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Abel Equation Of The First Kind
In mathematics, an Abel equation of the first kind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form :y'=f_3(x)y^3+f_2(x)y^2+f_1(x)y+f_0(x) \, where f_3(x)\neq 0. If f_3(x)=0 and f_0(x)=0, or f_2(x)=0 and f_0(x)=0, the equation reduces to a Bernoulli equation, while if f_3(x) = 0 the equation reduces to a Riccati equation In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form : y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2( .... Properties The substitution y=\dfrac brings the Abel equation of the first kind to the " Abel equation of the second kind" of the form :uu'=-f_0(x)u^3-f_1(x)u^2-f_2(x)u-f_3(x). \, The substitution : \begin \xi & = \int f_3(x)E^2~dx, \\ ptu & = \left(y+\dfrac\right)E^, \\ ptE & = \exp\left(\int\left(f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for over 250 years. He was also an innovator in the field of elliptic functions, discoverer of Abelian functions. He made his discoveries while living in poverty and died at the age of 26 from tuberculosis. Most of his work was done in six or seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "What a head the young Norwegian has!" The Abel Prize in mathematics, originally proposed in 1899 to complement the Nobel Prizes (but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree three, and a real function. In particular, the domain and the codomain are the set of the real numbers. Setting produces a cubic equation of the form :ax^3+bx^2+cx+d=0, whose solutions are called roots of the function. A cubic function has either one or three real roots (which may not be distinct); all odd-degree polynomials have at least one real root. The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic function is monotonic. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. Up to an affine transformation, there are only thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Differential Equation
In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form : y'+ P(x)y = Q(x)y^n, where n is a real number. Some authors allow any real n, whereas others require that n not be 0 or 1. The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation. Transformation to a linear differential equation When n = 0, the differential equation is linear. When n = 1, it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For n \neq 0 and n \neq 1, the substitution u = y^ reduces any Bernoulli equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riccati Equation
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form : y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2(x) where q_0(x) \neq 0 and q_2(x) \neq 0. If q_0(x) = 0 the equation reduces to a Bernoulli equation, while if q_2(x) = 0 the equation becomes a first order linear ordinary differential equation. The equation is named after Jacopo Riccati (1676–1754). More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation. Conversion to a second order linear equation The non-linear Riccati equation can always be converted to a second order linear ordinary differential equation (ODE): If :y'=q_0(x) + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Equation Of The Second Kind
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd who offered his firstborn flock up to God as an offering. God accepted his offering but not his brother's. Cain then killed Abel out of jealousy. According to Genesis, this was the first murder in the history of mankind. Genesis narrative Interpretations Jewish and Christian interpretations According to the narrative in Genesis, Abel ( ''Hébel'', in pausa ''Hā́ḇel''; grc-x-biblical, Ἅβελ ''Hábel''; ar, هابيل, ''Hābēl'') is Eve's second son. His name in Hebrew is composed of the same three consonants as a root meaning "breath". Julius Wellhausen has proposed that the name is independent of the root. Eberhard Schrader had previously put forward the Akkadian (Old Assyrian dialect) ''ablu'' ("son") as a more lik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimitrios E
Demetrius is the Latinized form of the Ancient Greek male given name ''Dēmḗtrios'' (), meaning “Demetris” - "devoted to goddess Demeter". Alternate forms include Demetrios, Dimitrios, Dimitris, Dmytro, Dimitri, Dimitrie, Dimitar, Dumitru, Demitri, Dhimitër, and Dimitrije, in addition to other forms (such as Russian Dmitry) descended from it. Demetrius and its variations may refer to the following: *Demetrius of Alopece (4th century BC), Greek sculptor noted for his realism *Demetrius of Phalerum ( – BC) *Demetrius, somatophylax of Alexander the Great (d. 330 BC) *Demetrius - brother of Antigonus I Monophthalmus, king of Macedonia 306-301 BC *Demetrius I of Macedon (337–283 BC), called ''Poliorcetes'', son of Antigonus I Monophthalmus, King of Macedonia 294–288 BC *Demetrius the Fair (Demetrius the Handsome, Demetrius of Cyrene) (285 BC-249/250 BC) - Hellenistic king of Cyrene *Demetrius II Aetolicus, son of Antigonus II, King of Macedonia 239–229 BC ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodoros I , Emperor of Nicaea in 1204–1221 or 1205–1222
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Theodoros I may refer to: * Patriarch Theodore I of Alexandria, Greek Patriarch of Alexandria in 607–609 * Theodore I of Constantinople, Ecumenical Patriarch in 677–679 * Pope Theodoros I of Alexandria, ruled in 730–742 * Theodore I Laskaris Theodore I Laskaris or Lascaris ( gr, Θεόδωρος Κομνηνὸς Λάσκαρις, Theodōros Komnēnos Laskaris; 1175November 1221) was the first emperor of Nicaea—a successor state of the Byzantine Empire—from 1205 to his d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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