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Algebraic Differential Equation
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used. The intention is to include equations formed by means of differential operators, in which the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer algebra and number theory. A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator. Formulations *Derivations ''D'' can be used as algebraic analogues of the formal part of differential calculus, so that algebraic differential equations make sense in commutative rings. *The theory of differential fields was set u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Invariant Theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. Introduction Let G be a group, and V a finite-dimensional vector space over a field k (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G in V is a group homomorphism \pi:G \to GL(V), which induces a group action of G on V. If k /math> is the space of polynomial functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include Graph of a function, graphs of Multivalued function, multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a Surface (topology), surface: a two-dimensional real manifold, but it contains more structure (specifically a Complex Manifold, complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and Metrizabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Monodromy Group
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''monodromy'' comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be ''single-valued'' as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a monodromy group: a group of transformations acting on the data that encodes what happens as we "run round" in one dimension. Lack of monodromy is sometimes called ''polydromy''. Definition Let X be a connected and locally connected based topological space with base point x, and let p: \tilde \to X be a covering with fiber F = p^(x). For a loop \gamma: , 1\to X based at x, denote a lift ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Schwarz List
{{disambiguation ...
Schwarz may refer to: * Schwarz, Germany, a municipality in Mecklenburg-Vorpommern, Germany * Schwarz (surname), a surname (and list of people with the surname) * Schwarz (musician), American DJ and producer * ''Schwarz'' (Böhse Onkelz album), released simultaneously with ''Weiß'', 1993 * ''Schwarz'' (Conrad Schnitzler album), a reissue of the 1971 Kluster album ''Eruption'' * Schwarz (cards), in some card games, a Schneider (low point score) in which no tricks are taken * Schwarz Gruppe, a multinational retail group * Schwarz Pharma, a German drug company See also * * * Schwartz (other) * Schwarzhorn (other) * Swartz (other) Swartz may refer to: ;Places *Swartz, Louisiana * Swartz Creek (other) *Swartz Bay, British Columbia on the north end of the Saanich Peninsula on Vancouver Island ** Swartz Bay Ferry Terminal * Swartz Nunataks, in Antarctica ;People * Swa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Transcendental Function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits). This is in contrast to an algebraic function. Examples of transcendental functions include the exponential function, the logarithm function, the hyperbolic functions, and the trigonometric functions. Equations over these expressions are called transcendental equations. Definition Formally, an analytic function f of one real or complex variable is transcendental if it is algebraically independent of that variable. This means the function does not satisfy any polynomial equation. For example, the function f given by :f(x)=\frac for all x is not transcendental, but algebraic, because it satisfies the polynomial equation :(ax+b)-(cx+d)f(x)=0. Similarly, the functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Function
In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are: * f(x) = 1/x * f(x) = \sqrt * f(x) = \frac Some algebraic functions, however, cannot be expressed by such finite expressions (this is the Abel–Ruffini theorem). This is the case, for example, for the Bring radical, which is the function implicitly defined by : f(x)^5+f(x)+x = 0. In more precise terms, an algebraic function of degree in one variable is a function y = f(x), that is continuous in its domain and satisfies a polynomial equation of positive degree : a_n(x)y^n+a_(x)y^+\cdots+a_0(x)=0 where the coefficients are polynomial functions of , with integer coefficients. It can be shown that the same class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
D-module
In mathematics, a ''D''-module is a module (mathematics), module over a ring (mathematics), ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara. The methods of ''D''-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. This approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems (holonomic systems), and on the cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Éléments De Géométrie Algébrique
The (''EGA''; from French: "Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné) is a rigorous treatise on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the . In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation and basic reference of modern algebraic geometry. Editions Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the '' Séminaire de géométrie algébrique'' (known as ''SGA''). Indeed, as explained by Grothendieck in the preface of the published version of ''SGA'', by 1970 it had become clear that incorporating all of the planned material in ''EGA'' would require significan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He received the Fields Medal in 1966 for advances in algebraic geometry, homological algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
