Chaotic root-finding for a small class of polynomials

In this paper we present a new closed-form solution to a chaotic difference equation, $y_{n+1} = a_2 y_{n}^2 + a_1 y_{n} + a_0$ with coefficient $a_0 = (a_1 - 4)(a_1 + 2) / (4 a_2)$, and using this solution, show how corresponding exact roots to a special set of related polynomials of order $2^p, p \in \mathbb{N}$ with two independent parameters can be generated, for any $p$

Escape rate and Hausdorff measure for entire functions

The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.Comment: 24 pages; some errors corrected, proof of Theorem 2 shortene

CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior

[EN] A family of fourth-order iterative methods without memory, for solving nonlinear systems, and its seventh-order extension, are analyzed. By using complex dynamics tools, their stability and reliability are studied by means of the properties of the rational function obtained when they are applied on quadratic polynomials. The stability of their fixed points, in terms of the value of the parameter, its critical points and their associated parameter planes, etc. give us important information about which members of the family have good properties of stability and whether in any of them appear chaos in the iterative process. The conclusions obtained in this dynamical analysis are used in the numerical section, where an academical problem and also the chemical problem of predicting the diffusion and reaction in a porous catalyst pellet are solved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Guasp, L.; Torregrosa Sánchez, JR. (2018). CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior. Journal of Mathematical Chemistry. 56(7):1902-1923. https://doi.org/10.1007/s10910-017-0814-0S19021923567S. Amat, S. Busquier, Advances in Iterative Methods for Nonlinear Equations (Springer, Berlin, 2016)S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)S. Amat, S. Busquier, S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods. Comput. Appl. Math. 189, 22–33 (2006)I.K. Argyros, Á.A. Magreñn, On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)D.K.R. Babajee, A. Cordero, J.R. Torregrosa, Study of multipoint iterative methods through the Cayley quadratic test. Comput. Appl. Math. 291, 358–369 (2016). doi: 10.1016/J.CAM.2014.09.020P. Blanchard, The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, Article ID 780153 (2013)C. Chun, M.Y. Lee, B. Neta, J. Džunić, On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)A. Cordero, E. Gómez, J.R. Torregrosa, Efficient high-order iterative methods for solving nonlinear systems and their application on heat conduction problems. Complexity 2017, Article ID 6457532 (2017)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley Publishing Company, Reading, 1989)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: formalism and first application to atomic problems. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. Math. Chem. 49, 1384–1415 (2011)Á.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry. Math. Chem. 51(9), 2361–2385 (2013)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving Hammerstein integral equation arisen in chemical phenomenon. Proc. Comput. Sci. 3, 361–364 (2011)B. Neta, C. Chun, M. Scott, Basins of attraction for optimal eighth-order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)M.S. Petković, B. Neta, L.D. Petković, J. Džunić, Multipoint Methods for Solving Nonlinear Equations (Elsevier, Amsterdam, 2013)R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. Math. Chem. 52(1), 255–267 (2014)R. Singh, G. Nelakanti, J. Kumar, A new effcient technique for solving two-point boundary value problems for integro-differential equations. Math. Chem. 52, 2030–2051 (2014

AKT1 polymorphisms are associated with risk for metabolic syndrome

Converging lines of evidence suggest that AKT1 is a major mediator of the responses to insulin, insulin-like growth factor 1 (IGF1), and glucose. AKT1 also plays a key role in the regulation of both muscle cell hypertrophy and atrophy. We hypothesized that AKT1 variants may play a role in the endophenotypes that make up metabolic syndrome. We studied a 12-kb region including the first exon of the AKT1 gene for association with metabolic syndrome-related phenotypes in four study populations [FAMUSS cohort (n = 574; age 23.7 ± 5.7 years), Strong Heart Study (SHS) (n = 2,134; age 55.5 ± 7.9 years), Dynamics of Health, Aging and Body Composition (Health ABC) (n = 3,075; age 73.6 ± 2.9 years), and Studies of a Targeted Risk Reduction Intervention through Defined Exercise (STRRIDE) (n = 175; age 40–65 years)]. We identified a three SNP haplotype that we call H1, which represents the ancestral alleles at the three loci and H2, which represents the derived alleles at the three loci. In young adult European Americans (FAMUSS), H1 was associated with higher fasting glucose levels in females. In middle age Native Americans (SHS), H1 carriers showed higher fasting insulin and HOMA in males, and higher BMI in females. In older African-American and European American subjects (Health ABC) H1 carriers showed a higher incidence of metabolic syndrome. Homozygotes for the H1 haplotype showed about twice the risk of metabolic syndrome in both males and females (p < 0.001). In middle-aged European Americans with insulin resistance (STRRIDE) studied by intravenous glucose tolerance test (IVGTT), H1 carriers showed increased insulin resistance due to the Sg component (p = 0.021). The 12-kb haplotype is a risk factor for metabolic syndrome and insulin resistance that needs to be explored in further populations

Biodiversity of the Deep-Sea Continental Margin Bordering the Gulf of Maine (NW Atlantic): Relationships among Sub-Regions and to Shelf Systems

Background: In contrast to the well-studied continental shelf region of the Gulf of Maine, fundamental questions regarding the diversity, distribution, and abundance of species living in deep-sea habitats along the adjacent continental margin remain unanswered. Lack of such knowledge precludes a greater understanding of the Gulf of Maine ecosystem and limits development of alternatives for conservation and management. Methodology/Principal Findings: We use data from the published literature, unpublished studies, museum records and online sources, to: (1) assess the current state of knowledge of species diversity in the deep-sea habitats adjacent to the Gulf of Maine (39–43uN, 63–71uW, 150–3000 m depth); (2) compare patterns of taxonomic diversity and distribution of megafaunal and macrofaunal species among six distinct sub-regions and to the continental shelf; and (3) estimate the amount of unknown diversity in the region. Known diversity for the deep-sea region is 1,671 species; most are narrowly distributed and known to occur within only one sub-region. The number of species varies by sub-region and is directly related to sampling effort occurring within each. Fishes, corals, decapod crustaceans, molluscs, and echinoderms are relatively well known, while most other taxonomic groups are poorly known. Taxonomic diversity decreases with increasing distance from the continental shelf and with changes in benthic topography. Low similarity in faunal composition suggests the deep-sea region harbours faunal communities distinct from those of the continental shelf. Non-parametric estimators of species richness suggest a minimum of 50% of the deep-sea species inventory remains to be discovered. Conclusions/Significance: The current state of knowledge of biodiversity in this deep-sea region is rudimentary. Our ability to answer questions is hampered by a lack of sufficient data for many taxonomic groups, which is constrained by sampling biases, life-history characteristics of target species, and the lack of trained taxonomists