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Gradient Conjecture
In mathematics, the gradient conjecture, due to René Thom (1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka (University of Savoie, France), Tadeusz Mostowski ( Warsaw University, Poland) and Adam Parusiński ( University of Angers, France). The conjecture states that given a real-valued analytic function ''f'' defined on R''n'' and a trajectory ''x''(''t'') of the gradient vector field of ''f'' having a limit point ''x''0 ∈ R''n'', where ''f'' has an isolated critical point at ''x''0, there exists a limit (in the projective space PR''n-1'') for the secant lines from ''x''(''t'') to ''x''0, as ''t'' tends to zero. The proof depends on a theorem due to Stanis%C5%82aw %C5%81ojasiewicz Stanisław Łojasiewicz (9 October 1926 – 14 November 2002) was a Polish mathematician.. Biography At the end of the 1950s, he solved the problem of distribution division by analytic functions, introducing the %C5%81ojasiewicz inequality. Its .... References * R. ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Erik Christopher Zeeman). Life and career René Thom grow up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940. After German invasion of France, his family took refuge in Switzerland and then in Lyon. In 1941 he moved to Paris to attend Lycée Saint-Louis and in 1943 he began studying mathematics at École Normale Supérieure, becoming agrégé in 1946. He received his PhD in 1951 from the University of Paris. His thesis, titled ''Espaces fibrés en sphères et carrés de Steenrod'' (''Sphere bundles and Steenrod squares''), was w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Savoie
Savoy Mont Blanc University (french: Université Savoie Mont Blanc, a.k.a. Chambéry University) is a public university in the region of Savoy, with one campus in Annecy and two around Chambéry. Campuses The university was officially founded in 1979 from several colleges founded in the 1960s and 1970s. To avoid a straight choice between the two biggest towns of the Savoie/Haute-Savoie region, the authorities decided to set up a campus in each city for different areas of study. The university has three campuses: * The Annecy-le-Vieux campus (near Annecy) is the university's "technology institute" ( IUT), and teaches engineering-related subjects and business and administration related subjects. There is either the faculty of economics and management (IMUS, ''Institut de Management de l'Université de Savoie''). * Jacob-Bellecombette (1.5 km south of Chambéry) is the campus for students of languages, literature, social sciences, law and economics. It has a library, sports ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warsaw University
The University of Warsaw ( pl, Uniwersytet Warszawski, la, Universitas Varsoviensis) is a public university in Warsaw, Poland. Established in 1816, it is the largest institution of higher learning in the country offering 37 different fields of study as well as 100 specializations in humanities, technical, and the natural sciences. The University of Warsaw consists of 126 buildings and educational complexes with over 18 faculties: biology, chemistry, journalism and political science, philosophy and sociology, physics, geography and regional studies, geology, history, applied linguistics and philology, Polish language, pedagogy, economics, law and public administration, psychology, applied social sciences, management and mathematics, computer science and mechanics. The University of Warsaw is one of the top Polish universities. It was ranked by '' Perspektywy'' magazine as best Polish university in 2010, 2011, 2014, and 2016. International rankings such as ARWU and University We ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Angers
The University of Angers (french: Université d'Angers; UA) is a public university in western France, with campuses in Angers, Cholet, and Saumur. It forms part of thAngers-Le Mans University Community History The University of Angers was initially established during the 11th century as the ''School of Angers''. It became known as the ''University of Angers'' in 1337 and was the fifth largest university in France at the time. The university existed until 1793 when all universities in France were closed. Nearly 2 centuries later, the university was reestablished in 1971 after a regrouping of several preexisting higher education establishments. It would go on to add additional campuses in Cholet and Saumur in 1987 and 2004, respectively. Today, the University of Angers counts more than 25,000 students across all campuses. Academics The University of Angers offers bachelors, vocational bachelors, masters, and doctoral degrees across its 8 faculties and institutes: *Faculty of Tou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trajectory
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously. The mass might be a projectile or a satellite. For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass. In control theory, a trajectory is a time-ordered set of states of a dynamical system (see e.g. Poincaré map). In discrete mathematics, a trajectory is a sequence (f^k(x))_ of values calculated by the iterated application of a mapping f to an element x of its source. Physics of trajectories A familiar example of a trajectory is the path of a projectile, such as a thrown ball or rock. In a significantly simplified model, the object moves only under the influence of a uniform gravitational force field. This can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\bf) may be defined by: :df=\nabla f \cdot d\bf where ''df'' is the total infinitesimal change in ''f'' for an infinitesimal displacement d\bf, and is seen to be maximal when d\bf is in the direction of the gradi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S. There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence (x_n)_ in a topological space X is a point x such that, for every neighbourhood V of x, there are infinitely many natural numbers n such that x_n \in V. This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. The similarly named notion of a (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is synon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Line
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciprocal) trigonometric function of the cosine * the secant method, a root-finding algorithm in numerical analysis, based on secant lines to graphs of functions * a secant ogive Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproc ... in nose cone design {{mathdab sr:Секанс ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanis%C5%82aw %C5%81ojasiewicz
Stanisław Łojasiewicz (9 October 1926 – 14 November 2002) was a Polish mathematician.. Biography At the end of the 1950s, he solved the problem of distribution division by analytic functions, introducing the %C5%81ojasiewicz inequality. Its solution opened the road to important results in the new theory of partial differential equations. The method established by Łojasiewicz led him to advance the theory of semianalytic sets, which opened an important chapter in modern analysis. Commemoration The ''Łojasiewicz Lectures'' are a series of annual lectures in mathematics given at the Jagiellonian University in honour of Łojasiewicz. See also * Real algebraic geometry In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomia ... References 1926 births 2002 deaths 20th-centur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
