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Painlevé Transcendents
In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. They were discovered by , , , and . History Painlevé transcendents have their origin in the study of special functions, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformations of linear differential equations. One of the most useful classes of special functions are the elliptic functions. They are defined by second order ordinary differential equations whose singularities have the Painlevé property: the only movable singularities are poles. This property is rare in nonlinear equations. Poincaré and L. Fuchs showed that any first order equation with the Painlevé property can be transformed into the Weierstrass elliptic function or the Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazarus Fuchs
Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a Jewish-German mathematician who contributed important research in the field of linear differential equations. He was born in Moschin (Mosina) (located in Grand Duchy of Posen) and died in Berlin, Germany. He was buried in Schöneberg in the St. Matthew's Cemetery. His grave in section H is preserved and listed as a grave of honour of the State of Berlin. He is the eponym of Fuchsian groups and functions, and the Picard–Fuchs equation. A singular point ''a'' of a linear differential equation :y''+p(x)y'+q(x)y=0 is called Fuchsian if ''p'' and ''q'' are meromorphic around the point ''a'', and have poles of orders at most 1 and 2, respectively. According to a theorem of Fuchs, this condition is necessary and sufficient for the regularity of the singular point, that is, to ensure the existence of two linearly independent solutions of the form : y_j=\sum_^\infty a_(x-x_0)^,\quad a_0\ne0\,\quad j=1,2. where the expo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Weyl Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standard re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra \mathfrak over a field of characteristic 0 . In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements ''x'' such that the adjoint endomorphism \operatorname(x) : \mathfrak \to \mathfrak is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.pg 231 In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Dihedral Group
In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p1m1'', seen as an infinite set of parallel reflections along an axis. Definition Every dihedral group is generated by a rotation ''r'' and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer ''n'' such that ''rn'' is the identity, and we have a finite dihedral group of order 2''n''. If the rotation is ''not'' a rational multiple of a full rotation, then there is no such ''n'' and the resulting group has infinitely many elements and is called Dih∞. It has presentations :\langle r, s \mid s^2 = 1, srs = r^ \rangle \,\! :\langle x, y \mid x^2 = y^2 = 1 \rangle \,\! and is isomorphic to a semidirect product of Z and Z/2, and to the free product Z/2 * Z/2. It is the automorphism group of the graph con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bäcklund Transformation
Backlund is a Swedish surname. Notable people with the surname include: * Albert Victor Bäcklund (1845-1922), mathematician * Bengt Backlund (1926–2006), Swedish flatwater canoer * Bob Backlund (born 1949), American professional wrestler * Filip Backlund (born 1990), Swedish motorcycle road racer * Göran Backlund (born 1957), Swedish sprint canoer * Gordon Backlund (born 1940), American politician and electrical engineer * Gösta Backlund (1893—1918), Swedish footballer * Gotthard Backlund, Swedish chess master * Ivar Backlund (1892—1969), Swedish officer * Johan Backlund (born 1981), Swedish ice hockey goaltender * Jukka Backlund (born 1982), Finnish music producer * Kaj Backlund (1945–2013), Finnish jazz trumpeter, composer, and bandleader * Mikael Backlund (born 1989), Swedish ice hockey player * Nils Backlund (1896–1964), Swedish water polo player * Oskar Backlund Johan Oskar Backlund (28 April 1846 – 29 August 1916) was a Swedish-Russian astronomer. His name ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian System
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory. Overview Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the initial value problem cannot be solved analytically. One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H(\boldsymbol,\boldsymbol,t), also known as the Hamiltonian. The state of the system, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite-Weber Function
In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates. The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling , called H. F. Weber's equations: and If f(a,z) is a solution, then so are f(a,-z), f(-a,iz)\textf(-a,-iz). If f(a,z)\, is a solution of equation (), then f(-ia,ze^) is a solution of (), and, by symmetry, f(-ia,-ze^), f(ia,-ze^)\textf(ia,ze^) are also solutions of (). Solutions There are independent even and odd solutions of the form (). These are given by (following the notation of Abramowitz and Stegun (1965)): y_1(a;z) = \exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac14; \; \tfrac12\; ; \; \frac\right)\,\,\,\,\,\, (\mathrm) and y_2(a;z) = z\exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac34; \; \tfra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airy Function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solutions to the differential equation \frac - xy = 0 , known as the Airy equation or the Stokes equation. This is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential). Definitions For real values of ''x'', the Airy function of the first kind can be defined by the improper Riemann integral: \operatorname(x) = \dfrac\int_0^\infty\cos\left(\dfrac + xt\right)\, dt\equiv \dfrac \lim_ \int_0^b \cos\left(\dfrac + xt\right)\, dt, which converges by Dirichlet's test. For any real number x there is positive real number M such that function \dfrac3 + xt is increasing, unbounded and convex with continuous and unbounded derivative on interval lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confluent Hypergeometric Function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term ''confluent'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. * Whittaker functions (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
