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Paraproduct
In mathematics, a paraproduct is a non-commutative bilinear operator acting on functions that in some sense is like the product of the two functions it acts on. According to Svante Janson and Jaak Peetre, in an article from 1988, "the name 'paraproduct' denotes an idea rather than a unique definition; several versions exist and can be used for the same purposes." The concept emerged in J.-M. Bony’s theory of paradifferential operators. This said, for a given operator \Lambda to be defined as a paraproduct, it is normally required to satisfy the following properties: * It should "reconstruct the product" in the sense that for any pair of functions (f, g) in its domain, :: fg = \Lambda(f, g) + \Lambda(g, f). * For any appropriate functions f and h with h(0)=0, it is the case that h(f) = \Lambda(f, h'(f)). * It should satisfy some form of the Leibniz rule. A paraproduct may also be required to satisfy some form of Hölder's inequality In mathematical analysis, Hölder's inequal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Michel Bony
Jean-Michel Bony (born 1 February 1942 in Paris) is a French mathematician, specializing in mathematical analysis. He is known for his work on microlocal analysis and pseudodifferential operators. Education and career Bony completed his undergraduate and graduate studies at the École Normale Supérieure, where he received his Ph.D in 1972 with thesis advisor Gustave Choquet. Bony became a professor at the University of Paris-Sud and is now a professor at the École Polytechnique. His doctoral students include Jean-Yves Chemin. Research Bony's research deals with microlocal analysis, partial differential equations and potential theory. In 1981 he published important results on paradifferential operators, extending the theory of pseudifferential operators published by Ronald Coifman and Yves Meyer in 1979. Bony applied his theory to the propagation of singularities in solutions of semilinear wave equations. Recognition In 1980, Bony received the Prix Paul Doistau–Émile Blutet ... [...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] |
Non-commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Operator
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function B : V \times W \to X such that for all w \in W, the map B_w v \mapsto B(v, w) is a linear map from V to X, and for all v \in V, the map B_v w \mapsto B(v, w) is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Such a map B satisfies the following properties. * For any \lambda \in F, B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). * The map B is additive in both components: if v_1, v_2 \in V and w_1, w_2 \in W, then B(v_1 + v_2, w) = B(v_1, w) + B(v_2, w) and B(v, w_1 + w_2) = B(v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product (mathematics)
In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called ''factors''. For example, 30 is the product of 6 and 5 (the result of multiplication), and x\cdot (2+x) is the product of x and (2+x) (indicating that the two factors should be multiplied together). The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the ''commutative law'' of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well. There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different algebraic structures. Product of two numbers Product of a seque ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Svante Janson
Carl Svante Janson (born 21 May 1955) is a Swedish mathematician. A member of the Royal Swedish Academy of Sciences since 1994, Janson has been the chaired professor of mathematics at Uppsala University since 1987. In mathematical analysis, Janson has publications in functional analysis (especially harmonic analysis) and probability theory. In mathematical statistics, Janson has made contributions to the theory of U-statistics. In combinatorics, Janson has publications in probabilistic combinatorics, particularly random graphs and in the analysis of algorithms: In the study of random graphs, Janson introduced U-statistics and the Hoeffding decomposition. Janson has published four books and over 300 academic papers (). He has an Erdős number of 1. Biography Svante Janson has already had a long career in mathematics, because he started research at a very young age. From prodigy to docent A child prodigy in mathematics, Janson took high-school and even university classes whil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + u \cdot v' or in Leibniz's notation as \frac (u\cdot v) = \frac \cdot v + u \cdot \frac. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let ''u''(''x'') and ''v''(''x'') be two differentiable functions of ''x''. Then the differential of ''uv'' is : \begin d(u\cdot v) & = (u + du)\cdot (v + dv) - u\cdot v \\ & = u\cdot dv + v\cdot du + du\cdot dv. \end Since the term ''du''·''dv'' is "negligi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hölder's Inequality
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of spaces. :Theorem (Hölder's inequality). Let be a measure space and let with . Then for all measurable real number, real- or complex number, complex-valued function (mathematics), functions and on , ::\, fg\, _1 \le \, f\, _p \, g\, _q. :If, in addition, and and , then Hölder's inequality becomes an equality if and only if and are Linear dependence, linearly dependent in , meaning that there exist real numbers , not both of them zero, such that -almost everywhere. The numbers and above are said to be Hölder conjugates of each other. The special case gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if is infinite, the right-hand side also being infinite in that case. Conversely, if is in and is in , then the pointwise product is in . Hölder's inequality is used to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
