(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 767479, 14996] NotebookOptionsPosition[ 763923, 14877] NotebookOutlinePosition[ 764935, 14916] CellTagsIndexPosition[ 764892, 14913] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Introduction to Computational Mathematics and Physics", "Title"], Cell["APAM E1601y (Spring 2009)", "Subsubtitle", CellChangeTimes->{{3.441539122363717*^9, 3.441539123921317*^9}}], Cell["\<\ Prof. Michael Mauel mauel@columbia.edu\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Course Description", "Section"], Cell[TextData[{ StyleBox["Introduction to computational methods in applied mathematics and \ physics", FontSlant->"Italic"], ": Students develop solutions in a small number of subject areas to acquire \ a first taste in the practical use of computers in solving mathematics and \ physics problems." }], "Text"], Cell[TextData[{ "APAM1601 does not require prior programming experience (but prior computer \ experience and talent are ", StyleBox["helpful", FontSlant->"Italic"], ".) Topics change from year-to-year, and only a limited number of topics \ (typically four per term) are selected for discussion and investigation. \ Topics range from classical and modern physics and applied mathematics, but \ the course is not meant to cover these areas broadly. Instead, each topic \ will be self-contained and limited in scope. We try to make topics \ interesting and absorbing, and they will amplify and expand on a student's \ knowledge acquired during your first year of physics and mathematics course \ work. The goal of this course is to provide some depth in select topics \ instead of providing a general (but shallow) overview of an entire subject \ area.\nExamples include elementary interpolation of functions, solution of \ nonlinear algebraic equations, curve-fitting and hypothesis testing, wave \ propagation, fluid motion, gravitational and celestial mechanics, chaotic \ dynamics. (APAM1601 is usually taught by a team of two professors, an applied \ physicist and an applied mathematician. This semester, the course will have a \ stronger computational physics emphasis because of the interests of the \ instructor.)\nThe basic requirement for this course is one year of college \ level Calculus and Physics; programming experience is desirable but not \ required. Students will gain confidence preparing, understanding, and \ presenting computer-assisted solutions to elementary but relevant problems. \ \n This course uses ", StyleBox["Mathematica", FontSlant->"Italic"], " combined with detailed classnotes. \n Why ", StyleBox["Mathematica", FontSlant->"Italic"], "? ", StyleBox["Mathematica", FontSlant->"Italic"], " is a very powerful system for evaluating expressions and graphing results. \ It's disadvantage is that it is commercial software, but it's advantage is \ that it is available for all major operating systems and available widely at \ Columbia University. The most important reason I selected ", StyleBox["Mathematica", FontSlant->"Italic"], " is that it is an interpretive system: ", StyleBox["Mathematica", FontSlant->"Italic"], " instantly responds to every expression you write. Additionally, ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks contain a perfect record of your interaction with the computer. \ Save your notebooks, edit them, and re-execute them. In the beginning, the \ syntax of ", StyleBox["Mathematica", FontSlant->"Italic"], " will seem complicated. Later, it will become second nature. Experience \ using ", StyleBox["Mathematica", FontSlant->"Italic"], " will be developed as the complexity of the subject material is advanced. \ In addition to classnotes distributed by the instructors, a textbook \ illustrating the use of ", StyleBox["Mathematica", FontSlant->"Italic"], " for science and engineering will be required." }], "Text"], Cell[TextData[{ "(For those interested in other numerical research packages (like MATLAB, \ IDL, or others), please see me. I ", StyleBox["will", FontSlant->"Italic"], " accept projects in any language or system, but I will have only time in \ class to discuss these other packages/programming systems.)" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Textbook and References", "Section"], Cell[TextData[{ "Textbooks are not required for APAM1601. ", StyleBox["Mathematica", FontSlant->"Italic"], " is fully documented in your browser. When you program with ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["always have the help browser window open", FontWeight->"Bold"], "." }], "Text"], Cell[TextData[{ "For those of you not able to learn by reading a computer display, I \ recommend a good basic book for learning ", StyleBox["Mathematica", FontSlant->"Italic"], " by Bruce and Eve Torrence:" }], "Text"], Cell[GraphicsData["CompressedBitmap", "\<\ eJzsvXd4HdW1Ni5k2Y7Dzb33l5sbIAQCSS7phB4IJIQUCN0GGxsbV2xsg7Fp xqa6d8u23CSrV6tXq1lW7+Wco6Peu6zeddR19H7v2nMkmxYgX5Lv+T3P/WPP 2TOzZ8/MWu9eba/Z5+nV72xY9/rqdza+vPqWx95a/caGjS+/fcujW97ioVnX WFlZpbN032IldbA6vZmDmBMbUX7xLErjXVEa54TSS86su8xFWYLrXJQkulqj KN7ZGsWXWCtNcLeahdJEd55wR3GCG/eK4l25ZV0dY7FS7azZgTQuS/Bg8ZRi Jcc8bVCW5MUT5cleLJ5SrGxQkeJlNVvtlyV5TB+XNryoIsX729z4WHPf21qr 8ZiqsaOK1POoSFOFHZWnytm089YztekmKT7qAksHleqs6qCSZyvTVJEnSTs/ V66diyrphSctV1SprmrSfVGX7ieFXdXIWZ6wkXaztJpqXK1OqO6rUs6jKlkV XlGd4mstVbnWj69cmeLLK3jzVF9UpfqxC+m3Ot3XWru/uoPcbBZq03xlV4rV 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(About \ $90.00 expensive; $47 used from ", ButtonBox["Amazon.com", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://www.amazon.com/gp/product/0131469908/qid=1137272841/sr=1-7/\ ref=sr_1_7/103-7053304-0739019?s=books&v=glance&n=283155"], None}], "). Also, now in", StyleBox[" 2nd Edition", FontWeight->"Bold"], " (published in 2005). ", StyleBox["Very popular and useful.", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.4415395157967777`*^9, 3.4415395464825706`*^9}}], Cell[TextData[{ "Remember, ", StyleBox["you do not required to purchase either of these textbooks", FontWeight->"Bold", FontSlant->"Italic"], ". ", StyleBox["All", FontSlant->"Italic"], " lectures will be available online, and ", StyleBox["Mathematica", FontSlant->"Italic"], " is fully and conveniently documented by the \"Help Browser\"." }], "Text", CellChangeTimes->{{3.441539578270219*^9, 3.441539580730555*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Class Organization and Grades (", StyleBox["Important", FontColor->RGBColor[1., 0., 0.]], ")" }], "Section", CellChangeTimes->{{3.441539712488453*^9, 3.441539714849742*^9}}], Cell[TextData[{ "This class combines lectures and computer demonstrations. During lectures, \ the fundamentals of each topics will be presented and discussed. During \ demonstrations, we will use ", StyleBox["Mathematica", FontSlant->"Italic"], " to explore and experiment. All of the classroom demonstrations will be \ available online. You should copy these notebooks to your own computer, load \ them into ", StyleBox["Mathematica,", FontSlant->"Italic"], " and experiment. \n", StyleBox["You will not make progress in this course without using \ Mathematica!", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "\nThe class notebooks are available on the our class webpage : \n\t", StyleBox["<", FontFamily->"Courier"], ButtonBox["http://www.apam.columbia.edu/courses/ap1601y/", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://www.apam.columbia.edu/courses/ap1601y/"], None}], StyleBox[">", FontFamily->"Courier"], StyleBox["\n", FontWeight->"Bold"], "Since your work can be conveniently saved as ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks, you can record and edit your work for later discussion and \ presentation. Remember, when you work with a Columbia University workstation, \ you need to use a file-transfer program (like WINSCP) to transfer your \ notebooks to and from your directory space on CUNIX. Setup a directory on \ your CUNIX account to place all of your work.\nThis semester, we will discuss \ four subjects. To help you get to know ", StyleBox["Mathematica", FontSlant->"Italic"], " and each topic, I will ask that you to prepare and submit ", StyleBox["simple exercises. ", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["Your final grade, however, will be based entirely on your four ", FontWeight->"Bold"], StyleBox["research projects.", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " These will be ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks. I prefer you to submit each by email (with your UNI in the \ filename):\n\t", StyleBox[""Courier"], ButtonBox["mauel@columbia.edu", BaseStyle->"Hyperlink", ButtonData:>{ URL["mailto:mauel@columbia.edu"], None}], StyleBox[">", FontFamily->"Courier"], "\n[", StyleBox["Please note", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ": unless I tell you otherwise, please execute the menu command (from within \ ", StyleBox["Mathematica", FontSlant->"Italic"], ") \"", StyleBox["Kernel>Delete All Output", FontWeight->"Bold"], "\" so that your ", StyleBox["large", FontSlant->"Italic"], " graphics files do not overwhelm my emailbox.]\nAfter receiving your \ projects by email, I can run your programs and correct them if needed. If you \ have trouble doing this, you may give me a hardcopy. (When sending email, you \ will probably need to login into your ", StyleBox["cunix", FontFamily->"Courier"], " account, transfer any files from your local computer to ", StyleBox["cunix", FontFamily->"Courier"], ", and email the message from ", StyleBox["cunix", FontFamily->"Courier"], ".) \nYou should discuss your projects with Prof. Mauel (1) before you \ begin, (2) during your investigations, and (3) as you begin to finalize your \ project, presentation, and results. To help you develop your ", StyleBox["Mathematica", FontSlant->"Italic"], " ability, we will assign weekly exercises, and these will help you in the \ preparation of your projects. \nYour grade will be based on entirely on the \ creativity, organization, and presentation of your research projects and your \ ability to complete the weekly exercises. \n", StyleBox["Grading will be based on primarily on your four research projects, \ but I will consider your exercises and your effort. There are no \ examinations.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text", CellChangeTimes->{{3.44153961332213*^9, 3.4415397065455933`*^9}, { 3.441539844202166*^9, 3.4415399142077913`*^9}}], Cell[CellGroupData[{ Cell["TA: Bing Qian \"Victoria\" Li", "Subsection", CellChangeTimes->{{3.441544148287395*^9, 3.441544159539462*^9}}], Cell[TextData[{ "This semester Victoria Li (", ButtonBox["bl2327@columbia.edu", BaseStyle->"Hyperlink", ButtonData->{ URL["mailto:bl2327@columbia.edu"], None}], ") will be available to assist students with their exercises, projects, and \ the various issues using ", StyleBox["Mathematica. Please contact ", FontSlant->"Italic"], "Victoria to find out the best time and place to meet with her. If you need \ some help and I am not available, ask Victoria to schedule some tutorials and \ problem solving sessions. " }], "Text", CellChangeTimes->{{3.4415441645429897`*^9, 3.441544167268043*^9}, { 3.4415442172077217`*^9, 3.441544237461681*^9}, {3.441544379719421*^9, 3.441544424623496*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Class Schedule", "Subsection"], Cell["\<\ Lectures will occur once a week at a day and time that best meets the needs \ of students. \ \>", "Text"], Cell[TextData[{ "We plan to cover four subjects in four three-week periods. Prior to this, \ we will spend two weeks getting up to speed with ", StyleBox["Mathematica", FontSlant->"Italic"], ". Although ", StyleBox["these subjects may change", FontWeight->"Bold"], ", presently, the class schedule is listed below. (Feel free to contact me \ with your ideas about topics. Physics and applied mathematics extends across \ a great many topics, and I would be very interested in preparing a three-week \ session on a topic of your choice!) 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0 g s .66477 .39222 m .67512 .4211 L .64536 .41529 L p .458 .886 .889 r F P 0 g s .5031 .32122 m .49545 .38599 L .52558 .39188 L p .762 .535 .59 r F P 0 g s .49545 .38599 m .5031 .32122 L .48815 .30257 L p .758 .584 .661 r F P 0 g s .31307 .35035 m .33291 .51298 L .35992 .50315 L p .648 .611 .798 r F P 0 g s .33291 .51298 m .31307 .35035 L .2824 .34435 L p .651 .624 .809 r F P 0 g s .64536 .41529 m .63645 .40253 L .66477 .39222 L p .458 .886 .889 r F P 0 g s .63645 .40253 m .64536 .41529 L .61553 .40946 L p 0 0 0 r F P 0 g s .71232 .66774 m .61056 .68417 L .62551 .70121 L p .417 .764 .996 r F P 0 g s .62551 .70121 m .74308 .6731 L .71232 .66774 L p .417 .764 .996 r F P 0 g s .61553 .40946 m .60825 .41279 L .63645 .40253 L p 0 0 0 r F P 0 g s .60825 .41279 m .61553 .40946 L .58562 .40361 L p .112 0 0 r F P 0 g s .47204 .59687 m .38704 .49329 L .35992 .50315 L p .523 .565 .839 r F P 0 g s .34366 .35633 m .35992 .50315 L .38704 .49329 L p .644 .596 .785 r F P 0 g s .35992 .50315 m .34366 .35633 L .31307 .35035 L p .648 .611 .798 r F P 0 g s .58562 .40361 m .58018 .42301 L .60825 .41279 L p .112 0 0 r F P 0 g s .58018 .42301 m .58562 .40361 L .55564 .39775 L p .373 .011 .215 r F P 0 g s .56503 .63227 m .48046 .57758 L .47204 .59687 L p .221 .523 .917 r F P 0 g s .47204 .59687 m .58032 .64971 L .56503 .63227 L p .221 .523 .917 r F P 0 g s .38704 .49329 m .47204 .59687 L .48046 .57758 L p .495 .532 .824 r F P 0 g s .48815 .30257 m .46524 .38009 L .49545 .38599 L p .758 .584 .661 r F P 0 g s .46524 .38009 m .48815 .30257 L .47308 .28378 L p .751 .614 .707 r F P 0 g s .55564 .39775 m .55223 .43318 L .58018 .42301 L p .373 .011 .215 r F P 0 g s .55223 .43318 m .55564 .39775 L .52558 .39188 L p .504 .246 .456 r F P 0 g s .48046 .57758 m .41427 .48337 L .38704 .49329 L p .495 .532 .824 r F P 0 g s .37417 .36229 m .38704 .49329 L .41427 .48337 L p .639 .576 .768 r F P 0 g s .38704 .49329 m .37417 .36229 L .34366 .35633 L p .644 .596 .785 r F P 0 g s .68148 .66237 m .5955 .66701 L .61056 .68417 L p .336 .741 .981 r F P 0 g s .61056 .68417 m .71232 .66774 L .68148 .66237 L p .336 .741 .981 r F P 0 g s .52558 .39188 m .5244 .4433 L .55223 .43318 L p .504 .246 .456 r F P 0 g s .5244 .4433 m .52558 .39188 L .49545 .38599 L p .567 .38 .587 r F P 0 g s .48896 .55813 m .44163 .47342 L .41427 .48337 L p .453 .483 .8 r F P 0 g s .41427 .48337 m .48046 .57758 L .48896 .55813 L p .453 .483 .8 r F P 0 g s .4046 .36824 m .41427 .48337 L .44163 .47342 L p .631 .549 .744 r F P 0 g s .41427 .48337 m .4046 .36824 L .37417 .36229 L p .639 .576 .768 r F P 0 g s .49545 .38599 m .49669 .45339 L .5244 .4433 L p .567 .38 .587 r F P 0 g s .49669 .45339 m .49545 .38599 L .46524 .38009 L p .6 .461 .663 r F P 0 g s .54961 .6147 m .48896 .55813 L .48046 .57758 L p .022 .394 .851 r F P 0 g s .48046 .57758 m .56503 .63227 L .54961 .6147 L p .022 .394 .851 r F P 0 g s .47308 .28378 m .43496 .37417 L .46524 .38009 L p .751 .614 .707 r F P 0 g s .43496 .37417 m .47308 .28378 L .4579 .26484 L p .745 .635 .738 r F P 0 g s .46524 .38009 m .4691 .46343 L .49669 .45339 L p .6 .461 .663 r F P 0 g s .4691 .46343 m .46524 .38009 L .43496 .37417 L p .619 .513 .712 r F P 0 g s .49753 .5385 m .4691 .46343 L .44163 .47342 L p .388 .408 .759 r F P 0 g s .44163 .47342 m .48896 .55813 L .49753 .5385 L p .388 .408 .759 r F P 0 g s .43496 .37417 m .44163 .47342 L .4691 .46343 L p .619 .513 .712 r F P 0 g s .44163 .47342 m .43496 .37417 L .4046 .36824 L p .631 .549 .744 r F P 0 g s .4579 .26484 m .4046 .36824 L .43496 .37417 L p .745 .635 .738 r F P 0 g s .65057 .65699 m .58032 .64971 L .5955 .66701 L p .202 .684 .93 r F P 0 g s .5955 .66701 m .68148 .66237 L .65057 .65699 L p .202 .684 .93 r F P 0 g s .4046 .36824 m .4579 .26484 L .44259 .24575 L p .739 .65 .761 r F P 0 g s .44259 .24575 m .37417 .36229 L .4046 .36824 L p .739 .65 .761 r F P 0 g s .50618 .51871 m .49669 .45339 L .4691 .46343 L p .275 .277 .681 r F P 0 g s .4691 .46343 m .49753 .5385 L .50618 .51871 L p .275 .277 .681 r F P 0 g s .53407 .59699 m .49753 .5385 L .48896 .55813 L p 0 .123 .641 r F P 0 g s .48896 .55813 m .54961 .6147 L .53407 .59699 L p 0 .123 .641 r F P 0 g s .37417 .36229 m .44259 .24575 L .42717 .22652 L p .733 .661 .778 r F P 0 g s .42717 .22652 m .34366 .35633 L .37417 .36229 L p .733 .661 .778 r F P 0 g s .34366 .35633 m .42717 .22652 L .41162 .20712 L p .728 .669 .792 r F P 0 g s .41162 .20712 m .31307 .35035 L .34366 .35633 L p .728 .669 .792 r F P 0 g s .61958 .65159 m .56503 .63227 L .58032 .64971 L p 0 .546 .784 r F P 0 g s .58032 .64971 m .65057 .65699 L .61958 .65159 L p 0 .546 .784 r F P 0 g s .5149 .49874 m .5244 .4433 L .49669 .45339 L p .056 .027 .504 r F P 0 g s .49669 .45339 m .50618 .51871 L .5149 .49874 L p .056 .027 .504 r F P 0 g s .66422 .98553 m .43309 .93927 L .44094 .92117 L p .449 .665 .954 r F P 0 g s .43309 .93927 m .24661 .79055 L .27386 .7808 L p .557 .622 .873 r F P 0 g s .27386 .7808 m .44094 .92117 L .43309 .93927 L p .557 .622 .873 r F P 0 g s .31307 .35035 m .41162 .20712 L .39595 .18757 L p .724 .676 .802 r F P 0 g s .39595 .18757 m .2824 .34435 L .31307 .35035 L p .724 .676 .802 r F P 0 g s .52369 .4786 m .55223 .43318 L .5244 .4433 L p 0 0 .091 r F P 0 g s .5244 .4433 m .5149 .49874 L .52369 .4786 L p 0 0 .091 r F P 0 g s .2824 .34435 m .39595 .18757 L .38015 .16787 L p .721 .681 .811 r F P 0 g s .38015 .16787 m .25166 .33835 L .2824 .34435 L p .721 .681 .811 r F P 0 g s .49753 .5385 m .53407 .59699 L .5184 .57913 L p .77 .308 0 r F P 0 g s .5184 .57913 m .50618 .51871 L .49753 .5385 L p .77 .308 0 r F P 0 g s .25166 .33835 m .38015 .16787 L .36422 .14801 L p .717 .686 .818 r F P 0 g s .36422 .14801 m .22084 .33232 L .25166 .33835 L p .717 .686 .818 r F P 0 g s .56503 .63227 m .61958 .65159 L .58852 .64618 L p .332 0 0 r F P 0 g s .58852 .64618 m .54961 .6147 L .56503 .63227 L p .332 0 0 r F P 0 g s .55223 .43318 m .52369 .4786 L .53257 .45828 L p .766 .887 .476 r F P 0 g s .53257 .45828 m .58018 .42301 L .55223 .43318 L p .766 .887 .476 r F P 0 g s .44094 .92117 m .64858 .96934 L .66422 .98553 L p .449 .665 .954 r F P 0 g s .64858 .96934 m .44094 .92117 L .44886 .90291 L p .411 .646 .954 r F P 0 g s .44094 .92117 m .27386 .7808 L .30123 .771 L p .544 .607 .867 r F P 0 g s .30123 .771 m .44886 .90291 L .44094 .92117 L p .544 .607 .867 r F P 0 g s .58018 .42301 m .53257 .45828 L .54152 .43778 L p .874 .994 .767 r F P 0 g s .54152 .43778 m .60825 .41279 L .58018 .42301 L p .874 .994 .767 r F P 0 g s .50618 .51871 m .5184 .57913 L .50262 .56114 L p .954 .612 .309 r F P 0 g s .50262 .56114 m .5149 .49874 L .50618 .51871 L p .954 .612 .309 r F P 0 g s .55738 .64075 m .53407 .59699 L .54961 .6147 L p .591 .064 0 r F P 0 g s .54961 .6147 m .58852 .64618 L .55738 .64075 L p .591 .064 0 r F P 0 g s .55055 .4171 m .63645 .40253 L .60825 .41279 L p .861 .971 .862 r F P 0 g s .60825 .41279 m .54152 .43778 L .55055 .4171 L p .861 .971 .862 r F P 0 g s .44886 .90291 m .63282 .95302 L .64858 .96934 L p .411 .646 .954 r F P 0 g s .63282 .95302 m .44886 .90291 L .45685 .88448 L p .356 .618 .95 r F P 0 g s .44886 .90291 m .30123 .771 L .32872 .76116 L p .527 .587 .858 r F P 0 g s .32872 .76116 m .45685 .88448 L .44886 .90291 L p .527 .587 .858 r F P 0 g s .55967 .39624 m .66477 .39222 L .63645 .40253 L p .834 .936 .894 r F P 0 g s .63645 .40253 m .55055 .4171 L .55967 .39624 L p .834 .936 .894 r F P 0 g s .4867 .543 m .52369 .4786 L .5149 .49874 L p .95 .722 .558 r F P 0 g s .5149 .49874 m .50262 .56114 L .4867 .543 L p .95 .722 .558 r F P 0 g s .52616 .63532 m .5184 .57913 L .53407 .59699 L p .713 .309 .282 r F P 0 g s .53407 .59699 m .55738 .64075 L .52616 .63532 L p .713 .309 .282 r F P 0 g s .24661 .79055 m .20977 .58021 L .24176 .58578 L p .648 .636 .824 r F P 0 g s .24176 .58578 m .27386 .7808 L .24661 .79055 L p .648 .636 .824 r F P 0 g s .56886 .37519 m .69322 .38187 L .66477 .39222 L p .81 .906 .907 r F P 0 g s .66477 .39222 m .55967 .39624 L .56886 .37519 L p .81 .906 .907 r F P 0 g s .81605 .94017 m .66422 .98553 L .64858 .96934 L p .417 .77 .996 r F P 0 g s .45685 .88448 m .61694 .93657 L .63282 .95302 L p .356 .618 .95 r F P 0 g s .61694 .93657 m .45685 .88448 L .46491 .8659 L p .274 .573 .939 r F P 0 g s .45685 .88448 m .32872 .76116 L .35634 .75128 L p .505 .561 .847 r F P 0 g s .35634 .75128 m .46491 .8659 L .45685 .88448 L p .505 .561 .847 r F P 0 g s .75667 .60801 m .77377 .67844 L .80439 .68378 L p .637 .869 .985 r F P 0 g s .80439 .68378 m .78628 .59741 L .75667 .60801 L p .637 .869 .985 r F P 0 g s .27386 .7808 m .24176 .58578 L .27368 .59134 L p .646 .625 .815 r F P 0 g s .27368 .59134 m .30123 .771 L .27386 .7808 L p .646 .625 .815 r F P 0 g s .57814 .35395 m .72179 .37148 L .69322 .38187 L p .791 .883 .912 r F P 0 g s .69322 .38187 m .56886 .37519 L .57814 .35395 L p .791 .883 .912 r F P 0 g s .49487 .62987 m .50262 .56114 L .5184 .57913 L p .751 .451 .475 r F P 0 g s .5184 .57913 m .52616 .63532 L .49487 .62987 L p .751 .451 .475 r F P 0 g s .7272 .61855 m .74308 .6731 L .77377 .67844 L p .625 .882 .988 r F P 0 g s .77377 .67844 m .75667 .60801 L .7272 .61855 L p .625 .882 .988 r F P 0 g s .47066 .52472 m .53257 .45828 L .52369 .4786 L p .911 .756 .677 r F P 0 g s .52369 .4786 m .4867 .543 L .47066 .52472 L p .911 .756 .677 r F P 0 g s .69787 .62905 m .71232 .66774 L .74308 .6731 L p .607 .896 .988 r F P 0 g s .74308 .6731 m .7272 .61855 L .69787 .62905 L p .607 .896 .988 r F P 0 g s .30123 .771 m .27368 .59134 L .30551 .59689 L p .643 .612 .804 r F P 0 g s .30551 .59689 m .32872 .76116 L .30123 .771 L p .643 .612 .804 r F P 0 g s .5875 .33251 m .75048 .36104 L .72179 .37148 L p .776 .864 .914 r F P 0 g s .72179 .37148 m .57814 .35395 L .5875 .33251 L p .776 .864 .914 r F P 0 g s .64858 .96934 m .78421 .93528 L .81605 .94017 L p .417 .77 .996 r F P 0 g s .78421 .93528 m .64858 .96934 L .63282 .95302 L p .353 .753 .984 r F P 0 g s .66868 .6395 m .68148 .66237 L .71232 .66774 L p .575 .907 .978 r F P 0 g s .71232 .66774 m .69787 .62905 L .66868 .6395 L p .575 .907 .978 r F P 0 g s .46491 .8659 m .60093 .91999 L .61694 .93657 L p .274 .573 .939 r F P 0 g s .60093 .91999 m .46491 .8659 L .47304 .84715 L p .143 .494 .905 r F P 0 g s .46491 .8659 m .35634 .75128 L .38409 .74135 L p .474 .526 .83 r F P 0 g s .38409 .74135 m .47304 .84715 L .46491 .8659 L p .474 .526 .83 r F P 0 g s .63962 .6499 m .65057 .65699 L .68148 .66237 L p .512 .903 .939 r F P 0 g s .68148 .66237 m .66868 .6395 L .63962 .6499 L p .512 .903 .939 r F P 0 g s .32872 .76116 m .30551 .59689 L .33726 .60242 L p .638 .595 .789 r F P 0 g s .33726 .60242 m .35634 .75128 L .32872 .76116 L p .638 .595 .789 r F P 0 g s .4635 .6244 m .4867 .543 L .50262 .56114 L p .759 .532 .589 r F P 0 g s .50262 .56114 m .49487 .62987 L .4635 .6244 L p .759 .532 .589 r F P 0 g s .59694 .31089 m .77931 .35055 L .75048 .36104 L p .765 .85 .915 r F P 0 g s .75048 .36104 m .5875 .33251 L .59694 .31089 L p .765 .85 .915 r F P 0 g s .45449 .50628 m .54152 .43778 L .53257 .45828 L p .875 .767 .74 r F P 0 g s .53257 .45828 m .47066 .52472 L .45449 .50628 L p .875 .767 .74 r F P 0 g s .6107 .66025 m .61958 .65159 L .65057 .65699 L p .385 .847 .827 r F P 0 g s .65057 .65699 m .63962 .6499 L .6107 .66025 L p .385 .847 .827 r F P 0 g s .63282 .95302 m .7523 .93038 L .78421 .93528 L p .353 .753 .984 r F P 0 g s .7523 .93038 m .63282 .95302 L .61694 .93657 L p .255 .715 .951 r F P 0 g s .58191 .67055 m .58852 .64618 L .61958 .65159 L p 0 0 0 r F P 0 g s .61958 .65159 m .6107 .66025 L .58191 .67055 L p 0 0 0 r F P 0 g s .35634 .75128 m .33726 .60242 L .36894 .60793 L p .633 .574 .771 r F P 0 g s .36894 .60793 m .38409 .74135 L .35634 .75128 L p .633 .574 .771 r F P 0 g s .47304 .84715 m .58479 .90328 L .60093 .91999 L p .143 .494 .905 r F P 0 g s .58479 .90328 m .47304 .84715 L .48124 .82824 L p 0 .342 .811 r F P 0 g s .47304 .84715 m .38409 .74135 L .41196 .73137 L p .429 .474 .804 r F P 0 g s .41196 .73137 m .48124 .82824 L .47304 .84715 L p .429 .474 .804 r F P 0 g s .55326 .6808 m .55738 .64075 L .58852 .64618 L p .159 0 0 r F P 0 g s .58852 .64618 m .58191 .67055 L .55326 .6808 L p .159 0 0 r F P 0 g s .43206 .61893 m .47066 .52472 L .4867 .543 L p .756 .581 .659 r F P 0 g s .4867 .543 m .4635 .6244 L .43206 .61893 L p .756 .581 .659 r F P 0 g s .38409 .74135 m .36894 .60793 L .40054 .61344 L p .624 .546 .746 r F P 0 g s .40054 .61344 m .41196 .73137 L .38409 .74135 L p .624 .546 .746 r F P 0 g s .52474 .69101 m .52616 .63532 L .55738 .64075 L p .381 .029 .236 r F P 0 g s .55738 .64075 m .55326 .6808 L .52474 .69101 L p .381 .029 .236 r F P 0 g s .43819 .4877 m .55055 .4171 L .54152 .43778 L p .846 .769 .778 r F P 0 g s .54152 .43778 m .45449 .50628 L .43819 .4877 L p .846 .769 .778 r F P 0 g s .49635 .70117 m .49487 .62987 L .52616 .63532 L p .5 .244 .458 r F P 0 g s .52616 .63532 m .52474 .69101 L .49635 .70117 L p .5 .244 .458 r F P 0 g s .61694 .93657 m .72031 .92547 L .7523 .93038 L p .255 .715 .951 r F P 0 g s .72031 .92547 m .61694 .93657 L .60093 .91999 L p .1 .632 .866 r F P 0 g s .48124 .82824 m .41196 .73137 L .43996 .72135 L p .359 .393 .76 r F P 0 g s .41196 .73137 m .40054 .61344 L .43206 .61893 L p .612 .509 .712 r F P 0 g s .43206 .61893 m .43996 .72135 L .41196 .73137 L p .612 .509 .712 r F P 0 g s .46809 .71128 m .4635 .6244 L .49487 .62987 L p .56 .374 .586 r F P 0 g s .49487 .62987 m .49635 .70117 L .46809 .71128 L p .56 .374 .586 r F P 0 g s .48124 .82824 m .56852 .88644 L .58479 .90328 L p 0 .342 .811 r F P 0 g s .56852 .88644 m .48124 .82824 L .48951 .80915 L p 0 .049 .556 r F P 0 g s .43996 .72135 m .48951 .80915 L .48124 .82824 L p .359 .393 .76 r F P 0 g s .40054 .61344 m .45449 .50628 L .47066 .52472 L p .75 .613 .706 r F P 0 g s .47066 .52472 m .43206 .61893 L .40054 .61344 L p .75 .613 .706 r F P 0 g s .43996 .72135 m .43206 .61893 L .4635 .6244 L p .593 .455 .662 r F P 0 g s .4635 .6244 m .46809 .71128 L .43996 .72135 L p .593 .455 .662 r F P 0 g s .48951 .80915 m .43996 .72135 L .46809 .71128 L p .241 .255 .675 r F P 0 g s .42175 .46897 m .55967 .39624 L .55055 .4171 L p .823 .769 .802 r F P 0 g s .55055 .4171 m .43819 .4877 L .42175 .46897 L p .823 .769 .802 r F P 0 g s .60093 .91999 m .68824 .92054 L .72031 .92547 L p .1 .632 .866 r F P 0 g s .68824 .92054 m .60093 .91999 L .58479 .90328 L p .133 0 0 r F P 0 g s .45449 .50628 m .40054 .61344 L .36894 .60793 L p .744 .635 .739 r F P 0 g s .46809 .71128 m .49786 .7899 L .48951 .80915 L p .241 .255 .675 r F P 0 g s .49786 .7899 m .46809 .71128 L .49635 .70117 L p .019 0 .49 r F P 0 g s .43819 .4877 m .36894 .60793 L .33726 .60242 L p .738 .651 .763 r F P 0 g s .36894 .60793 m .43819 .4877 L .45449 .50628 L p .744 .635 .739 r F P 0 g s .48951 .80915 m .55211 .86945 L .56852 .88644 L p 0 .049 .556 r F P 0 g s .55211 .86945 m .48951 .80915 L .49786 .7899 L p .811 .349 0 r F P 0 g s .42175 .46897 m .33726 .60242 L .30551 .59689 L p .732 .663 .781 r F P 0 g s .33726 .60242 m .42175 .46897 L .43819 .4877 L p .738 .651 .763 r F P 0 g s 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1.18677 L p .405 .662 .964 r F P 0 g s .44299 1.18677 m .65737 1.23807 L .64062 1.22244 L p .405 .662 .964 r F P 0 g s .26815 1.0459 m .44299 1.18677 L .45067 1.16901 L p .539 .623 .884 r F P 0 g s .44299 1.18677 m .26815 1.0459 L .24029 1.05562 L p .55 .637 .889 r F P 0 g s .58191 .67055 m .53202 .71111 L .52336 .73108 L p .863 .993 .745 r F P 0 g s .53202 .71111 m .58191 .67055 L .6107 .66025 L p .858 .982 .854 r F P 0 g s .50629 .77047 m .51891 .83507 L .53558 .85233 L p .952 .613 .328 r F P 0 g s .51891 .83507 m .50629 .77047 L .51479 .75086 L p .948 .718 .557 r F P 0 g s .55211 .86945 m .59159 .9057 L .62389 .91066 L p .612 .106 .025 r F P 0 g s .59159 .9057 m .55211 .86945 L .53558 .85233 L p .711 .319 .303 r F P 0 g s .37164 .41185 m .5875 .33251 L .57814 .35395 L p .779 .765 .84 r F P 0 g s .57814 .35395 m .38848 .43105 L .37164 .41185 L p .779 .765 .84 r F P 0 g s .6107 .66025 m .54076 .69096 L .53202 .71111 L p .858 .982 .854 r F P 0 g s .54076 .69096 m .6107 .66025 L .63962 .6499 L p .834 .951 .894 r F P 0 g s .62373 1.20667 m .45841 1.15109 L .45067 1.16901 L p .36 .639 .961 r F P 0 g s .45067 1.16901 m .64062 1.22244 L .62373 1.20667 L p .36 .639 .961 r F P 0 g s .29615 1.03612 m .45067 1.16901 L .45841 1.15109 L p .525 .607 .877 r F P 0 g s .45067 1.16901 m .29615 1.03612 L .26815 1.0459 L p .539 .623 .884 r F P 0 g s .63962 .6499 m .54957 .67063 L .54076 .69096 L p .834 .951 .894 r F P 0 g s .54957 .67063 m .63962 .6499 L .66868 .6395 L p .81 .922 .911 r F P 0 g s .51479 .75086 m .50211 .81767 L .51891 .83507 L p .948 .718 .557 r F P 0 g s .50211 .81767 m .51479 .75086 L .52336 .73108 L p .913 .756 .674 r F P 0 g s .53558 .85233 m .55922 .90072 L .59159 .9057 L p .711 .319 .303 r F P 0 g s .55922 .90072 m .53558 .85233 L .51891 .83507 L p .747 .449 .478 r F P 0 g s .19803 .84524 m .24029 1.05562 L .26815 1.0459 L p .644 .648 .839 r F P 0 g s .35466 .3925 m .59694 .31089 L .5875 .33251 L p .769 .763 .847 r F P 0 g s .5875 .33251 m .37164 .41185 L .35466 .3925 L p .769 .763 .847 r F P 0 g s .66868 .6395 m .55847 .65011 L .54957 .67063 L p .81 .922 .911 r F P 0 g s .55847 .65011 m .66868 .6395 L .69787 .62905 L p .791 .899 .919 r F P 0 g s .60671 1.19078 m .46623 1.13301 L .45841 1.15109 L p .295 .604 .952 r F P 0 g s .45841 1.15109 m .62373 1.20667 L .60671 1.19078 L p .295 .604 .952 r F P 0 g s .32429 1.0263 m .45841 1.15109 L .46623 1.13301 L p .506 .585 .868 r F P 0 g s .45841 1.15109 m .32429 1.0263 L .29615 1.03612 L p .525 .607 .877 r F P 0 g s .79534 1.21082 m .65737 1.23807 L .67399 1.25359 L p .403 .773 .993 r F P 0 g s .67399 1.25359 m .82848 1.21526 L .79534 1.21082 L p .403 .773 .993 r F P 0 g s .23126 .85035 m .26815 1.0459 L .29615 1.03612 L p .642 .638 .831 r F P 0 g s .26815 1.0459 m .23126 .85035 L .19803 .84524 L p .644 .648 .839 r F P 0 g s .79368 .86241 m .81605 .94017 L .78421 .93528 L p .595 .892 .989 r F P 0 g s .51891 .83507 m .52678 .89574 L .55922 .90072 L p .747 .449 .478 r F P 0 g s .52678 .89574 m 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1.19299 m .58954 1.17475 L .60671 1.19078 L p 0 .558 .772 r F P 0 g s .60671 1.19078 m .69545 1.19746 L .662 1.19299 L p 0 .558 .772 r F P 0 g s .39622 .87569 m .45086 .76461 L .4335 .74663 L p .737 .652 .765 r F P 0 g s .4335 .74663 m .36339 .87064 L .39622 .87569 L p .737 .652 .765 r F P 0 g s .50641 1.04004 m .49599 .96635 L .46702 .97646 L p .202 .23 .667 r F P 0 g s .46702 .97646 m .49822 1.05898 L .50641 1.04004 L p .202 .23 .667 r F P 0 g s .36339 .87064 m .4335 .74663 L .41598 .7285 L p .732 .665 .785 r F P 0 g s .41598 .7285 m .33047 .86559 L .36339 .87064 L p .732 .665 .785 r F P 0 g s .49011 1.07775 m .55479 1.1423 L .53719 1.12588 L p .541 .028 0 r F P 0 g s .53719 1.12588 m .49822 1.05898 L .49011 1.07775 L p .541 .028 0 r F P 0 g s .33047 .86559 m .41598 .7285 L .39833 .71022 L p .727 .675 .8 r F P 0 g s .39833 .71022 m .29748 .86052 L .33047 .86559 L p .727 .675 .8 r F P 0 g s .51467 1.02092 m .5251 .95618 L .49599 .96635 L p 0 0 .475 r F P 0 g s .49599 .96635 m .50641 1.04004 L .51467 1.02092 L p 0 0 .475 r F P 0 g s .29748 .86052 m .39833 .71022 L .38052 .69178 L p .722 .684 .812 r F P 0 g s .38052 .69178 m .26441 .85544 L .29748 .86052 L p .722 .684 .812 r F P 0 g s .58954 1.17475 m .662 1.19299 L .62847 1.1885 L p .238 0 0 r F P 0 g s .62847 1.1885 m .57224 1.15859 L .58954 1.17475 L p .238 0 0 r F P 0 g s .55847 .65011 m .41598 .7285 L .4335 .74663 L p .808 .776 .823 r F P 0 g s .41598 .7285 m .55847 .65011 L .56745 .62939 L p .793 .775 .836 r F P 0 g s .68441 1.54094 m .43786 1.48797 L .4452 1.47102 L p .348 .653 .969 r F P 0 g s .43786 1.48797 m .2336 1.33919 L .26212 1.32957 L p .517 .623 .896 r F P 0 g s .26212 1.32957 m .4452 1.47102 L .43786 1.48797 L p .517 .623 .896 r F P 0 g s .26441 .85544 m .38052 .69178 L .36257 .67319 L p .718 .69 .822 r F P 0 g s .36257 .67319 m .23126 .85035 L .26441 .85544 L p .718 .69 .822 r F P 0 g s .52301 1.00162 m .55435 .94597 L .5251 .95618 L p 0 0 .085 r F P 0 g s .5251 .95618 m .51467 1.02092 L .52301 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