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Differential Galois Theory
In mathematics, differential Galois theory is the field that studies extensions of differential fields. Whereas algebraic Galois theory studies extensions of field (mathematics), algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation (abstract algebra), derivation, ''D''. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory. Motivation and basic concepts In mathematics, some types of elementary functions cannot express the indefinite integrals of other elementary functions. A well-known example is e^, whose indefinite integral is the error function \operatornamex, familiar in statistics. Other examples include the sinc function \tfrac and x^x. It's import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Field
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebra, algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered as belonging to differential algebra. More specifically, ''differential algebra'' refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are ring (mathematics), rings, field (mathematics), fields, and algebra over a field, algebras equipped with finitely many derivation (differential algebra), derivations. A natural example of a differential field is the field of rational functions in one variable over the complex numbers, \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull Topology
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists d\in\N such that every group in the system can be generated by d elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Algebra
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered as belonging to differential algebra. More specifically, ''differential algebra'' refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations. A natural example of a differential field is the field of rational functions in one variable over the complex numbers, \mathbb(t), where the derivation is differentiation with respect to t. More generally, every differential e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Notices Of The American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume was published in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Beginning with the January 2025 issue, the editor-in-chief is Mark C. Wilson, succeeding past editor Erica Flapan. The cover regularly features mathematical visualizations. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom reside outside the United States. By publishing high-level exposition, the ''Notices'' provides opportunities for mathematicians to find out what is going on in the field. Each is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *'' Memoirs of the American Mathematical Society'' *'' Notices of the American Mathematical Society'' *'' Proceedings of the Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard–Vessiot Theory
In differential algebra, Picard–Vessiot theory is the study of the differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major goal is to describe when the differential equation can be solved by quadratures in terms of properties of the differential Galois group. The theory was initiated by Émile Picard and Ernest Vessiot from about 1883 to 1904. and give detailed accounts of Picard–Vessiot theory. History The history of Picard–Vessiot theory is discussed by . Picard–Vessiot theory was developed by Picard between 1883 and 1898 and by Vessiot from 1892 to 1904 (summarized in and ). The main result of their theory says very roughly that a linear differential equation can be solved by quadratures if and only if its differential Galois group is connected and solvable. Unfortunately it is hard to tell exactly what they proved as the concept of being "solvable by quadratur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Linear Group
In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel (algebra), kernel of the determinant :\det\colon \operatorname(n, R) \to R^\times. where R^\times is the multiplicative group of R (that is, R excluding 0 when R is a field). These elements are "special" in that they form an Algebraic variety, algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When R is the finite field of order q, the notation \operatorname(n,q) is sometimes used. Geometric interpretation The special linear group \operatorname(n,\R) can be characterized as the group of ''volume and orientation (mathematics), orientation preserving'' linear tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airy Equation
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 Ai(''x'') and the related function Bi(''x''), are linearly independent solutions to the differential equation \frac - xy = 0 , known as the Airy equation or the Stokes equation. Because the solution of the linear differential equation \frac - ky = 0 is oscillatory for and exponential for , the Airy functions are oscillatory for and exponential for . In fact, the Airy equation 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 , 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)\, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Group
In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle group forms a subgroup of , the multiplicative group of all nonzero complex numbers. Since \C^\times is abelian, it follows that \mathbb T is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure : \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, \mathbb T^n (the direct product of \mathbb T with itself n times) is geometrically an n-toru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine Integral
In mathematics, trigonometric integrals are a indexed family, family of nonelementary integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int_x^\infty\frac\,dt~. Note that the integrand \frac is the sinc function, and also the zeroth Bessel function#Spherical Bessel functions: jn.2C yn, spherical Bessel function. Since is an even function, even entire function (holomorphic over the entire complex plane), is entire, odd, and the integral in its definition can be taken along Cauchy's integral theorem, any path connecting the endpoints. By definition, is the antiderivative of whose value is zero at , and is the antiderivative whose value is zero at . Their difference is given by the Dirichlet integral, \operatorname(x) - \operatorname(x) = \int_0^\infty\frac\,dt = \frac \quad \text \quad \operatorname(x) = \frac + \operatorname(x) ~. In signal processing, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
