Ojlerova karakteristika
U matematici, a tačnije u algebarskoj topologiji i poliedarskoj kombinatorici, Ojlerova karakteristika (u pojedinim granama matematike ponekad referisana i samo kao karakteristika ili Ojlerov broj — ne treba mešati sa Ojlerovom konstantom, na koju se, takođe, često referiše kao na Ojlerov broj) je invarijantna vrednost koja zavisi od topološkog oblika i osobina objekta koji opisuje. Najčešće se obeležava malim grčkim slovom χ (hi). Naziv zahvaljuje Leonardu Ojleru, poznatom švajcarskom matematičaru i fizičaru.
Originalno se upotrebljavala u geometriji za opisivanje poliedara, ali je svoju primenu pronašla u topologiji i kasnije u teoriji grafova. To je navedeno za platonska tela 1537. godine u neobjavljenom rukopisu Frančeska Maurolika.[1] Leonard Ojler, po kome je koncept dobio ime, uveo ga je generalno za konveksne poliedre, ali nije uspeo da rigorozno dokaže da je on invarijanta. U savremenoj matematici, Ojlerova karakteristika proizilazi iz homologije i, apstraktnije, homološke algebre.[2][3][4][5]
Ojlerova karakteristika u geometriji i topologiji[uredi | uredi izvor]
Ojlerova karakteristika geometrijske figure u geometriji označava sumu , gde je T broj temena figure, I broj ivica a P broj pljosni date figure. Upravo ovaj identitet[6] je prvi dokazao Ojler.
Jasno, svaki trougao ima karakteristiku 1 (3 temena, 3 ivice i jedna pljosan). Odavde sledi da i svaka ravanska figura ima Ojlerovu karakteristiku 1 (svaka figura u ravni se može triangulisati[7], tj. razložiti na više manjih trouglova — sada se spajanjem dva trougla po zajedničkoj ivici karakteristika ne menja, jer se broj temena povećava za 1, broj ivica za 2, a broj pljosni za 1). Kako se i svaki poliedar može razložiti na lanac povezanih poliedara, to je karakteristika celog poliedra upravo 2 (nastavljanjem poliedara jedan na drugi se karakteristika ne menja, slično kao malopre, ali se pri dodavanju „poslednjeg” poliedra broj ivica i temena ne menja, a dobija se dodatna pljosan).[8] Uopšteno, za pravilan poliedar sa n „rupa” važi da mu je karakteristika 2(1-n) (npr. torus je karakteristike 0). Ispod je data tabela nekih konveksnih i nekih nekonveksnih trodimenzionalnih geometrijskih figura sa svojim karakteristikama.
Naziv | Slika | Konveksnost | Broj temena (T) |
Broj ivica (I) |
Broj pljosni (P) |
Karakteristika |
---|---|---|---|---|---|---|
Tetraedar | konveksan | 4 | 6 | 6 | 2 | |
Heksaedar (kocka) |
konveksan | 8 | 12 | 6 | 2 | |
Oktaedar | konveksan | 6 | 12 | 8 | 2 | |
Dodekaedar | konveksan | 20 | 30 | 12 | 2 | |
Ikosaedar | konveksan | 12 | 30 | 20 | 2 | |
Tetrahemiheksaedar | konkavan | 6 | 12 | 7 | 1 | |
Oktahemioktaedar | konkavan | 12 | 24 | 12 | 0 | |
Mali zvezdasti dodekaedar | konkavan | 12 | 30 | 12 | -6 | |
Veliki zvezdasti dodekaedar | konkavan | 20 | 30 | 12 | 2 |
Slično kao u geometriji se definiše Ojlerova karakteristika i u topologiji. Ispod se nalazi tabela sa nekim topološkim oblicima sa svojim karakteristikama.
Naziv | Slika | Konveksnost | Karakteristika |
---|---|---|---|
Sfera | konveksan | 2 | |
Torus | konkavan | 0 | |
Dupli (dvorupi) torus |
konkavan | -2 | |
Trorupi torus | konkavan | -4 |
Ojlerova karakteristika u teoriji grafova[uredi | uredi izvor]
Ojlerova karakteristika planarnog grafa G u teoriji grafova je rezultat , gde je V(G) skup čvorova grafa G, E(G) skup grana grafa G, a f(G’) broj oblasti na koje planarno utapanje G’ grafa G razdeljuje ravan ℝ × ℝ svojim granama i čvorovima.
Može se pokazati da svi planarni grafovi imaju Ojlerovu karakteristiku 2 (u teoriji grafova je ovo tvrđenje poznato kao Ojlerova teorema[9]). U opštem slučaju će važiti, za proizvoljan graf G, , gde je ω(G) broj komponenti povezanosti grafa G.
Ispod je data tabela sa nekoliko grafova i njihovim karakteristikama.
Vidi još[uredi | uredi izvor]
Reference[uredi | uredi izvor]
- ^ Friedman, Michael (2018). A History of Folding in Mathematics: Mathematizing the Margins. Science Networks. Historical Studies. 59. Birkhäuser. str. 71. ISBN 978-3-319-72486-7. doi:10.1007/978-3-319-72487-4.
- ^ Cartan, Henri Paul; Eilenberg, Samuel (1956). Homological Algebra. Princeton mathematical series. 19. Princeton University Press. ISBN 9780674079779. OCLC 529171.
- ^ Eilenberg, Samuel; Moore, J.C. (1965). Foundations of relative homological algebra. Memoirs of the American Mathematical Society number. 55. American Mathematical Society. ISBN 9780821812556. OCLC 1361982.
- ^ Pellikka, M; S. Suuriniemi; L. Kettunen; C. Geuzaine (2013). „Homology and Cohomology Computation in Finite Element Modeling” (PDF). SIAM J. Sci. Comput. 35 (5): B1195—B1214. CiteSeerX 10.1.1.716.3210 . doi:10.1137/130906556.
- ^ Arnold, Douglas N.; Richard S. Falk; Ragnar Winther (16. 5. 2006). „Finite element exterior calculus, homological techniques, and applications”. Acta Numerica. 15: 1—155. Bibcode:2006AcNum..15....1A. S2CID 122763537. doi:10.1017/S0962492906210018.
- ^ „Euler's Formula”. Encyclopaedia Britannica.
- ^ „Computational Geometry” (PDF).
- ^ „Euler's Characteristic in Algebraic Topolgy”. San José State University. Arhivirano iz originala 25. 02. 2020. g. Pristupljeno 12. 02. 2020.
- ^ „Euler's Formula”.
Literatura[uredi | uredi izvor]
- Richeson, David S.; Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press 2008.
- Flegg, H. Graham; From Geometry to Topology, Dover 2001, p. 40.
- Bott, Raoul and Tu, Loring W. (1982). Differential Forms in Algebraic Topology. Springer-Verlag. ISBN 0-387-90613-4.
- Bredon, Glen E. (1993). Topology and Geometry. Springer-Verlag. ISBN 0-387-97926-3.
- Milnor, John W.; Stasheff, James D. (1974). Characteristic Classes. Princeton University Press. ISBN 0-691-08122-0.
- Cromwell, Peter R. (1997), Polyhedra, Cambridge: Cambridge University Press, ISBN 978-0-521-55432-9, MR 1458063.
- Grünbaum, Branko (1994), „Polyhedra with hollow faces”, Ur.: Bisztriczky, Tibor; Schneider, Peter McMullen;Rolf; Weiss, A., Proceedings of the NATO Advanced Study Institute on Polytopes: Abstract, Convex and Computational, Dordrecht: Kluwer Acad. Publ., str. 43—70, ISBN 978-94-010-4398-4, MR 1322057, doi:10.1007/978-94-011-0924-6_3.
- Grünbaum, Branko (2003), „Are your polyhedra the same as my polyhedra?” (PDF), Ur.: Aronov, Boris; Basu, Saugata; Pach, János; Sharir, Micha, Discrete and Computational Geometry. Algorithms and Combinatorics (PDF), Algorithms and Combinatorics, 25, Berlin: Springer, str. 461—488, CiteSeerX 10.1.1.102.755 , ISBN 978-3-642-62442-1, MR 2038487, doi:10.1007/978-3-642-55566-4_21, Arhivirano iz originala (PDF) 03. 08. 2016. g., Pristupljeno 12. 02. 2020.
- Richeson, David S. (2008), Euler's Gem: The polyhedron formula and the birth of topology, Princeton, NJ: Princeton University Press, ISBN 978-0-691-12677-7, MR 2440945
- Henri Cartan, Samuel Eilenberg, Homological algebra. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp. ISBN 0-691-04991-2
- Grothendieck, Alexander (1957). „Sur quelques points d'algèbre homologique, I”. Tohoku Mathematical Journal. 9 (2): 119—221. doi:10.2748/tmj/1178244839.
- Saunders Mac Lane, Homology. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8
- Peter Hilton; Stammbach, U. A course in homological algebra. Second edition. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp. ISBN 0-387-94823-6
- Gelfand, Sergei I.; Yuri Manin, Methods of homological algebra. Translated from Russian 1988 edition. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xx+372 pp. ISBN 3-540-43583-2
- Gelfand, Sergei I.; Yuri Manin, Homological algebra. Translated from the 1989 Russian original by the authors. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences (Algebra, V, Encyclopaedia Math. Sci., 38, Springer, Berlin, 1994). Springer-Verlag, Berlin, 1999. iv+222 pp. ISBN 3-540-65378-3
- Serge Lang: Algebra. 3rd edition, Springer 2002, ISBN 978-0-387-95385-4, pp. 157–159 (online copy, str. 157, na sajtu Gugl knjige)
- M. F. Atiyah; I. G. Macdonald: Introduction to Commutative Algebra. Oxford 1969, Addison–Wesley Publishing Company, Inc. ISBN 0-201-00361-9.
- Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory, Progress in Mathematics, 174, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1
- Hovey, Mark (1999), Model categories, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1359-1
- Quillen, Daniel (1967), Homotopical Algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-03914-5
- R.L. Taylor, Covering groups of non connected topological groups, Proceedings of the American Mathematical Society, vol. 5 (1954), 753–768.
- R. Brown and O. Mucuk, Covering groups of non-connected topological groups revisited, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 115 (1994), 97–110.
- R. Brown and T. Porter, On the Schreier theory of non-abelian extensions: generalisations and computations, Proceedings of the Royal Irish Academy, vol. 96A (1996), 213–227.
- G. Janelidze and G. M. Kelly, Central extensions in Malt'sev varieties, Theory and Applications of Categories, vol. 7 (2000), 219–226.
- P. J. Morandi, Group Extensions and H3. From his collection of short mathematical notes.
- Buchsbaum, David A. (1955), „Exact categories and duality”, Transactions of the American Mathematical Society, 80 (1): 1—34, ISSN 0002-9947, JSTOR 1993003, MR 0074407, doi:10.1090/S0002-9947-1955-0074407-6
- Freyd, Peter (1964), Abelian Categories, New York: Harper and Row
- Mitchell, Barry (1965), Theory of Categories, Boston, MA: Academic Press
- Popescu, Nicolae (1973), Abelian categories with applications to rings and modules, Boston, MA: Academic Press
- Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90613-3
- Hatcher, Allen (2002). Algebraic Topology. Cambridge: Cambridge University Press. ISBN 0-521-79540-0.
- Cox, David (2004). Galois Theory. Wiley-Interscience. ISBN 9781118031339. MR 2119052.
- Jacobson, Nathan (2009). Basic Algebra I (2nd izd.). Dover Publications. ISBN 978-0-486-47189-1.
- Rose, John S. (2012). A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8. Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.
- Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York, Heidelberg: Springer-Verlag, ISBN 0-387-90244-9, MR 0463157
- May, J. Peter (1999), A Concise Course in Algebraic Topology (PDF), University of Chicago Press, ISBN 0-226-51182-0, MR 1702278
- Switzer, Robert (1975), Algebraic Topology — Homology and Homotopy, Springer-Verlag, ISBN 3-540-42750-3, MR 0385836
- Thom, René (1954), „Quelques propriétés globales des variétés différentiables”, Commentarii Mathematici Helvetici, 28: 17—86, MR 0061823, S2CID 120243638, doi:10.1007/BF02566923[mrtva veza]
Spoljašnje veze[uredi | uredi izvor]
- Weisstein, Eric W. „Euler characteristic”. MathWorld.
- Weisstein, Eric W. „Polyhedral formula”. MathWorld.
- Matveev, S.V. (2001) [1994], „Euler characteristic”, Ur.: Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Euler Characteristic of the Barycentric Subdivision of an n-Simplex. In math.stackexchange.
- Euler characteristic constant under barycentric subdivision. In math.stackexchange.