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In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. [1]
If the discriminant is zero the fraction converges to the single root of multiplicity two. If the discriminant is positive the equation has two real roots, and the continued fraction converges to the larger (in absolute value) of these. The rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ...
[1] [2] Fractions are collected based on differences in a specific property of the individual components. A common trait in fractionations is the need to find an optimum between the amount of fractions collected and the desired purity in each fraction. Fractionation makes it possible to isolate more than two components in a mixture in a single run.
In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order. [51] [52] In these works, the classical Darcy law is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of ...
The convergents of the continued fraction for φ are ratios of successive Fibonacci numbers: φ n = F n+1 / F n is the n-th convergent, and the (n + 1)-st convergent can be found from the recurrence relation φ n+1 = 1 + 1 / φ n. [31] The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The fractional distillation of organic substances played an important role in the 9th-century works attributed to the Islamic alchemist Jabir ibn Hayyan, as for example in the Kitāb al-Sabʿīn ('The Book of Seventy'), translated into Latin by Gerard of Cremona (c. 1114–1187) under the title Liber de septuaginta. [1]
Further, Gallup reported that: "Americans consistently say that jobs and the economy are the most important problems facing the country, with 26% citing jobs specifically as the nation's most important problem in March." Republicans and Democrats agreed that bringing the jobs home was the number one solution approach, but differed on other poll ...
The Columbia Encyclopedia 5th ed. says of him "at times Mill came close to socialism, a theory repugnant to his predecessors". [ citation needed ] He was a proponent of utilitarianism , an ethical theory developed by his predecessor Jeremy Bentham .