zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Shannon, Rényie and Tsallis entropy analysis of DNA using phase plane. (English) Zbl 1231.92034
Summary: This paper analyzes DNA information using entropy and phase plane concepts. First, the DNA code is converted into a numerical format by means of histograms that capture DNA sequence lengths ranging from one up to ten bases. This strategy measures dynamical evolutions from 4 up to 4 10 signal states. The resulting histograms are analyzed using three distinct entropy formulations namely the Shannon, Rényie and Tsallis definitions. Charts of entropy versus sequence length are applied to a set of twenty four species, characterizing 486 chromosomes. The information is synthesized and visualized by adapting phase plane concepts leading to a categorical representation of chromosomes and species.
MSC:
92C40Biochemistry, molecular biology
94A17Measures of information, entropy
Keywords:
chromosomes
References:
[1]Schuh, R. T.; Brower, A. V. Z.: Biological systematics: principles and applications, (2009)
[2], Advances in biochemical engineering biotechnology (2007)
[3]Pearson, H.: Genetics: what is a gene?, Nature 441, No. 7092, 398-401 (2006)
[4]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
[5]Costa, António C.; Machado, J. A. Tenreiro; Quelhas, Maria Dulce: Histogram-based DNA analysis for the visualization of chromosome, genome and species information, , 1207-1214 (2011)
[6]Machado, J. A. Tenreiro; Costa, António C.; Quelhas, Maria Dulce: Entropy analysis of the DNA code dynamics in human chromosomes, Computers & mathematics with applications (2011)
[7]UCSC Genome Bioinformatics. http://hgdownload.cse.ucsc.edu/downloads.html.
[8]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)
[9]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)
[10]Zhao, Hao; Bourque, Guillaume: Recovering genome rearrangements in the mammalian phylogeny, Genome research 19, 934-942 (2009)
[11]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)
[12]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)
[13]Dunn, Casey W.: Broad phylogenomic sampling improves resolution of the animal tree of life, Nature 452, 745-750 (2008)
[14]Li, Chun; Ma, Hong; Zhou, Yang; Wang, Xiaolei; Zheng, Xiaoqi: Similarity analysis of DNA sequences based on the weighted pseudo-entropy, Journal of computational chemistry 32, No. 4, 675-680 (2011)
[15]Georgiou, D. N.; Karakasidis, T. E.; Nieto, Juan J.; Torres, A.: A study of entropy/clarity of genetic sequences using metric spaces and fuzzy sets, Journal of theoretical biology 267, No. 1, 95-105 (2010)
[16]Georgiou, D. N.; Karakasidis, T. E.; Nieto, J. J.; Torres, A.: Use of fuzzy clustering technique and matrices to classify amino acids and its impact to chou’s pseudo amino acid composition, Journal of theoretical biology 257, No. 1, 17-26 (2009)
[17]Lachowicz, Mirosław: Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear analysis: real world applications 12, No. 4, 2396-2407 (2011) · Zbl 1225.93101 · doi:10.1016/j.nonrwa.2011.02.014
[18]Nieto, J. J.; Torres, A.; Vázquez-Trasande, M. M.: A metric space to study differences between polynucleotides, Applied mathematics letters 16, No. 8, 1289-1294 (2003) · Zbl 1106.92307 · doi:10.1016/S0893-9659(03)90131-5
[19]Tanaka, Y.; Yoshimura, I.; I; Nakai, K.: Positional variations among heterogeneous nucleosome maps give dynamical information on chromatin, Chromosoma 119, No. 4, 391-404 (2010)
[20]Shannon, C. E.: A mathematical theory of communication, Bell system technical journal 27, No. July, October (1948) · Zbl 1154.94303
[21]Jaynes, E. T.: Information theory and statistical mechanics, Physics review 106, 620 (1957) · Zbl 0084.43701
[22]Khinchin, A. I.: Mathematical foundations of information theory, (1957) · Zbl 0088.10404
[23]Plastino, A.; Plastino, A. R.: Tsallis entropy and Jaynes information theory formalism, Brazilian journal of physics 29, No. 1, 50-60 (1999)
[24]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)
[25]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
[26]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)
[27]T. Carter, An introduction to information theory and entropy, Complex Systems Summer School, Santa Fe, June 2007.
[28]Rathie, P.; Da Silva, S.: Shannon, Lévy, and Tsallis: a note, Applied mathematical sciences 2, No. 28, 1359-1363 (2008) · Zbl 1154.94351 · doi:http://www.m-hikari.com/ams/ams-password-2008/ams-password25-28-2008/dasilvaAMS25-28-2008.pdf
[29]Beck, C.: Generalised information and entropy measures in physics, Contemporary physics 50, No. 4, 495-510 (2009)
[30]Gray, R. M.: Entropy and information theory, (2009)
[31]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
[32]Taneja, I. G.: On measures of information and inaccuracy, Journal of statistical physics 14, 203-270 (1976)
[33]Sharma, B. D.; Taneja, R. K.: Three generalized additive measures of entropy, Elektronische informations und kybernetik 13, 419-433 (1977) · Zbl 0372.94021
[34]Wehrl, A.: General properties of entropy, Reviews of modern physics 50, 221-260 (1978)