Differential Test for Series of Positive Terms, New Tree-Field Representations in Graph Theory and New Number Field, Extension of Dirac Extraction, and Their Applications

Differential Test for Series of Positive Terms, New Tree-Field Representations in Graph Theory and New Number Field, Extension of Dirac Extraction, and Their Applications

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Author(s): Yi-Fang Chang

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673 1108 121-133 Volume 1 - Dec 2012


First, a new differential test for series of positive terms is proved. Let f(x) be a positive continuous function corresponded to a series of positive terms , and g(x) is a derivative of reciprocal of f(x), i.e., . Then, if for enough large x, the series converges; if , the series diverges. The rest may make the limit form, and is universal and complete. Next, the tree in the graph theory is extended to a new tree-field representation. It includes two parts: tree and field. A field is a set of legion small trees. They can transform each other between tree and field. This is a unification of simplicity (tree) and complexity (field), and may be applied to various complex systems on science, politics, economy, philosophy and so on. Further, it may be extended to the whole graph theory G=(V,E,F), here F is a set of small graphs. Third, based on a brief review on developments of number system, a new developed pattern is proposed. The quaternion is extended to a matrix form aI+bC+cB+dA, in which the unit matrix I and three special matrices C,B,A correspond to number 1 and three units of imaginary number i,j,k, respectively. They form usually a ring. But some fields may be composed of some special 2-rank, even n-rank matrices, for example, three matrices aI+bC, aI+cB, aI+dA and so on. It is a new type of hypercomplex number fields. The physical applications and possible meaning of the new number system is researched. Finally, the Dirac extraction is extended to any terms whose extraction should be and , etc. Moreover, the general complexity is also discussed.


series of positive terms, convergence and divergence, differential, infinite integral, graph theory, tree, number system, matrix, complex number, ring, field, extraction, application, complexity


  1. A.Robinson, Non-Standard Analysis. North-Holland Publishing Company. 1974
  2. L.H.Loomis, Calculus (Third Edition). Addison-Wesley Publishing Company. 1982
  3. G.H.Hardy, A Course of Pure Mathematics (Tenth Edition). Cambridge Mathematical Library. 1993
  4. T.J.I’A.Bromwich, An Introduction to the Theory of Infinite Series. AMS Chelsea Publishing. 2005
  5. Yi-Fang Chang, Differential test for series of positive terms. J.Yunnan Normal University. 1999, 19(3):5-7
  6. R.Diestel, Graph Theory (Second Edition). Springer.2000
  7. B.Bollobas, Modern Graph Theory. Springer-Verlag.2002
  8. A.P.Wolff. About Philosophy. Prentice-Hall.2005
  9. M.Eigen and P.Schuster, The Hypercycle. Springer. 1978
  10. Yi-Fang Chang, Matrix representation of graph theory on hypercycle and its development and application. J.Jishou University. 2011,32(2):36-41
  11. Yi-Fang Chang, Economic crisis, nonlinear economic growth theory and its three laws. J.Jishou University. 2012,33(3):94-99
  12. Yi-Fang Chang, Multiply connected topological economics, nonlinear theory of economic growth and its three laws, and four theorems on knowledge economic theory. Global Journal of Science Frontier Research. 2012, 12(13):1-13
  13. Yi-Fang Chang, Multiply connected topology and basic principles and mathematical analysis of east economics. J.Jishou University. 2010,41(4):59-66
  14. M.Klein, Mathematical Thought From Ancient to Modern Times. Oxford Univ.Press. 1972
  15. I.R.Shafarevich, Basic Notions of Algebra. Springer-Verlag. 2005
  16. L.E.Dickson, Linear Algebras. Cambridge Univ.Press.1914
  17. M.Artin, Algebra. Prentice-Hall,Inc.1991.p492-500
  18. P.Roman, Theory of Elementary Particles. North-Holland Publ.Co.1964
  19. Q.H.Jr.Good, Properties of the Dirac matrices. Rev.Mod.Phys., 1955,27(2):187-211
  20. T.D.Lee, Particle Physics and Introduction to Field Theory. Harwood Academic Publishers. 1983
  21. L.A.Segel, The infinite and the infinitesimal in models for natural phenomena. Rev.Mod.Phys. 1991,63(2):225-238
  22. Yi-Fang Chang, Physical meaning of non-normal complex number. J.Potential Science (China). 1982,2,13
  23. P.Fayet and S.Ferrara, Supersymmetry. Phys.Reports. 1977, 32(5):249-334
  24. Yi-Fang Chang, High energy behavious of particles and unified statistics. Hadronic J. 1984,7(5):1118-1133
  25. Yi-Fang Chang, Some possible tests of the inapplicability of Pauli’s exclusion principle. Hadronic J. 1984,7(6):1469-1473
  26. Yi-Fang Chang, New Research of Particle Physics and Relativity. Yunnan Science and Technology Press, 1989; Phys.Abst. 93,1371(1990)
  27. Yi-Fang Chang, Test of Pauli’s exclusion principle in particle physics, astrophysics and other fields. Hadronic J. 1999,22(3):257-268
  28. Yi-Fang Chang, Possible Outlet of Test on Violation of Pauli Exclusion Principle. arXiv:0709. 0550
  29. Yi-Fang Chang, Supersymmetry, super-unification and higher-dimensional complex space in particle physics. J.Yunnan University. 2003,25(1):37-40.
  30. Yi-Fang Chang, Some New Unifications in Supersymmetry and Higher Dimensional Complex Space. arXiv:0804.0267
  31. Yi-Fang Chang, Fractals model of particle, complex dimension and its meaning. Exploration of Nature (China). 1988,7(2):21-23
  32. Yi-Fang Chang, In mathematics and physics development of fractal dimension and fractal space-time theory. Exploration of Nature (China). 1991,10(2):49-54
  33. Yi-Fang Chang, Fractal relativity, generalized Noether’s theorem and new research on space-time. Galilean Electrodynamics. 2010,21(6):112-116
  34. P.A.M.Dirac, The Principles of Quantum Mechanics. Oxford. 1958
  35. Yi-Fang Chang, Unification of quantum and general relativity. J.Yunnan University. 2010,32(5): 537-541
  36. J.T.Bonner, The Evolution of Complexity. Princeton University Press. 1988
  37. J.H.Holland, Hidden Order: How Adaptation Builds Complexity. Reading, MA: Addison Wesley. 1995
  38. I.Prigogine and I.Stengers, Order Out of Chaos. New York: Bantam Books Inc. 1984
  39. N.Rescher, Complexity: A Philosophical Overview. Transaction Publishers. 1998
  40. Yi-Fang Chang, Social Synergetics, Social Physics and Research of Fundamental Laws in Social Complex Systems. arXiv:0911.1155(2009)
  41. Yi-Fang Chang, Nonlinear whole biology and its basic laws. Chinese Science Abstracts. 2001,7: 227-228
  42. Chang Yi-Fang. Nonlinear whole biology and loop quantum theory applied to biology. NeuroQuantology. 2012,10(2):190-197

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