Stability analysis of fourth-order iterative method for finding multiple roots of nonlinear equations

[EN] The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studies dealt with the analysis of iterative schemes for solving non-linear equations with simple roots; however, the case involving multiple roots remains almost unexplored. The main objective of this paper was to discuss the dynamical analysis of the rational map associated with an existing class of iterative procedures for multiple roots. This study was performed for cases of double and triple multiplicities, giving as a conjecture that the wideness of the convergence regions of the multiple roots increases when the multiplicity is higher and also that this family of parametric methods includes some specially fast and stable elements with global convergence.This research was partially supported by Ministerio de Ciencia, Innovación y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089Cordero Barbero, A.; Jaiswal, J.; Torregrosa Sánchez, JR. (2019). Stability analysis of fourth-order iterative method for finding multiple roots of nonlinear equations. Applied Mathematics and Nonlinear Sciences. 4(1):43-56. https://doi.org/10.2478/AMNS.2019.1.00005S435641Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.32186

Dynamical Techniques for Analyzing Iterative Schemes with Memory

[EN] We construct a new biparametric three-point method with memory to highly improve the computational efficiency of its original partner, without adding functional evaluations. In this way, through different estimations of self-accelerating parameters, we have modified an existing seventh-order method. The parameters have been defined by Hermite interpolating polynomial that allows the accelerating effect. In particular, the R-order of the proposed iterative method with memory is increased from seven to ten. A real multidimensional analysis of the stability of this method with memory is made, in order to study its dependence on the initial estimations. Taking into account that usually iterative methods with memory are more stable than their derivative-free partners and the obtained results in this study, the behavior of this scheme shows to be excellent, but for a small domain. Numerical examples and comparison are also provided, confirming the theoretical results.This research has been partially supported by grants from Spanish Ministerio de Economia y Competitividad (MTM2014-52016-C2-2-P) and by Generalitat Valenciana (PROMETEO/2016/089).Choubey, N.; Cordero Barbero, A.; Jaiswal, J.; Torregrosa Sánchez, JR. (2018). Dynamical Techniques for Analyzing Iterative Schemes with Memory. Complexity. https://doi.org/10.1155/2018/1232341

Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces

[EN] Semilocal convergence for an iteration of order five for solving nonlinear equations in Banach spaces is established under second-order Fr,chet derivative satisfying the Lipschitz condition. It is done by deriving a number of recurrence relations. A theorem for the existence-uniqueness along with the estimation of error bounds of the solution is established. Its R-order is shown to be equal to five. Both efficiency and computational efficiency indices are given. A variety of examples are worked out to show its applicability. In comparison to existing methods having similar R-orders, improved results in terms of computational efficiency index and error bounds are found using our methodology.The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.Singh, S.; Gupta, D.; Martínez Molada, E.; Hueso Pagoaga, JL. (2016). Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces. Mediterranean Journal of Mathematics. 13(6):4219-4235. doi:10.1007/s00009-016-0741-5S42194235136Cordero A., Hueso J.L., Martinez E., Torregrosa J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)Chen, L., Gu, C., Ma Y.: Semilocal convergence for a fifth order Newton’s method using Recurrence relations in Banach spaces. J. Appl. Math. 2011, 1–15 (2011)Wang X., Kou J., Gu C.: Semilocal convergence of a sixth order Jarrat method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)Zheng L., Gu C.: Semilocal convergence of a sixth order method in Banach spaces. Numer. Algorithms 61, 413–427 (2012)Zheng L., Gu C.: Recurrence relations for semilocal convergence of a fifth order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)Proinov P.D., Ivanov S.I.: On the convergence of Halley’s method for multiple polynomial zeros. Mediterr. J. Math. 12, 555–572 (2015)Ezquerro, J.A., Hernández-Verón M.A.: On the domain of starting points of Newton’s method under center lipschitz conditions. Mediterr. J. Math. (2015). doi: 10.1007/s00009-015-0596-1Cordero A., Hernández-Verón M.A., Romero N., Torregrosa J.R.: Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces. J. Comput. Appl. Math. 273, 205–213 (2015)Parida P.K., Gupta D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)Hueso J.L., Martínez E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)Argyros, I.K., Hilout S.: Numerical methods in nonlinear analysis. World Scientific Publ. Comp., New Jersey (2013)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical modelling with applications in biosciences and engineering. Nova Publishers, New York (2011)Argyros I.K., Khattri S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2004)Argyros I.K., Hilout A.S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Kantorovich, L.V., Akilov G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros I.K., George S., Magreñán A.A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of higher convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)Argyros I.K., Magreñán A.A.: A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2015)Amat S., Hernández M.A., Romero N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Chun, C., St $${\breve{a}}$$ a ˘ nic $${\breve{a}}$$ a ˘ , P., Neta, B.: Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)Ostrowski, A.M.: Solution of equations in Euclidean and Banach spaces, 3rd edn. Academic Press, New-York (1977)Jaiswal J.P.: Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency. Numer. Algorithms 71, 933–951 (2015)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964

Differences in the pattern and regulation of mineral deposition in human cell lines of osteogenic and non-osteogenic origin

Bone marrow-derived mesenchymal stem cells (MSCs) are widely used as a cellular model of bone formation, and can mineralize in vitro in response to osteogenic medium (OM). It is unclear, however, whether this property is specific to cells of mesenchymal origin. We analysed the OM response in 3 non-osteogenic lines, HEK293, HeLa and NTera, compared to MSCs. Whereas HEK293 cells failed to respond to OM conditions, the 2 carcinoma-derived lines NTera and HeLa deposited a calcium phosphate mineral comparable to that present in MSC cultures. However, unlike MSCs, HeLa and NTera cultures did so in the absence of dexamethasone. This discrepancy was confirmed, as bone morphogenetic protein inhibition obliterated the OM response in MSCs but not in HeLa or NTera, indicating that these 2 models can deposit mineral through a mechanism independent of established dexamethasone or bone morphogenetic protein signalling

Nanoparticles for Applications in Cellular Imaging

In the following review we discuss several types of nanoparticles (such as TiO2, quantum dots, and gold nanoparticles) and their impact on the ability to image biological components in fixed cells. The review also discusses factors influencing nanoparticle imaging and uptake in live cells in vitro. Due to their unique size-dependent properties nanoparticles offer numerous advantages over traditional dyes and proteins. For example, the photostability, narrow emission peak, and ability to rationally modify both the size and surface chemistry of Quantum Dots allow for simultaneous analyses of multiple targets within the same cell. On the other hand, the surface characteristics of nanometer sized TiO2allow efficient conjugation to nucleic acids which enables their retention in specific subcellular compartments. We discuss cellular uptake mechanisms for the internalization of nanoparticles and studies showing the influence of nanoparticle size and charge and the cell type targeted on nanoparticle uptake. The predominant nanoparticle uptake mechanisms include clathrin-dependent mechanisms, macropinocytosis, and phagocytosis

Doing synthetic biology with photosynthetic microorganisms

The use of photosynthetic microbes as synthetic biology hosts for the sustainable production of commodity chemicals and even fuels has received increasing attention over the last decade. The number of studies published, tools implemented, and resources made available for microalgae have increased beyond expectations during the last few years. However, the tools available for genetic engineering in these organisms still lag those available for the more commonly used heterotrophic host organisms. In this mini-review, we provide an overview of the photosynthetic microbes most commonly used in synthetic biology studies, namely cyanobacteria, chlorophytes, eustigmatophytes and diatoms. We provide basic information on the techniques and tools available for each model group of organisms, we outline the state-of-the-art, and we list the synthetic biology tools that have been successfully used. We specifically focus on the latest CRISPR developments, as we believe that precision editing and advanced genetic engineering tools will be pivotal to the advancement of the field. Finally, we discuss the relative strengths and weaknesses of each group of organisms and examine the challenges that need to be overcome to achieve their synthetic biology potential.Peer reviewe

Observation of inverse Compton emission from a long γ-ray burst.

Long-duration γ-ray bursts (GRBs) originate from ultra-relativistic jets launched from the collapsing cores of dying massive stars. They are characterized by an initial phase of bright and highly variable radiation in the kiloelectronvolt-to-megaelectronvolt band, which is probably produced within the jet and lasts from milliseconds to minutes, known as the prompt emission1,2. Subsequently, the interaction of the jet with the surrounding medium generates shock waves that are responsible for the afterglow emission, which lasts from days to months and occurs over a broad energy range from the radio to the gigaelectronvolt bands1-6. The afterglow emission is generally well explained as synchrotron radiation emitted by electrons accelerated by the external shock7-9. Recently, intense long-lasting emission between 0.2 and 1 teraelectronvolts was observed from GRB 190114C10,11. Here we report multi-frequency observations of GRB 190114C, and study the evolution in time of the GRB emission across 17 orders of magnitude in energy, from 5 × 10-6 to 1012 electronvolts. We find that the broadband spectral energy distribution is double-peaked, with the teraelectronvolt emission constituting a distinct spectral component with power comparable to the synchrotron component. This component is associated with the afterglow and is satisfactorily explained by inverse Compton up-scattering of synchrotron photons by high-energy electrons. We find that the conditions required to account for the observed teraelectronvolt component are typical for GRBs, supporting the possibility that inverse Compton emission is commonly produced in GRBs

The global burden of cancer attributable to risk factors, 2010–19: a systematic analysis for the Global Burden of Disease Study 2019

BACKGROUND: Understanding the magnitude of cancer burden attributable to potentially modifiable risk factors is crucial for development of effective prevention and mitigation strategies. We analysed results from the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019 to inform cancer control planning efforts globally. METHODS: The GBD 2019 comparative risk assessment framework was used to estimate cancer burden attributable to behavioural, environmental and occupational, and metabolic risk factors. A total of 82 risk–outcome pairs were included on the basis of the World Cancer Research Fund criteria. Estimated cancer deaths and disability-adjusted life-years (DALYs) in 2019 and change in these measures between 2010 and 2019 are presented. FINDINGS: Globally, in 2019, the risk factors included in this analysis accounted for 4·45 million (95% uncertainty interval 4·01–4·94) deaths and 105 million (95·0–116) DALYs for both sexes combined, representing 44·4% (41·3–48·4) of all cancer deaths and 42·0% (39·1–45·6) of all DALYs. There were 2·88 million (2·60–3·18) risk-attributable cancer deaths in males (50·6% [47·8–54·1] of all male cancer deaths) and 1·58 million (1·36–1·84) risk-attributable cancer deaths in females (36·3% [32·5–41·3] of all female cancer deaths). The leading risk factors at the most detailed level globally for risk-attributable cancer deaths and DALYs in 2019 for both sexes combined were smoking, followed by alcohol use and high BMI. Risk-attributable cancer burden varied by world region and Socio-demographic Index (SDI), with smoking, unsafe sex, and alcohol use being the three leading risk factors for risk-attributable cancer DALYs in low SDI locations in 2019, whereas DALYs in high SDI locations mirrored the top three global risk factor rankings. From 2010 to 2019, global risk-attributable cancer deaths increased by 20·4% (12·6–28·4) and DALYs by 16·8% (8·8–25·0), with the greatest percentage increase in metabolic risks (34·7% [27·9–42·8] and 33·3% [25·8–42·0]). INTERPRETATION: The leading risk factors contributing to global cancer burden in 2019 were behavioural, whereas metabolic risk factors saw the largest increases between 2010 and 2019. Reducing exposure to these modifiable risk factors would decrease cancer mortality and DALY rates worldwide, and policies should be tailored appropriately to local cancer risk factor burden

A note on the convergence rate of Kumar–Singh–Srivastava methods for solving nonlinear equations

In the present article, it is shown that both the methods presented in Kumar et al. (2013) do not possess the order of convergence as claimed. One of the two methods, derivative involved method possesses the convergence rate of eighth order whereas the other derivative free method possesses sixth order convergence. The theoretical convergence rate is also validated by computational order of convergence