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Mollifiers
In mathematics, mollifiers (also known as ''approximations to the identity'') are particular smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a (generalized) function, convolving it with a mollifier "mollifies" it, that is, its sharp features are smoothed, while still remaining close to the original. They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them. Historical notes Mollifiers were introduced by Kurt Otto Friedrichs in his paper , which is considered a watershed in the modern theory of partial differential equations.See the commentary of Peter Lax on the paper in . The name of this mathematical object has a curious genesis, and Peter Lax tells the story in his commentary on that paper published in Friedrichs' "''Selecta''". According to him, at that time, the mathematician Donald Alexander Flanders ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurt Otto Friedrichs
Kurt Otto Friedrichs (September 28, 1901 – December 31, 1982) was a German-American mathematician. He was the co-founder of the Courant Institute at New York University, and a recipient of the National Medal of Science. Biography Friedrichs was born in Kiel, Schleswig-Holstein on September 28, 1901. His family soon moved to Düsseldorf, where he grew up. He attended several different universities in Germany studying the philosophical works of Heidegger and Husserl, but finally decided that mathematics was his real calling. During the 1920s, Friedrichs pursued this field in Göttingen, which had a renowned Mathematical Institute under the direction of Richard Courant. Courant became a close colleague and lifelong friend of Friedrichs. In 1931, Friedrichs became a full professor of mathematics at the Technische Hochschule in Braunschweig. In early February 1933, a few days after Hitler became the Chancellor of Germany, Friedrichs met and immediately fell in love with a young J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Operator
An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms involving integrals * Integral transforms, which are maps between two function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...s, which involve integrals {{Authority control Integral calculus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Historical Notes
History is the systematic study of the past, focusing primarily on the human past. As an academic discipline, it analyses and interprets evidence to construct narratives about what happened and explain why it happened. Some theorists categorize history as a social science, while others see it as part of the humanities or consider it a hybrid discipline. Similar debates surround the purpose of history—for example, whether its main aim is theoretical, to uncover the truth, or practical, to learn lessons from the past. In a more general sense, the term ''history'' refers not to an academic field but to the past itself, times in the past, or to individual texts about the past. Historical research relies on primary and secondary sources to reconstruct past events and validate interpretations. Source criticism is used to evaluate these sources, assessing their authenticity, content, and reliability. Historians strive to integrate the perspectives of several sources to develop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Identity
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert ae_\lambda - a \rVert = 0. Similarly, a left approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert e_\lambda a - a \rVert = 0. An approximate identity is a net which is both a right approximate identity and a left approximate identity. C*-algebras For C*-algebras, a right (or left) approximate identity consisting of self-adjoint elements is the same as an approximate identity. The net of all positive elements in ''A'' of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra. This is called the canonical approximate identity of a C*-algebra. Approx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence (mathematics)
In mathematics, a series (mathematics), series is the summation, sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_1, a_2, a_3, \ldots) defines a series (mathematics), series that is denoted :S=a_1 + a_2 + a_3 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = a_1 +a_2 + \cdots + a_n = \sum_^n a_k. A series is convergent (or converges) if and only if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit of a sequence, limit; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if and only if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , 1 produce a convergent series: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Schwartz
Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of Distribution (mathematics), 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 Alsace, 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 Pediatrics, 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 Schwartz, Marie-Hélène Lévy, daughter of the probabilist Paul Lévy (mathematician), Paul Lévy who was then teaching at the École polytechniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitely Differentiable Function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (mathematics)
Distributions, also known as Schwartz distributions are a kind of generalized function in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than Solution of a differential equation, classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function, Dirac delta function. A Function (mathematics), function f is normally thought of as on the in the function Domain (function), domain by "sending" a point x in the domain t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be Heuristic, represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limit (mathematics), limits or, as is common in mathematics, measure theory and the theory of distribution (mathematics), distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
