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Laurent-Moïse Schwartz
Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in 1950 for his work on the theory of distributions. For several years he taught at the École polytechnique. Biography Family Laurent Schwartz came from a Jewish family of Alsatian origin, with a strong scientific background: his father was a well-known surgeon, his uncle Robert Debré (who contributed to the creation of UNICEF) was a famous pediatrician, and his great-uncle-in-law, Jacques Hadamard, was a famous mathematician. During his training at Lycée Louis-le-Grand to enter the École Normale Supérieure, he fell in love with Marie-Hélène Lévy, daughter of the probabilist Paul Lévy who was then teaching at the École polytechnique. They married in 1938. Later they had two children, Marc-André and Claudine. Marie-Hélène w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. Since the 17th century, Paris has been one of the world's major centres of finance, diplomacy, commerce, fashion, gastronomy, and science. For its leading role in the arts and sciences, as well as its very early system of street lighting, in the 19th century it became known as "the City of Light". Like London, prior to the Second World War, it was also sometimes called the capital of the world. The City of Paris is the centre of the Île-de-France region, or Paris Region, with an estimated population of 12,262,544 in 2019, or about 19% of the population of France, making the region France's primate city. The Paris Region had a GDP of €739 billion ($743 billion) in 2019, which is the highest in Europe. According to the Economist Intelli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Hogbe Nlend
Henri Hogbe Nlend (born 23 December 1939) is a Cameroonian mathematician, university professor, former government minister and presidential candidate. Biography Henri Hogbe Nlend was a professor at the University of Yaoundé, and at the University of Bordeaux. In 1976, at a meeting of the International Mathematical Union it was decided to form an African Mathematical Union. Hogbe Nlend was elected as its first president, a post he held until 1986. nternational Handbook of Mathematics Education Alan J. Bishop, , accessed 1 August 2008 The AMU was partially funded from another organization in Paris, which was also chaired by Hogbe Nlend. It is said that he was good at raising funds and that meetings were held twice a year. Hogbe Nlend was a candidate in the presidential election held on 12 October 1997, which was boycotted by the major opposition parties, and placed second, although he received only 2.9% of the vote. The winning candidate, incumbent President Paul Biya, appointed N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., f(x)) to its value at zero of its domain (f(0)), or as the weak limit of a sequence of bump functions (e.g., \delta(x) = \lim_ \frace^), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions. The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be scientific, belong to a non-scientific discipline, or no discipline at all. Depending on the context, a theory's assertions might, for example, include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several related meanings. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction ("falsify") of it. Scientific theories are the most reliable, rigorous, and compr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award honours the Canadian mathematician John Charles Fields. The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the Nobel Prize of Mathematics, although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria. According to the annual Academic Excellence Survey by ARWU, the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics. The prize includes a monetary award which, since 2006, has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Per Enflo
Per H. Enflo (; born 20 May 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved problems that had been considered fundamental. Three of these problems had been open for more than forty years: * The basis problem and the approximation problem and later * the invariant subspace problem for Banach spaces. In solving these problems, Enflo developed new techniques which were then used by other researchers in functional analysis and operator theory for years. Some of Enflo's research has been important also in other mathematical fields, such as number theory, and in computer science, especially computer algebra and approximation algorithms. Enflo works at Kent State University, where he holds the title of University Professor. Enflo has earlier held positions at the Miller Institute for Basic Research in Science at the University of California, Berkeley, Stanford University, École Polytechnique, (Paris) and The Royal Institute of Techn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylinder Set Measure
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space. Definition Let E be a separable real topological vector space. Let \mathcal (E) denote the collection of all surjective continuous linear maps T : E \to F_T defined on E whose image is some finite-dimensional real vector space F_T: \mathcal (E) := \left\. A cylinder set measure on E is a collection of probability measures \left\. where \mu_T is a probability measure on F_T. These measures are required to satisfy the following consistency condition: if \pi_ : F_S \to F_T is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radonifying Function
In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the second space. It acquired its name because the pushforward measure on the second space was historically thought of as a Radon measure. Definition Given two separable Banach spaces E and G, a CSM \ on E and a continuous linear map \theta \in \mathrm (E; G), we say that \theta is ''radonifying'' if the push forward CSM (see below) \left\ on G "is" a measure, i.e. there is a measure \nu on G such that ::\left( \theta_ (\mu_) \right)_ = S_ (\nu) for each S \in \mathcal (G), where S_ (\nu) is the usual push forward of the measure \nu by the linear map S : G \to F_. Push forward of a CSM Because the definition of a CSM on G requires that the maps in \mathcal (G) be surjective, the definition of the push forward for a CSM requires careful attention. The CSM ::\left\ is defined by ::\left( \thet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwartz–Bruhat Function
In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the space of Schwartz–Bruhat functions. Definitions *On a real vector space \mathbb^n, the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space \mathcal(\mathbb^n). *On a torus, the Schwartz–Bruhat functions are the smooth functions. *On a sum of copies of the integers, the Schwartz–Bruhat functions are the rapidly decreasing functions. *On an elementary group (i.e., an abelian locally compact group that is a product of copies of the reals, the integers, the circle group, and finite groups), the Schwartz–Bruhat functions are the smooth functions all of whose derivatives are rapidly decreasing. *O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwartz Space
In mathematics, Schwartz space \mathcal is the function space of all Function (mathematics), functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space \mathcal^* of \mathcal, that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French mathematician Laurent Schwartz. Definition Let \mathbb be the Set (mathematics), set of non-negative integers, and for any n \in \mathbb, let \mathbb^n := \underbrace_ be the ''n''-fold Cartesian product. The ''Schwartz space'' or space of rapidly decreasing functions on \mathbb^n is the function spaceS \left(\mathbb^n, \mathbb\right) := \left \,where C^(\mathbb^n, \mathbb) is the function space of smooth functions from \mathbb^n into \mathbb, and\, f\, _:= \sup_ \lef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwartz Kernel Theorem
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space \mathcal of test functions. The space \mathcal itself consists of smooth functions of compact support. Statement of the theorem Let X and Y be open sets in \mathbb^n. Every distribution k \in \mathcal'(X \times Y) defines a continuous linear map K \colon \mathcal(Y) \to \mathcal'(X) such that for every u \in \mathcal(X), v \in \mathcal(Y). Conversely, for every such continuous linear map K there exists one and only one distribution k \in \mathcal'(X \times Y) such that () holds. The distribution k is the kernel of the map K. Note Given a distribution k \in \mathcal'(X \times Y) one can always write the linear map K informally as :Kv = \int_ k(\c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
