Loading document ...

Page of

Loading page ...

Author(s): Yi-Fang Chang

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

- A.Robinson, Non-Standard Analysis. North-Holland Publishing Company. 1974
- L.H.Loomis, Calculus (Third Edition). Addison-Wesley Publishing Company. 1982
- G.H.Hardy, A Course of Pure Mathematics (Tenth Edition). Cambridge Mathematical Library. 1993
- T.J.Iâ€™A.Bromwich, An Introduction to the Theory of Infinite Series. AMS Chelsea Publishing. 2005
- Yi-Fang Chang, Differential test for series of positive terms. J.Yunnan Normal University. 1999, 19(3):5-7
- R.Diestel, Graph Theory (Second Edition). Springer.2000
- B.Bollobas, Modern Graph Theory. Springer-Verlag.2002
- A.P.Wolff. About Philosophy. Prentice-Hall.2005
- M.Eigen and P.Schuster, The Hypercycle. Springer. 1978
- Yi-Fang Chang, Matrix representation of graph theory on hypercycle and its development and application. J.Jishou University. 2011,32(2):36-41
- Yi-Fang Chang, Economic crisis, nonlinear economic growth theory and its three laws. J.Jishou University. 2012,33(3):94-99
- 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
- Yi-Fang Chang, Multiply connected topology and basic principles and mathematical analysis of east economics. J.Jishou University. 2010,41(4):59-66
- M.Klein, Mathematical Thought From Ancient to Modern Times. Oxford Univ.Press. 1972
- I.R.Shafarevich, Basic Notions of Algebra. Springer-Verlag. 2005
- L.E.Dickson, Linear Algebras. Cambridge Univ.Press.1914
- M.Artin, Algebra. Prentice-Hall,Inc.1991.p492-500
- P.Roman, Theory of Elementary Particles. North-Holland Publ.Co.1964
- Q.H.Jr.Good, Properties of the Dirac matrices. Rev.Mod.Phys., 1955,27(2):187-211
- T.D.Lee, Particle Physics and Introduction to Field Theory. Harwood Academic Publishers. 1983
- L.A.Segel, The infinite and the infinitesimal in models for natural phenomena. Rev.Mod.Phys. 1991,63(2):225-238
- Yi-Fang Chang, Physical meaning of non-normal complex number. J.Potential Science (China). 1982,2,13
- P.Fayet and S.Ferrara, Supersymmetry. Phys.Reports. 1977, 32(5):249-334
- Yi-Fang Chang, High energy behavious of particles and unified statistics. Hadronic J. 1984,7(5):1118-1133
- Yi-Fang Chang, Some possible tests of the inapplicability of Pauliâ€™s exclusion principle. Hadronic J. 1984,7(6):1469-1473
- Yi-Fang Chang, New Research of Particle Physics and Relativity. Yunnan Science and Technology Press, 1989; Phys.Abst. 93,1371(1990)
- Yi-Fang Chang, Test of Pauliâ€™s exclusion principle in particle physics, astrophysics and other fields. Hadronic J. 1999,22(3):257-268
- Yi-Fang Chang, Possible Outlet of Test on Violation of Pauli Exclusion Principle. arXiv:0709. 0550
- Yi-Fang Chang, Supersymmetry, super-unification and higher-dimensional complex space in particle physics. J.Yunnan University. 2003,25(1):37-40.
- Yi-Fang Chang, Some New Unifications in Supersymmetry and Higher Dimensional Complex Space. arXiv:0804.0267
- Yi-Fang Chang, Fractals model of particle, complex dimension and its meaning. Exploration of Nature (China). 1988,7(2):21-23
- 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
- Yi-Fang Chang, Fractal relativity, generalized Noetherâ€™s theorem and new research on space-time. Galilean Electrodynamics. 2010,21(6):112-116
- P.A.M.Dirac, The Principles of Quantum Mechanics. Oxford. 1958
- Yi-Fang Chang, Unification of quantum and general relativity. J.Yunnan University. 2010,32(5): 537-541
- J.T.Bonner, The Evolution of Complexity. Princeton University Press. 1988
- J.H.Holland, Hidden Order: How Adaptation Builds Complexity. Reading, MA: Addison Wesley. 1995
- I.Prigogine and I.Stengers, Order Out of Chaos. New York: Bantam Books Inc. 1984
- N.Rescher, Complexity: A Philosophical Overview. Transaction Publishers. 1998
- Yi-Fang Chang, Social Synergetics, Social Physics and Research of Fundamental Laws in Social Complex Systems. arXiv:0911.1155(2009)
- Yi-Fang Chang, Nonlinear whole biology and its basic laws. Chinese Science Abstracts. 2001,7: 227-228
- Chang Yi-Fang. Nonlinear whole biology and loop quantum theory applied to biology. NeuroQuantology. 2012,10(2):190-197

**Cite this Article:**

International Journal of Sciences is Open Access Journal.

This article is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License.

Author(s) retain the copyrights of this article, though, publication rights are with Alkhaer Publications.

World-wide Delivery is FREE

Share this Issue with Friends:

Submit your Paper

© Copyright (c) 2012 Alkhaer Publications. All rights reserved. This work is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License.