/2 - 1\/\(2 n\), 1\/2 + 1\/\(2 n\)]\), \((1\/2 + 1\/\(2 n\), 1\)\)\(]\)\), FontColor->GrayLevel[0]]], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[BoxData[ \(with\ respect\ to\ \(u\_n\) \((0)\) = \(u\_n\) \((1)\), \ then\ we\ have\)], "Text", FontFamily->"Times New Roman"], Cell[BoxData[{ \(\(u\_n\) \((x)\) = \ ax, \ if\ \ \ 0 \[LessEqual] x < 1\/2 - 1\/\(2 n\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(u\_n\) \((x)\) = \ n\/2\ x\^2 + \ bx\ + \ c, \ if\ \ \ \ \ \ 1\/2 - 1\/\(2 n\) \[LessEqual] \ x\ \ \[LessEqual] \ 1\/2 + 1\/\(2 n\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(u\_n\) \((x)\) = \ d \((x - 1)\), \ if\ \ \ 1\/2 + 1\/\(2 n\) < x \[LessEqual] 1\), "\[IndentingNewLine]", \(\)}], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "We can compute the coefficients a,b, c, d from the conditions for the \ continuity of values and the first derivates of the function at x = ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "-", Cell[BoxData[ \(TraditionalForm\`1\/\(2 n\)\)]], " and x = ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "+", Cell[BoxData[ \(TraditionalForm\`1\/\(2 n\)\)]], ". \nThe solution is the function" }], "Text"], Cell[BoxData[{ \(\(u\_n\) \((x)\) = \(-\(1\/2\)\) x, \ if\ \ \ 0 \[LessEqual] x < 1\/2 - 1\/\(2 n\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(u\_n\) \((x)\) = \(n\/2\)[ x \((x - 1)\) + \((1\/2 - 1\/\(2 n\))\)\^2]\ , \ if\ \ \ \ \ \ 1\/2 - 1\/\(2 n\) \[LessEqual] \ x\ \ \[LessEqual] \ 1\/2 + 1\/\(2 n\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \(u\_n\) \((x)\) = \ \(1\/2\) \((x - 1)\), \ \ if\ \ \ 1\/2 + 1\/\(2 n\) < x \[LessEqual] 1\)}], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "which depends on n, is once continuously differenciable in the interval \ [0, 1] and satisfies the boundary conditions u(0) = u(1) =0.\n\nUsing ", StyleBox["Mathematica", FontSlant->"Italic"], " it is easy to plot the function ", Cell[BoxData[ \(TraditionalForm\`u\_n\)]], "(x) and its first derivative according to x as follows:" }], "Text"], Cell[BoxData[ \(u[n_, x_] 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Orthogonal systems, Fourier series ", "Section", TextAlignment->Left, FontSize->16], Cell[TextData[{ StyleBox["E.13.1", "Section", FontSize->14, FontColor->RGBColor[1, 0, 0]], StyleBox[".", FontColor->RGBColor[1, 0, 0]], " The sequence " }], "Text"], Cell[BoxData[ \({\[Phi]\_k = \(\@\(2\/\[Pi]\)\) sin \((\((2\ k - 1)\)\ x)\), \ \ \ \ k = 1, 2, ... }\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ "is", " ", "orthonormal", " ", "in", " ", "the", " ", "Hilbert", " ", "space", " ", StyleBox["H", FontSlant->"Italic"]}], " ", "=", " ", \(\(L\_2\) \((0, \ \[Pi])\) . \ \ Indeed\)}], ",", " ", \(using\ Mathematica\ we\ have\ got\)}]], "Text", FontFamily->"Times New Roman"], Cell[BoxData[ \(\[Phi]\_k_ := \(\@\(2\/\[Pi]\)\) Sin[\((2\ k - 1)\)\ x]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[CellGroupData[{ Cell[BoxData[ \(Assuming[{m, n} \[Element] Integers, {\[Integral]\_0\%\[Pi] \[Phi]\_m\ \(\[Phi]\_n\) \ \[DifferentialD]x, \[Integral]\_0\%\[Pi] \[Phi]\_n\ \(\[Phi]\_n\) \ \[DifferentialD]x}]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \({0, 1}\)], "Output"] }, Open ]], Cell["The function ", "Text"], Cell[BoxData[ \(u[x_] := Sin[2\ x]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ RowBox[{ "is", " ", "orthogonal", " ", "to", " ", "every", " ", \(\[Phi]\_k\), " ", "and", " ", "hence", " ", "the", " ", "Fourier", " ", "coefficients", " ", "of", " ", StyleBox["u", FontSlant->"Italic"]}]], "Text", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]\_k = Assuming[k \[Element] Integers, {\[Integral]\_0\%\[Pi] Sin[ 2\ x]\ \(\[Phi]\_k\) \[DifferentialD]x}]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \({0}\)], "Output"] }, Open ]], Cell["\<\ consequently the limit function is u = \[CapitalTheta] and not u = sin 2x.\ \>", "Text"], Cell[TextData[{ "However this does not occure, if the sequence would contain the function \ ", Cell[BoxData[ StyleBox[\(\@\(2\/\[Pi]\)\ Sin \((2\ x)\)\), FontFamily->"Times New Roman"]]], ". \nIt is orthonormal" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Assuming[ n \[Element] Integers, {\[Integral]\_0\%\[Pi]\@\( 2\/\[Pi]\)\ Sin[ 2\ x]\ \(\[Phi]\_n\) \[DifferentialD]x, \[Integral]\_0\%\[Pi]\(\ \((\@\(2\/\[Pi]\)\ Sin[2\ x])\)\^2\) \[DifferentialD]x}]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \({0, 1}\)], "Output"] }, Open ]], Cell["The coresponding Fourier coefficient is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]\_0 = \[Integral]\_0\%\[Pi] Sin[2\ x] \@\(2\/\[Pi]\)\ Sin[ 2\ x]\ \[DifferentialD]x\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \(\@\(\[Pi]\/2\)\)], "Output"] }, Open ]], Cell["then", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[Alpha]\_0\) \@\(2\/\[Pi]\)\ Sin[2\ x]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \(Sin[2\ x]\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["E.13.2", "Section", FontSize->14, FontColor->RGBColor[1, 0, 0]], ". Recall, that the sequence of functions " }], "Text"], Cell[BoxData[ \({1, \ x, \ x\^2, \ x\^3, ... }\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "forms a base for the Hilbert space ", Cell[BoxData[ \(L\_2\)]], "(-1, 1). But this base is not orthonormal. To orthonormalize it, we may \ apply the process of Gram-Schmidt orthonormalization which can be realized \ using ", StyleBox["Mathematica", FontSlant->"Italic"], " as follows " }], "Text"], Cell[BoxData[ \(<< LinearAlgebra`Orthogonalization`\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\({y\_1, y\_2, y\_3, y\_4, y\_5}\)\(=\)\(GramSchmidt[{1, x, x\^2, x\^3, x\^4}, Normalized \[Rule] True, InnerProduct \[Rule] \((\[Integral]\_\(-1\)\%1\((#1\ #2)\) \ \[DifferentialD]x &)\)]\)\(\ \)\)\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \({1\/\@2, \@\(3\/2\)\ x, 3\/2\ \@\(5\/2\)\ \((\(-\(1\/3\)\) + x\^2)\), 5\/2\ \@\(7\/2\)\ \((\(-\(\(3\ x\)\/5\)\) + x\^3)\), \(105\ \((\(-\(1\/5\)\) + x\^4 - 6\/7\ \((\(-\(1\/3\)\) \ + x\^2)\))\)\)\/\(8\ \@2\)}\)], "Output"] }, Open ]], Cell["Indeed, i.e.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[\[Integral]\_\(-1\)\%1 y\_i\ \(y\_j\) \[DifferentialD]x, {i, 1, 5}, {j, 1, i}] // TableForm\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ InterpretationBox[GridBox[{ {"1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"0", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"0", "0", "1", "\<\"\"\>", "\<\"\"\>"}, {"0", "0", "0", "1", "\<\"\"\>"}, {"0", "0", "0", "0", "1"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{1}, {0, 1}, {0, 0, 1}, {0, 0, 0, 1}, {0, 0, 0, 0, 1}}]]], "Output"] }, Open ]], Cell["\<\ The following example applies the Gram-Schmidt orthonormalization to the \ given list of three-dimensional vectors.\ \>", "Text"], Cell[BoxData[ \(u1 = {3, 4, 2}; u2 = {2, 5, 2}; u3 = {1, 2, 6};\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[CellGroupData[{ Cell[BoxData[ \({v\_1, v\_2, v\_3} = GramSchmidt[{u1, u2, u3}]\)], "Input", Background->RGBColor[0.776471, 0.941176, 0.992157]], Cell[BoxData[ \({{3\/\@29, 4\/\@29, 2\/\@29}, 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