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Newton's method

In numerical analysis, Newton's method known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f ′, an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close x 1 = x 0 − f f ′. is a better approximation of the root than x0. Geometrically, is the intersection of the x-axis and the tangent of the graph of f at: that is, the improved guess is the unique root of the linear approximation at the initial point; the process is repeated. This algorithm is first in the class of Householder's methods, succeeded by Halley's method; the method can be extended to complex functions and to systems of equations. The idea is to start with an initial guess, reasonably close to the true root to approximate the function by its tangent line using calculus, to compute the x-intercept of this tangent line by elementary algebra.

This x-intercept will be a better approximation to the original function's root than the first guess, the method can be iterated. More formally, suppose f: → ℝ is a differentiable function defined on the interval with values in the real numbers ℝ, we have some current approximation xn. We can derive the formula for a better approximation, xn + 1 by referring to the diagram on the right; the equation of the tangent line to the curve y = f at x = xn is y = f ′ + f, where f′ denotes the derivative. The x-intercept of this line is taken as the next approximation, xn + 1, to the root, so that the equation of the tangent line is satisfied when =: 0 = f ′ + f. Solving for xn + 1 gives x n + 1 = x n − f f ′. We start the process with some arbitrary initial value x0; the method will converge, provided this initial guess is close enough to the unknown zero, that f ′ ≠ 0. Furthermore, for a zero of multiplicity 1, the convergence is at least quadratic in a neighbourhood of the zero, which intuitively means that the number of correct digits doubles in every step.

More details can be found in the analysis section below. Householder's methods are similar but have higher order for faster convergence. However, the extra computations required for each step can slow down the overall performance relative to Newton's method if f or its derivatives are computationally expensive to evaluate; the name "Newton's method" is derived from Isaac Newton's description of a special case of the method in De analysi per aequationes numero terminorum infinitas and in De metodis fluxionum et serierum infinitarum. However, his method differs from the modern method given above: Newton applies the method only to polynomials, he does not compute the successive approximations xn, but computes a sequence of polynomials, only at the end arrives at an approximation for the root x. Newton views the method as purely algebraic and makes no mention of the connection with calculus. Newton may have derived his method from a less precise method by Vieta; the essence of Vieta's method can be found in the work of the Persian mathematician Sharaf al-Din al-Tusi, while his successor Jamshīd al-Kāshī used a form of Newton's method to solve xP − N = 0 to find roots of N.

A special case of Newton's method for calculating square roots was know

Crystallin, gamma D

Gamma-crystallin D is a protein that in humans is encoded by the CRYGD gene. Crystallins are separated into two classes: taxon-specific, or enzyme, ubiquitous; the latter class constitutes the major proteins of vertebrate eye lens and maintains the transparency and refractive index of the lens. Since lens central fiber cells lose their nuclei during development, these crystallins are made and retained throughout life, making them stable proteins. Mammalian lens crystallins are divided into alpha and gamma families. Alpha and beta families are further divided into basic groups. Seven protein regions exist in crystallins: four homologous motifs, a connecting peptide, N- and C-terminal extensions. Gamma-crystallins are a homogeneous group of symmetrical, monomeric proteins lacking connecting peptides and terminal extensions, they are differentially regulated after early development. Four gamma-crystallin genes and three pseudogenes are tandemly organized in a genomic segment as a gene cluster. Whether due to aging or mutations in specific genes, gamma-crystallins have been involved in cataract formation

Waltham Forest Guardian

Your Local Guardian the Waltham Forest Guardian, is a weekly local newspaper sold every Thursday in the London boroughs of Waltham Forest and Redbridge. The newspaper's weekly circulation is 2,508 copies, according to ABC figures for July to December 2017; the paper is part of the Guardian Series of local newspapers, which included the Chingford Guardian and Woodford Guardian, the Epping Forest Guardian. In September 2018 the newspaper was rebranded as Your Local Guardian, the result of a merger between the Chingford and Woodford and Waltham Forest editions; the group is owned by Newsquest, which prints dozens of local and regional papers across the UK. The newspaper was founded in 1876 under the name The Leyton Guardian. In 1935 the Walthamstow Guardian opened new headquarters in Forest Road; the newspaper group merged with the Epping Gazette series in 1942. In 1978 the company moved to new headquarters in Walthamstow. Production moved to an office in Larkshall Road, Highams Park. In 2009 production of the newspaper moved out of Waltham Forest for the first time in its history, with staff relocating to an office in nearby Epping in Essex.

The newspaper subsequently relocated again to a Newsquest office in Watford. In May 2015 Newsquest announced it was moving some of the newspaper's production to a'subbing hub' in Weymouth, Dorset; the publisher said the move was an investment in the'installation of a new editorial system to improve operational efficiency within the business and save costs'. But just over a year in August 2016, the move was dubbed a'failed experiment' by the National Union of Journalists after Newsquest announced it was cutting 19 jobs at the Weymouth site and again moving production of its newspapers back to local regions; the Weymouth office was closed in June 2017. In September 2018 the newspaper was relaunched as Your Local Guardian. Peter Dyke 2000s Pat Stannard 2000s - 2008 Amanda Patterson 2008 - 200? Anthony Longden 200?- 2012 Tim Jones 2012 - 2017 Victoria Birch 2017–present Waltham Forest Guardian www.facebook.com/newsguardianseries

Acrochordus arafurae

Common names: Arafura File snake, Elephant Trunk Snake or wrinkle file snake. Acrochordus arafurae is an aquatic snake species found in northern New Guinea. No subspecies are recognized. Adults grown to 8.25 ft in length. They are known to prey on large fish, such as eel-tailed catfish. Females are larger than males and they have been known to give birth to up to 17 young; the indigenous peoples of northern Australia hunt these snakes as they are quite common. As the snakes are near immobilized without the support of water the hunters throw each newly caught snake on the bank and continue hunting until they have enough. In New Guinea the skin is used to make drums. Acrochordidae by common name Acrochordidae by taxonomic synonyms Acrochordus.com Species Acrochordus arafurae at The Reptile Database. Accessed 16 August 2007. Arafura Filesnakes at Life is Short but Snakes are Long

Vertebrate Paleontology (Romer)

Vertebrate Paleontology is an advanced textbook on vertebrate paleontology by Alfred Sherwood Romer, published by the University of Chicago Press. It went through three editions and for many years constituted a authoritative work and the definitive coverage of the subject. A condensed version centering on comparative anatomy, coauthored by T. S. Parson came in 1977, remaining in print until 1985; the 1987 book Vertebrate Paleontology and Evolution by Robert L. Carroll is based on Romer's book; the book provides a detailed and comprehensive technical account of every main group of living and fossil vertebrates, though the mammal-like reptiles are covered in particular, these being Romer's main interest. At the rear of the book is a classification list which includes every genus known at the time of publication, along with locality and stratigraphic range

Archaeopriapulida

Archaeopriapulida is a group of priapulid-like worms known from Cambrian lagerstätte. The group is related to, similar to, the modern Priapulids, it is unclear. Despite a remarkable morphological similarity to their modern cousins, they fall outside of the priapulid crown group, not unambiguously represented in the fossil record until the Carboniferous, they are closely related or paraphyletic to the palaeoscolecids. Genus Acosmia maotiania Chen & Zhoi 1997 Acosmia maotiania Chen & Zhoi 1997 Genus Archotuba Hou et al. 1999 Archotuba conoidalis Hou et al. 1999 Genus Baltiscalida Slater et al. 2017 Baltiscalida njorda Slater et al. 2017 Genus Eopriapulites Liu & al 2014Eopriapulites sphinx Liu & al 2014 Genus Eximipriapulus Ma et al. 2014 Eximipriapulus globocaudatus Ma et al. 2014 Genus Gangtoucunia Luo & Hu 1999 Gangtoucunia aspera Luo & Hu 1999 Genus Lagenula Luo & Hu 1999 nomen dubium Lagenula striolata Luo & Hu 1999 nomen dubium Genus Laojieella Han et al. 2006 Laojieella thecata Han et al. 2006 Genus Lecythioscopa Conway Morris 1977 Lecythioscopa simplex Conway Morris 1977 Genus Oligonodus Luo & Hu 1999 nomen dubium Oligonodus specialis Luo & Hu 1999 nomen dubium Genus Omnidens Hou & al 2006 Omnidens amplus Hou & al 2006 Genus Sandaokania Luo & Hu 1999 nomen dubium Sandaokania latinodosa Luo & Hu 1999 nomen dubium Genus Singuuriqia Peel 2017 Singuuriqia simoni Peel 2017 Genus Sullulika Peel & Willman, 2018Sullulika broenlundi Peel & Willman, 2018 Genus Xishania Hong 1981 Xishania fusiformis Hong 1981 Xishania jiangxiensis Hong 1988 Genus Paratubiluchus Han, Zhang et Liu, 2004 Paratubiluchus bicaudatus Han, Zhang et Liu, 2004 Genus Xiaoheiqingella Hu 2002 Xiaoheiqingella peculiaris Hu 2002 Genus Priapulites Schram 1973 Priapulites konecniorum Schram 1973 Family Palaeopriapulitidae Hou et al. 1999 Genus Sicyophorus Luo & Hu 1999 Sicyophorus rara Luo & Hu 1999 Sicyophorus sp.

Genus Paraselkirkia Luo & Hu 1999Paraselkirkia sinica Luo & Hu 1999 Family Selkirkiidae Conway Morris 1977 Genus Selkirkia Walcott 1911 Selkirkia elongata Luo & Hu 1999 Selkirkia columbia Walcott 1911 Selkirkia pennsylvanica Resser & Howell 1938 Selkirkia spencei Resser 1939 Selkirkia willoughbyi Conway Morris & Robison 1986 Order Ancalagonida Adrianov & Malakhov 1995 Family Ancalagonidae Conway Morris 1977 Genus Ancalagon Conway Morris 1977 Ancalagon minor Conway Morris 1977 Family Fieldiidae Conway Morris 1977 Genus Fieldia Walcott 1912 Fieldia lanceolata Walcott 1912 Genus Scolecofurca Conway Morris 1977 Scolecofurca rara Conway Morris 1977 Family Ottoiidae Walcott 1911 Genus Ottoia Walcott 1911 Ottoia cylindrica Ottoia guizhouenis Yang, Zhao & Zhang 2016 Ottoia prolifica Walcott 1911 Ottoia tenuis Walcott 1911 Ottoia tricuspida Smith, Harvey & Butterfield 2015 Family Corynetidae Huang, Vannier & Chen 2004 Genus Corynetis Luo & Hu 1999 Corynetis brevis Luo & Hu 1999 [Anningvermis multispinosus Huang et al 2004) Corynetis fortis Hu et al. 2012 Corynetis pusillus Family Miskoiidae Walcott 1911 Genus Louisella Conway Morris 1977 Louisella pedunculata Conway Morris 1977 Genus Miskoia Walcott 1911 Miskoia placida Walcott 1931 Miskoia preciosa Walcott 1911