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Brillouin And Langevin Functions
The Brillouin and Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics. Brillouin function The Brillouin functionC. Kittel, ''Introduction to Solid State Physics'' (8th ed.), pages 303-4 is a special function defined by the following equation: :B_J(x) = \frac \coth \left ( \frac x \right ) - \frac \coth \left ( \frac x \right ) The function is usually applied (see below) in the context where x'' is a real variable and J is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as x \to +\infty and -1 as x \to -\infty. The function is best known for arising in the calculation of the magnetization of an ideal paramagnet. In particular, it describes the dependency of the magnetization M on the applied magnetic field B and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expectation Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Non-Newtonian Fluid Mechanics
A journal, from the Old French ''journal'' (meaning "daily"), may refer to: *Bullet journal, a method of personal organization *Diary, a record of what happened over the course of a day or other period *Daybook, also known as a general journal, a daily record of financial transactions * Logbook, a record of events important to the operation of a vehicle, facility, or otherwise *Record (other) *Transaction log, a chronological record of data processing *Travel journal In publishing, ''journal'' can refer to various periodicals or serials: *Academic journal, an academic or scholarly periodical ** Scientific journal, an academic journal focusing on science ** Medical journal, an academic journal focusing on medicine **Law review, a professional journal focusing on legal interpretation * Magazine, non-academic or scholarly periodicals in general **Trade magazine, a magazine of interest to those of a particular profession or trade ** Literary magazine, a magazine devoted to li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J Non-Newton Fluid Mech
J, or j, is the tenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its usual name in English is ''jay'' (pronounced ), with a now-uncommon variant ''jy'' ."J", ''Oxford English Dictionary,'' 2nd edition (1989) When used in the International Phonetic Alphabet for the ''y'' sound, it may be called ''yod'' or ''jod'' (pronounced or ). History The letter ''J'' used to be used as the swash letter ''I'', used for the letter I at the end of Roman numerals when following another I, as in XXIIJ or xxiij instead of XXIII or xxiii for the Roman numeral twenty-three. A distinctive usage emerged in Middle High German. Gian Giorgio Trissino (1478–1550) was the first to explicitly distinguish I and J as representing separate sounds, in his ''Ɛpistola del Trissino de le lettere nuωvamente aggiunte ne la lingua italiana'' ("Trissino's epistle about the letters recently added in the It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohen And Jedynak Approximations
Cohen may refer to: Places *Cohen-kuhi Tau/4, a star 420 light-years away from Earth in the Taurus Constellation *The Cohen Building of ''The Judd School'' in Tonbridge, England People * Cohen (surname), a common Jewish surname Arts, entertainment, and media *Matt Cohen Prize, an award given annually by the Writers' Trust of Canada to a Canadian writer * Shaughnessy Cohen Award, a Canadian literary award Law * Clinger–Cohen Act, a United States federal law that is designed to improve the way the federal government acquires and manages information technology *'' Cohen v. California'', a U.S. Supreme Court case dealing with freedom of speech *'' Cohen v. Cowles Media Co.'', a U.S. Supreme Court case establishing that freedom of the press does not exempt newspapers from generally applicable laws *''Cohens v. Virginia'', a U.S. Supreme Court decision most noted for the Marshall Court's assertion of its power to review state supreme court decisions in criminal law matters *''Flast ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods— in some sense inspired by the Padé theory— typically replace them. Since Padé approximant is a rational function, an artificial singul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Applied Physics
The ''Journal of Applied Physics'' is a peer-reviewed scientific journal with a focus on the physics of modern technology. The journal was originally established in 1931 under the name of ''Physics'', and was published by the American Physical Society for its first 7 volumes. In January 1937, ownership was transferred to the American Institute of Physics "in line with the efforts of the American Physical Society to enhance the standing of physics as a profession". The journal's current editor-in-chief is André Anders (Lawrence Berkeley National Laboratory). According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 2.877. References External links * Physics journals Weekly journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catastrophic Cancellation
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L_1 = 254.5\,\text long and the other L_2 = 253.5\,\text long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be \tilde L_1 = 255\,\text and \tilde L_2 = 253\,\text. These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 2% of the true lengths, , L_1 - \tilde L_1, /, L_1, < 2\%. However, if the ''approximate'' lengths are subtracted, the difference will be , even though the true difference between the lengths is . The difference of the approximations, |
Gauss's Continued Fraction
In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several important elementary functions, as well as some of the more complicated transcendental functions. History Lambert published several examples of continued fractions in this form in 1768, and both Euler and Lagrange investigated similar constructions, but it was Carl Friedrich Gauss who utilized the algebra described in the next section to deduce the general form of this continued fraction, in 1813. Although Gauss gave the form of this continued fraction, he did not give a proof of its convergence properties. Bernhard Riemann and L.W. Thomé obtained partial results, but the final word on the region in which this continued fraction converges was not given until 1901, by Edward Burr Van Vleck. Derivation Let f_0, f_1, f_2, \dots be a seque ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Langevin
Paul Langevin (; ; 23 January 1872 – 19 December 1946) was a French physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the ''Comité de vigilance des intellectuels antifascistes'', an anti-fascist organization created after the 6 February 1934 far right riots. Being a public opponent of fascism in the 1930s resulted in his arrest and being held under house arrest by the Vichy government for most of World War II. Langevin was also president of the Human Rights League (LDH) from 1944 to 1946, having recently joined the French Communist Party. He was a doctoral student of Pierre Curie and later a lover of widowed Marie Curie. He is also known for his two US patents with Constantin Chilowsky in 1916 and 1917 involving ultrasonic submarine detection. He is entombed at the Panthéon. Life Langevin was born in Paris, and studied at the '' École de Physique et Chimie'' and the ''École Normale Supérieure''. He then went to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
