zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Entropy analysis of the DNA code dynamics in human chromosomes. (English) Zbl 1228.94032
Summary: Deoxyribonucleic acid, or DNA, is the most fundamental aspect of life but present day scientific knowledge has merely scratched the surface of the problem posed by its decoding. While experimental methods provide insightful clues, the adoption of analysis tools supported by the formalism of mathematics will lead to a systematic and solid build-up of knowledge. This paper studies human DNA from the perspective of system dynamics. By associating entropy and the Fourier transform, several global properties of the code are revealed. The fractional order characteristics emerge as a natural consequence of the information content. These properties constitute a small piece of scientific knowledge that will support further efforts towards the final aim of establishing a comprehensive theory of the phenomena involved in life.

92D20Protein sequences, DNA sequences
Full Text: DOI
[1] Oldham, Keith B.; Spanier, Jerome: The fractional calculus: theory and application of differentiation and integration to arbitrary order, (1974) · Zbl 0292.26011
[2] Samko, Stefan G.; Kilbas, Anatoly A.; Marichev, Oleg I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[3] Miller, Kenneth S.; Ross, Bertram: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002
[4] Oustaloup, Alain: La commande CRONE: commande robuste d’ordre non entier, (1991) · Zbl 0864.93003
[5] Podlubny, Igor: Fractional differential equations, (1999)
[6] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006) · Zbl 1092.45003
[7] Magin, Richard L.: Fractional calculus in bioengineering, (2006)
[8] Sabatier, J.; Agrawal, O. P.; Machado, J. A. Tenreiro: Advances in fractional calculus: theoretical developments and applications in physics and engineering, (2007) · Zbl 1116.00014
[9] Mainardi, Francesco: Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, (2010) · Zbl 1210.26004
[10] Caponetto, Riccardo; Dongola, Giovanni; Fortuna, Luigi; Petráš, Ivo: Fractional order systems: modeling and control applications, (2010) · Zbl 1207.82058
[11] Monje, Concepcion Alicia; Chen, Yangquan; Vinagre, Blas Manuel; Xue, Dingyu; Feliu, Vicente: Fractional order systems and controls: fundamentals and applications, Series: advances in industrial control (2010)
[12] Diethelm, Kai: The analysis of fractional differential equations, Lecture notes in mathematics series (2010) · Zbl 1215.34001
[13] Machado, J. Tenreiro; Kiryakova, Virginia; Mainardi, Francesco: Recent history of fractional calculus, Communications in nonlinear science and numerical simulations 16, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[14] Shannon, C. E.: A mathematical theory of communication, Bell system technical journal 27, 379-423 (1948) · Zbl 1154.94303
[15] Jaynes, E. T.: Information theory and statistical mechanics, Physical review 106, 620 (1957) · Zbl 0084.43701
[16] Khinchin, A. I.: Mathematical foundations of information theory, (1957) · Zbl 0088.10404
[17] Plastino, A.; Plastino, A. R.: Tsallis entropy and Jaynes information theory formalism, Brazilian journal of physics 29, No. 1, 50-60 (1999)
[18] Li, X.; Essex, C.; Davison, M.; Hoffmann, K. H.; Schulzky, C.: Fractional diffusion, irreversibility and entropy, Journal of non-equilibrium thermodynamics 28, No. 3, 279-291 (2003)
[19] Haubold, H. J.; Mathai, A. M.; Saxena, R. K.: Boltzmann--Gibbs entropy versus Tsallis entropy: recent contributions to resolving the argument of Einstein concerning neither herr Boltzmann nor herr Planck has given a definition of W?, Astrophysics and space science 290, No. 3--4, 241-245 (2004) · Zbl 1115.82300 · doi:10.1023/B:ASTR.0000032616.18776.4b
[20] Mathai, A. M.; Haubold, H. J.: Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy, Physica A: statistical mechanics and its applications 375, No. 1, 110-122 (2007)
[21] T. Carter, An Introduction to Information Theory and Entropy, Complex Systems Summer School, Santa Fe, 2007.
[22] Rathie, P.; Da Silva, S.: Shannon, Lévy, and Tsallis: a note, Applied mathematical science 2, No. 28, 1359-1363 (2008) · Zbl 1154.94351 · http://www.m-hikari.com/ams/ams-password-2008/ams-password25-28-2008/dasilvaAMS25-28-2008.pdf
[23] Beck, C.: Generalised information and entropy measures in physics, Contemporary physics 50, No. 4, 495-510 (2009)
[24] Gray, R. M.: Entropy and information theory, (2009)
[25] Ubriaco, M. R.: Entropies based on fractional calculus, Physics letters A 373, No. 30, 2516-2519 (2009) · Zbl 1231.82024 · doi:10.1016/j.physleta.2009.05.026
[26] Taneja, I. G.: On measures of information and inaccuracy, Journal of statistical physics 14, 203-270 (1976)
[27] Sharma, B. D.; Taneja, R. K.: Three generalized additive measures of entropy, Elektronische informationsverarbeitung und kybernetik (EIK) 13, 419-433 (1977) · Zbl 0372.94021
[28] Wehrl, A.: General properties of entropy, Reviews of modern physics 50, 221-260 (1978)
[29] Micklos, David; Freyer, Greg A.: DNA science: A first course, (2003)
[30] Watson, James D.: DNA: the secret of life, (2004)
[31] Schuh, R. T.; Brower, A. V. Z.: Biological systematics: principles and applications, (2009)
[32] , Advances in biochemical engineering biotechnology (2007)
[33] Pearson, H.: Genetics: what is a gene?, Nature 441, No. 7092, 398-401 (2006)
[34] UCSC Genome Bioinformatics. http://hgdownload.cse.ucsc.edu/downloads.html.
[35] Sims, Gregory E.; Jun, Se-Ran; Wu, Guohong A.; Kim, Sung-Hou: Alignment-free genome comparison with feature frequency profiles (FFP) and optimal resolutions, Proceedings of the national Academy of sciences of the united states of America 106, No. 8, 2677-2682 (2009)
[36] Murphy, William J.; Pringle, Thomas H.; Crider, Tess A.; Springer, Mark S.; Miller, Webb: Using genomic data to unravel the root of the placental mammal phylogeny, Genome research 17, 413-421 (2007)
[37] Zhao, Hao; Bourque, Guillaume: Recovering genome rearrangements in the mammalian phylogeny, Genome research 19, 934-942 (2009)
[38] Prasad, Arjun B.; Allard, Marc W.: Confirming the phylogeny of mammals by use of large comparative sequence data sets, Molecular biology and evolution 25, No. 9, 1795-1808 (2008)
[39] Ebersberger, Ingo; Galgoczy, Petra; Taudien, Stefan; Taenzer, Simone; Platzer, Matthias; Von Haeseler, Arndt: Mapping human genetic ancestry, Molecular biology and evolution 24, No. 10, 2266-2276 (2007)
[40] Dunn, Casey W.: Broad phylogenomic sampling improves resolution of the animal tree of life, Nature 452, 745-750 (2008)
[41] Machado, J. A. Tenreiro; Costa, António C.; Quelhas, Maria Dulce: Fractional dynamics in DNA, Communications in nonlinear science and numerical simulations 16, No. 8, 2963-2969 (2011) · Zbl 1218.92038 · doi:10.1016/j.cnsns.2010.11.007
[42] Oppenheim, Alan V.; Willsky, Alan S.; Nawab, S. Hamid: Signals and systems, (1996)