Four Color Theorem and Applied Geophysics

Abstract

A selection of a number of geophysical methods to solve different geological, geodynamical, environmental, archaeological and other problems usually has no theoretical substantiation. The solution to this “four color” mathematical problem is able to assume that two independent geophysical methods are sufficient theoretically to characterize the geological-geophysical peculiarities of the area under study.

Share and Cite:

Eppelbaum, L. (2014) Four Color Theorem and Applied Geophysics. Applied Mathematics, 5, 658-666. doi: 10.4236/am.2014.54062.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Tikhonov, A.N. and Arsenin, V.Y. (1977) Solutions of Ill-Posed Problems. Winston, New York.
[2] Zhdanov, M.S. (2002) Geophysical Inverse Theory and Regularization Problems. In: Ser.: Methods in Geochemistry and Geophysics, 36, Elsevier, Amsterdam.
[3] Hadamard, J. (1902) Sur les Problèmes aux Dérivées Partielles et Leur Signification Physique. Princeton University Bulletin, 13, 49-52.
[4] Khesin, B.E., Alexeyev, V.V. and Eppelbaum, L.V. (1996) Interpretation of Geophysical Fields in Complicated Environments. In: Bookseries: Modern Approaches in Geophysics, Kluwer Academic Publishers, Dordrecht.
[5] Eppelbaum, L.V., Khesin, B.E. and Itkis, S.E. (2001) Prompt Magnetic Investigations of Archaeological Remains in Areas of Infrastructure Development: Israeli Experience. Archaeological Prospection, 8, 163-185.
http://dx.doi.org/10.1002/arp.167
[6] Parasnis, D.S. (1997) Principles of Applied Geophysics. Chapman & Hall, London.
[7] Telford, W.M., Geldart, L.P. and Sheriff, R.E. (1990) Applied Geophysics. 2nd Edition, Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9781139167932
[8] Eppelbaum, L.V., Eppelbaum, V.M. and Ben-Avraham, Z. (2003) Formalization and Estimation of Integrated Geological Investigations: Informational Approach. Geoinformatics, 14, 233-240.
http://dx.doi.org/10.6010/geoinformatics.14.233
[9] Shannon, C.E. (1948) A Mathematical Theory of Communication. Bell System Technical Journal, 27, 379-432, 623-656. http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x
[10] Brillouin, L.N. (1962) Science and Information Theory. 2nd Edition, Academic Press, New York.
[11] Khalfin, L.A. (1958) Information Theory of Geophysical Interpretation. Doklady Akademii Nauk SSSR, 122, 10071010.
[12] Svetov, B.S. (1992) Information Theory Basis of Geophysics. In: Bookseries: Advanced Geophysics. Russian Experience, Issue No. 1, Electromagnetic Research Centre, Moscow.
[13] Nikitin, A.A. (1993) Statistical Processing of Geophysical Data. In: Bookseries: Advanced Geophysics. Russian Experience, Issue No. 2, Electromagnetic Research Centre, Moscow.
[14] Gadallah, M.R. and Fisher, R. (2009) Exploration Geophysics. Springer, Berlin.
[15] Eppelbaum, L.V. and Khesin, B.E. (2012) Geophysical Studies in the Caucasus. Springer, Berlin.
http://dx.doi.org/10.1007/978-3-540-76619-3
[16] Ventsel, E.S. (1969) The Probability Theory. 3rd Edition, Nauka, Moscow. (in Russian)
[17] Duda, R.O. and Hart, P.E. (1973) Pattern Classification and Scene Analysis. John Wiley & Sons, Hoboken.
[18] Appel, K. and Haken, W. (1977) Every Planar Map Is Four Colorable: Part 1, Discharging. Illinois Journal of Mathematics, 21, 429-490.
[19] Appel, K., Haken, W. and Koch, S. (1977) Every Planar Mar Is Four Colorable: Part 2, Reducibility. Illinois Journal of Mathematics, 21, 491-567.
[20] Robertson, N., Sanders, D.P., Seymour, P.D. and Thomas, R. (1997) The Four Color Theorem. Journal of Combinatorial Theory, Series B, 70, 2-44. http://dx.doi.org/10.1006/jctb.1997.1750
[21] Thomas, R. (1998) An Update on the Four-Color Theorem. Notices of the AMS, 45, 848-859.
[22] Hansen, W.R. (1978) Suggestions to Authors of the Reports of the United States Geological Survey. 6th Edition, US Government Publishing Office, Washington DC.
[23] Nikitin, A.A. (1986) Foundation of Integrated Interpretation of Geophysical Data. In: Brodovoi, V.V. and Nikitin, A.A., Eds., Integration of Geophysical Methods, Nedra, Moscow, 42-57. (in Russian)
[24] Eppelbaum, L.V. (1989) The Development of Methods for Processing and Interpretation of Natural Geophysical Fields in Prospecting for Pyrite Ores in Mountainous Conditions. PhD Thesis, Institute of Geophysics (Georgia Academy of Science), Tbilisi. (in Russian)

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.