{"id":5,"date":"2017-06-22T10:58:06","date_gmt":"2017-06-22T07:58:06","guid":{"rendered":"https:\/\/coursepages.uta.fi\/mttms11\/?page_id=5"},"modified":"2018-12-17T16:06:42","modified_gmt":"2018-12-17T14:06:42","slug":"paasivu","status":"publish","type":"page","link":"https:\/\/coursepages2.tuni.fi\/mttms11\/","title":{"rendered":"P\u00e4\u00e4sivu"},"content":{"rendered":"<p>&nbsp;<\/p>\n<h2>Vaadittavat opintosuoritukset<\/h2>\n<p>Aktiivinen osallistuminen (esitelm\u00e4t, opponoinnit, itsearviointi, keskustelu). Tarkemmat ohjeet ja tietoa kirjallisuudesta annetaan seminaaritapaamisissa.<\/p>\n<h2>Aikataulu<\/h2>\n<p>Aikataulu tarkentuu ensimm\u00e4isell\u00e4 tapaamiskerralla, se riippuu osallistujam\u00e4\u00e4r\u00e4st\u00e4. Aikataulua voidaan sovittaa opiskelijoiden tarpeiden mukaan. Poissaoloja voi korvata muilla suorituksilla.<\/p>\n<h2>Arvostelu<\/h2>\n<p>Hyv\u00e4ksytty\/hyl\u00e4tty<\/p>\n<h2>Esitelm\u00e4aiheita<\/h2>\n<p>Esitelm\u00e4aiheista sovitaan tarkemmin seminaarissa. Alla on joitakin vaihtoehtoja, mutta omiakin aiheita saa ehdottaa.<\/p>\n<ul>\n<li>Trigonometristen funktioiden eri m\u00e4\u00e4rittelytavat<\/li>\n<li>Kompleksiset trigonometriset funktiot, trigonometristen kaavojen todistuksia<\/li>\n<li>Sinin summakaavan todistuksia (geometrinen ja kompleksianalyyttinen)<\/li>\n<li>Reaalianalyysi\u00e4 kompleksianalyysin keinoin, esim. integraali<\/li>\n<li>Yleinen mittaintegraali ja Riemannin integraali<\/li>\n<li>Analyysi\u00e4 metrisiss\u00e4 avaruuksissa<\/li>\n<li>Arkhimedinen ja ei-arkhimedinen valuaatio (metriikka, ultrametriikka)<\/li>\n<li>Funktioiden jatkuvuus ja lukujonon raja-arvo euklidisissa avaruuksissa R ja R^n sek\u00e4 metrisiss\u00e4 ja topologisissa avaruuksissa<\/li>\n<li>Todenn\u00e4k\u00f6isyyden aksioomat ja klassinen todenn\u00e4k\u00f6isyys<\/li>\n<li>Todenn\u00e4k\u00f6isyyslaskennan ongelmia, esim. kolmen vangin ongelma<\/li>\n<li>Geometrian lauseita: Apolloniuksen lause<\/li>\n<li>S\u00e4\u00e4nn\u00f6llisen monikulmion dihedraaliryhm\u00e4\u00a0<strong>\u00a0<\/strong><\/li>\n<li>Rationaaliluvun desimaaliesitykset ja ryhm\u00e4teoria<\/li>\n<li>Algebralliset ja transkendenttiset luvut, algebralliset ja transkendenttiset funktiot<\/li>\n<li>Irrartionalisuustodistuksia , esim. luvun \u00a0\u03c0 \u00a0irrationaalisuus<\/li>\n<li>Transkendenttisuustodistuksia; esim. luvun\u00a0\u00a0e\u00a0\u00a0transkendenttisuus<\/li>\n<li>Kompleksiluvuista eteenp\u00e4in (kvaterniot, oktoniot)<\/li>\n<li>Lukukunnat: sovelluksena nimitt\u00e4j\u00e4n rationalisointi<\/li>\n<li>Reaalilukujen vastine: p-adiset luvut<\/li>\n<li>Matriisin aste ja lineaarisen yht\u00e4l\u00f6ryhm\u00e4n ratkaisuavaruus<\/li>\n<li>Differentiaaliyht\u00e4l\u00f6iden ratkaisuavaruudet<\/li>\n<li>P\u00e4\u00e4akselilause tasossa ja avaruudessa<\/li>\n<li>\u00c4\u00e4rellinen summa ja tulo, potenssi ja monikerta algebrassa (MAOL:n taulukoiden algebran kaavat yleisiss\u00e4 algebrallisissa struktuureissa)<\/li>\n<li>Aritmetiikkaa kunnissa<\/li>\n<li>Binomikaavan yleistyksi\u00e4: multinomilause, binomityypin polynomit<\/li>\n<li>UFD (faktoriaalinen kokonaisalue) ja Eukleideen alueet (kokonaislukujen ja polynomien ominaisuuksien vertailua, jaollisuus kommutatiivisessa renkaassa)<\/li>\n<li>Polynomin nollakohdat, erityisesti rationaaliset nollakohdat<\/li>\n<li>Algebran peruslauseen vastine eksponenttifunktioille<\/li>\n<li>Polynomiyht\u00e4l\u00f6n juuret\u00a0ja radikaalit<\/li>\n<li>Polynomiyht\u00e4l\u00f6n juuret ja symmetriset funktiot (Vietan kaavat ja Newtonin identiteetit)<\/li>\n<li>Resultantti ja diskriminantti<\/li>\n<li>Eisensteinin jaottomuuskriteeri<\/li>\n<li>Descartes\u2019n merkkis\u00e4\u00e4nt\u00f6 ja Sturmin lause<\/li>\n<li>3. asteen yht\u00e4l\u00f6iden ratkaiseminen<\/li>\n<li>4. asteen yht\u00e4l\u00f6iden ratkaiseminen<\/li>\n<li>Erilaisia keskiarvoja, keskiarvon aksiomatisointi<\/li>\n<li>Aksiomaattiset systeemit ja todistaminen\u00a0kouluopetuksessa<\/li>\n<li>Numeerisia menetelmi\u00e4 (esim. numeerinen integrointi, Newtonin menetelm\u00e4, approksimointi, interpolointi)<\/li>\n<li>Algebran sovelluksia (esim. koodausteoria)<\/li>\n<li>Analyysin sovelluksia (esim. Keplerin lait)<\/li>\n<li>Matemaattisia pelej\u00e4 (esim. Solitaire-peli, Angry Birds)<\/li>\n<li>Matematiikka ja teknologia (esim. Google-hakujen tai GPS-paikannuksen\u00a0matematiikkaa)<\/li>\n<\/ul>\n<p><a href=\"http:\/\/www.jstor.org\/journals\/00029890.html\">The American Mathematical Monthly\u00a0<\/a><\/p>\n<ul>\n<li>J. P. Jones and S. Toporowski, Irrational Numbers, The American Mathematical Monthly Vol. 80, No. 4 (Apr., 1973), pp. 423-424.<\/li>\n<li>N. Calkin and H.S. Wilf, Recounting the Rationals, The American Mathematical Monthly, Vol. 107, No. 4 (Apr., 2000), pp. 360-363.<\/li>\n<li>Xiaoshen Wang,\u00a0 A Simple Proof of Descartes&#8217;s Rule of Signs, The American Mathematical Monthly, Vol. 111, No. 6 (Jun. &#8211; Jul., 2004), pp. 525-526.<\/li>\n<li>Zsolt Lengv\u00e1rszky, Proving the Pythagorean Theorem via Infinite Dissections, The American Mathematical Monthly, Vol. 120, No. 8 (Oct, 2013), pp. 751-753.<\/li>\n<li>Doreen De Leon, Using Undetermined Coefficients to Solve Certain Classes of Variable-Coefficient Equations, American Mathematical Monthly, Vol. 122, No. 3 (March 2015), pp. 246-255<\/li>\n<\/ul>\n<p><a href=\"http:\/\/www.jstor.org\/journals\/07468342.html\">The College Mathematics Journal<\/a><\/p>\n<ul>\n<li>Xun-Cheng Huang, A Discrete L&#8217;Hopital&#8217;s Rule, College Mathematics Journal, Vol. 19, No. 4 (Sep., 1988), pp. 321-329<\/li>\n<li>Airton von Sohsten de Medeiros: Elementary Linear Algebra and the Division Algorithm, The College Mathematics Journal, Vol. 33, No. 1 (Jan., 2002), pp. 51-53<\/li>\n<li>Sharon Barrs, James Braselton and Lorraine Braselton: \u00a0A Rational Root Theorem for Imaginary Roots, The College Mathematics Journal, Vol. 34, No. 5 (Nov., 2003), pp. 380-382<\/li>\n<li>P. K. Subramanian: Successive Differentiation and Leibniz&#8217;s Theorem, The College Mathematics Journal, Vol. 35, No. 4 (Sep., 2004), pp. 274-282<\/li>\n<li>Noah Samuel Brannen and Ben Ford: Logarithmic Differentiation: Two Wrongs Make a Right, The College Mathematics Journal, Vol. 35, No. 5 (Nov., 2004), pp. 388-390<\/li>\n<li>\u00a0Michel Helfgott: Placing the Natural Logarithm and the Exponential Function on an Equal Footing, The College Mathematics Journal, Vol. 35, No. 5 (Nov., 2004), pp. 390-393<\/li>\n<li>Ayshhyah Khazad and Allen J. SchwenkSource:\u00a0 Irrational Roots of Integers, The College Mathematics Journal, Vol. 36, No. 1 (Jan., 2005), pp. 56-57<\/li>\n<li>Jennifer Switkes: A Quotient Rule Integration by Parts Formula, The College Mathematics Journal, Vol. 36, No. 1 (Jan., 2005), pp. 58-60<\/li>\n<li>Horst Martini and Walter Wenzel: The Computation of Derivatives of Trigonometric Functions via the Fundamental Theorem of Calculus, The College Mathematics Journal, Vol. 36, No. 2 (Mar., 2005), pp. 154-158<\/li>\n<li>Jeffrey A. Graham:\u00a0 Self-Integrating Polynomials, The College Mathematics Journal, Vol. 36, No. 4 (Sep., 2005), pp. 318-320<\/li>\n<li>Philip M. Anselone and John W. Lee: Differentiability of Exponential Functions, The College Mathematics Journal, Vol. 36, No. 5 (Nov., 2005), pp. 388-393<\/li>\n<li>Robert Dawson: Differentiate Early, Differentiate Often! The College Mathematics Journal, Vol. 36, No. 5 (Nov., 2005), pp. 404-407<\/li>\n<li>O-Yeat Chan and James Smoak:\u00a0 More Designer Decimals: The Integers and Their Geometric Extensions, The College Mathematics Journal, Vol. 37, No. 5 (Nov., 2006), pp. 355-363.<\/li>\n<li>David Rose: The Pearson and Cauchy-Schwarz Inequalities, The College Mathematics Journal, Vol. 39, No. 1 (Jan., 2008), p. 64<\/li>\n<li>M. Leigh Lunsford, Marcus Pendergrass, Phillip Poplin and David Shoenthal: The Na\u00efve Chain Rule, The College Mathematics Journal, Vol. 39, No. 2 (Mar., 2008), pp. 142-145<\/li>\n<li>Carter C. Gay, Akalu Tefera and Aklilu: The Na\u00efve Product Rule for Derivatives, The College Mathematics Journal, Vol. 39, No. 2 (Mar., 2008), pp. 145-148<\/li>\n<li>Victor H. Moll: An Elementary Trigonometric Equation, The College Mathematics Journal, Vol. 39, No. 5 (Nov., 2008), pp. 395-399<\/li>\n<li>Michael Sheard:\u00a0 Trick or Technique? The College Mathematics Journal, Vol. 40, No. 1 (Jan., 2009), pp. 10-14<\/li>\n<li>Robert D. Poodiack and Kevin J. LeClair: Fundamental Theorems of Algebra for the Perplexes, The College Mathematics Journal, Vol. 40, No. 5 (November 2009), pp. 322-335<\/li>\n<li>Greg Oman:\u00a0 An Independent Axiom System for the Real Numbers, The College Mathematics Journal, Vol. 40, No. 2 (March 2009), pp. 78-86<\/li>\n<li>Vania Mascioni: An Area Approach to the Second Derivative, The College Mathematics Journal, Vol. 38, No. 5 (Nov., 2007), pp. 378-380<\/li>\n<li>Alan Sultan: CORDIC: How Hand Calculators Calculate, College Mathematics Journal, Vol. 40, No. 2 (March 2009), pp. 87-92<\/li>\n<\/ul>\n<p><a href=\"http:\/\/www.tandfonline.com\/loi\/tmes20#.UrK_yj7xvIU\">International Journal of Mathematical Education in Science and Technology\u00a0<\/a><\/p>\n<ul>\n<li>T.\u00a0Eisenberg (2003): On an unknown algorithm for computing square roots. International Journal of Mathematical Education in Science and Technology, 34:1, 153\u2013158<\/li>\n<li>Kin-Keung Poon &amp; Wai-Chee Shiu (2008): On the Dirichlet&#8217;s box principle, International Journal of Mathematical Education in Science and Technology, 39:6, 833-838<\/li>\n<li>Jan Benacka (2009): Power series solution to the pendulum equation, International Journal of Mathematical Education in Science and Technology, 40:8, 1109-1117<\/li>\n<li>Mohamed Allali (2010): Linear algebra and image processing, International Journal of Mathematical Education in Science and Technology, 41:6, 725-741<\/li>\n<li>Manuel A.P. Segurado , Margarida F.B. Silva &amp; Rita Castro (2011): Mathematics in chemistry: indeterminate forms and their meaning, International Journal of Mathematical Education in Science and Technology, 42:5, 664-679<\/li>\n<li>Manuel A.P. Segurado , Margarida F.B. Silva &amp; Rita Castro (2011): Mathematics in chemistry: indeterminate forms and their meaning, International Journal of Mathematical Education in Science and Technology, 42:5, 664-679<\/li>\n<li>Yukio Kobayashi (2013): Recursion formulae of 1^l+2^l+\u2026+n^l and their combinatoric meaning in terms of the tree diagrams, International Journal of Mathematical Education in Science and Technology, 44:1, 132-142<\/li>\n<li>Weng Kin Ho, Foo Him Ho &amp; Tuo Yeong Lee (2013): Exponential function and its derivative revisited, International Journal of Mathematical Education in Science and Technology\u00a044:3, 423-428<\/li>\n<li>Jozef Dobo\u0161 (2013): A continuous, nowhere monotone function, International Journal of Mathematical Education in Science and Technology, 44:4,\u00a0617-620<\/li>\n<li>V.K. Srinivasan (2013): Normals to a parabola, International Journal of Mathematical Education in Science and Technology,\u00a044:4, 568-579<\/li>\n<li>Casey Horgan &amp; Kurt Herzinger (2014) A further investigation of using Theon\u2019s ladder to find roots of quadratic equations, International Journal of Mathematical Education in Science and Technology, 45:1, 150-158\u00a0(Ks. my\u00f6s T. J. Osler, Using Theon\u2019s ladder to find roots of quadratic equations, \u00a0Mathematics and Computer Education, 2008)<\/li>\n<li>Gavin M Abernethy &amp; Mark McCartney (2016): Cannibalism and chaos in the classroom, International Journal of Mathematical Education in Science and Technology<\/li>\n<\/ul>\n<p><a href=\"http:\/\/www.jstor.org\/action\/showPublication?journalCode=mathgaze\">The Mathematical Gazette\u00a0<\/a><\/p>\n<ul>\n<li>Ann E. Hirst, What shape is an ellipse?\u00a0<em>The Mathematical Gazette\u00a0<\/em>Vol. 83, No. 498 (Nov., 1999), pp. 400-409<\/li>\n<li>G. Chambers, An Algorithm for the pth Root,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 83, No. 497 (Jul., 1999), pp. 258-260<\/li>\n<li>Mike Grant and Malcolm Perella, Descending to the Irrational,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 83, No. 497 (Jul., 1999), pp. 263-267<\/li>\n<li>A. Sofo, Generalisation of a Radical Identity,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 83, No. 497 (Jul., 1999), pp. 274-276<\/li>\n<li>Ladislav Beran, The Complex Roots of a Quadratic from a Circle,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 83, No. 497 (Jul., 1999), pp. 287-291<\/li>\n<li>Karel Stroethoff, Heron&#8217;s Formula via Complex Numbers,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 83, No. 497 (Jul., 1999), pp. 292-293<\/li>\n<li>Bob Ardler, How the Scalar and Vector Products Are Derived,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 82, No. 495 (Nov., 1998), pp. 454-456<\/li>\n<li>Robert J. Clarke, The Quadratic Equation Formula,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 82, No. 495 (Nov., 1998), pp. 460-462<\/li>\n<li>Mark Harvey, Ever Decreasing Circles and Inversion,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 82, No. 495 (Nov., 1998), pp. 472-475<\/li>\n<li>Murray Humphreys and Nicholas Macharia, Tests for Divisibility by 19,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 82, No. 495 (Nov., 1998), pp. 475-477<\/li>\n<li>Joerg Meyer, A Further Look at Pythagoras,\u00a0<em>The Mathematical Gazette<\/em>\u00a0Vol. 82, No. 495 (Nov., 1998), p. 488<\/li>\n<li>Pythagoraan lauseen todistuksia: Esim. Rebecca M. Conley and John H. Jaroma, Pythagoras by the Cross Ratio, The College Mathematics Journal, Vol. 37, No. 1 (Jan., 2006), pp. 50-52.<\/li>\n<\/ul>\n<p><a href=\"http:\/\/search.proquest.com.helios.uta.fi\/publication\/35418\/citation?accountid=14242\">Mathematics &amp; Computer Education<\/a><\/p>\n<ul>\n<li>David J. Sprows, Antiderivatives as Inverse Linear Transformations, Mathematics &amp; Computer Education , 44, 1 (2010).<\/li>\n<\/ul>\n<p><a href=\"http:\/\/www.jstor.org\/action\/showPublication?journalCode=mathmaga\">Mathematics Magazine\u00a0<\/a><\/p>\n<ul>\n<li>James E. Ward,\u00a0Vector Spaces of Magic Squares, Mathematics Magazine, Vol. 53, No. 2 (Mar., 1980), pp. 108-111<\/li>\n<li>Hansheng Yang and Heng Yang, The Arithmetic-Geometric Mean Inequality and the Constant e,\u00a0Mathematics Magazine, Vol. 74, No. 4 (Oct., 2001), pp. 321-323<\/li>\n<li>George I. Bell, A Fresh Look at Peg Solitaire,\u00a0<em>Mathematics Magazine<\/em>, Vol. 80, No. 1 (Feb., 2007), pp. 16-28<\/li>\n<li>JOE DeMAIO and ANDY LIGHTCAP: A Graph Theoretic Summation of the Cubes of the First n Integers, Mathematics Magazine, Vol. 82, No. 5 (December 2009), pp. 363-364<\/li>\n<li>Carlos Arcos, Gary Brookfield and Mike Krebs, Mini-Sudokus and Groups,\u00a0<em>Mathematics Magazine<\/em>, Vol. 83, No. 2 (April 2010), pp. 111-122<\/li>\n<li>Christopher Frayer,\u00a0More Polynomial Root Squeezing,\u00a0Mathematics Magazine, Vol. 83, No. 3 (June 2010), pp. 218-221<\/li>\n<li>Alexander Kheifets and James Propp, A Counterexample to Integration by Parts,\u00a0Mathematics Magazine, Vol. 83, No. 3 (June 2010), pp. 222-225<\/li>\n<li>John Franks, Cantor\u2019s Other Proofs that R Is Uncountable, Mathematics Magazine, Vol. 83, No. 4 (October 2010), pp. 283-289<\/li>\n<\/ul>\n<p><a href=\"http:\/\/www.jstor.org\/action\/showPublication?journalCode=mathteacher\">Mathematics Teacher<\/a><\/p>\n<ul>\n<li>John H. Lamb: Angry Birds Mathematics: Parabolas and Vectors,\u00a0<cite>The Mathematics Teacher\u00a0<\/cite>Vol. 107, No. 5 (December 2013\/January 2014), pp. 334-340<\/li>\n<\/ul>\n<p><a href=\"http:\/\/www.tandfonline.com\/loi\/upri20#.UrMHMz7xvIU\">PRIMUS\u00a0(Problems, Resources, and Issues in Mathematics Undergraduate Studies)<\/a><\/p>\n<ul>\n<li>David J. Sprows, Using Linear Algebra to do an Integration by Parts Example, PRIMUS 15, 4 (2005), 303-306.<\/li>\n<li>Barbara A. Shipman, A Comparative Study of Definitions on Limit and Continuity of Functions, PRIMUS 22, 8 (2012), 609-633<\/li>\n<li>L. Jeneva Moseley (2014) Cartooning in Algebra and Calculus, PRIMUS, 24:3, 232-246.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp; Vaadittavat opintosuoritukset Aktiivinen osallistuminen (esitelm\u00e4t, opponoinnit, itsearviointi, keskustelu). Tarkemmat ohjeet ja tietoa kirjallisuudesta annetaan seminaaritapaamisissa. Aikataulu Aikataulu tarkentuu ensimm\u00e4isell\u00e4 tapaamiskerralla, se riippuu osallistujam\u00e4\u00e4r\u00e4st\u00e4. Aikataulua voidaan sovittaa opiskelijoiden tarpeiden mukaan. Poissaoloja voi korvata muilla suorituksilla. Arvostelu Hyv\u00e4ksytty\/hyl\u00e4tty Esitelm\u00e4aiheita Esitelm\u00e4aiheista sovitaan tarkemmin seminaarissa. Alla on joitakin vaihtoehtoja, mutta omiakin aiheita saa ehdottaa. Trigonometristen funktioiden eri m\u00e4\u00e4rittelytavat &hellip; <a href=\"https:\/\/coursepages2.tuni.fi\/mttms11\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">P\u00e4\u00e4sivu<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-5","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/coursepages2.tuni.fi\/mttms11\/wp-json\/wp\/v2\/pages\/5","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/coursepages2.tuni.fi\/mttms11\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/coursepages2.tuni.fi\/mttms11\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/coursepages2.tuni.fi\/mttms11\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/coursepages2.tuni.fi\/mttms11\/wp-json\/wp\/v2\/comments?post=5"}],"version-history":[{"count":12,"href":"https:\/\/coursepages2.tuni.fi\/mttms11\/wp-json\/wp\/v2\/pages\/5\/revisions"}],"predecessor-version":[{"id":34,"href":"https:\/\/coursepages2.tuni.fi\/mttms11\/wp-json\/wp\/v2\/pages\/5\/revisions\/34"}],"wp:attachment":[{"href":"https:\/\/coursepages2.tuni.fi\/mttms11\/wp-json\/wp\/v2\/media?parent=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}