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Whitney Extension Theorem
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if ''A'' is a closed subset of a Euclidean space, then it is possible to extend a given function of ''A'' in such a way as to have prescribed derivatives at the points of ''A''. It is a result of Hassler Whitney. Statement A precise statement of the theorem requires careful consideration of what it means to prescribe the derivative of a function on a closed set. One difficulty, for instance, is that closed subsets of Euclidean space in general lack a differentiable structure. The starting point, then, is an examination of the statement of Taylor's theorem. Given a real-valued ''C''''m'' function ''f''(x) on R''n'', Taylor's theorem asserts that for each a, x, y ∈ R''n'', there is a function ''R''''α''(x,y) approaching 0 uniformly as x,y → a such that where the sum is over multi-indices ''α''. L ... [...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] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor's Theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ''k'' of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration theory. Biography Life Hassler Whitney was born on March 23, 1907, in New York City, where his father Edward Baldwin Whitney was the First District New York Supreme Court judge. His mother, A. Josepha Newcomb Whitney, was an artist and active in politics. He was the paternal nephew of Connecticut Governor and Chief Justice Simeon Eben Baldwin, his paternal grandfather was William Dwight Whitney, professor of Ancient Languages at Yale University, linguist and Sanskrit scholar. Whitney was the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-grandson of American founding father Roger Sherman. His maternal grandparents were astronomer and mathematician Simon Newcomb (1835-1909), a Steeves desce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-index
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. Definition and basic properties An ''n''-dimensional multi-index is an ''n''-tuple :\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n) of non-negative integers (i.e. an element of the ''n''-dimensional set of natural numbers, denoted \mathbb^n_0). For multi-indices \alpha, \beta \in \mathbb^n_0 and x = (x_1, x_2, \ldots, x_n) \in \mathbb^n one defines: ;Componentwise sum and difference :\alpha \pm \beta= (\alpha_1 \pm \beta_1,\,\alpha_2 \pm \beta_2, \ldots, \,\alpha_n \pm \beta_n) ;Partial order :\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\ ;Sum of components (absolute value) :, \alpha , = \alpha_1 + \alpha_2 + \cdots + \alpha_n ;Factorial :\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n! ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Extension
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel's Lemma
In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations. Statement Suppose ''U'' is an open set in the Euclidean space R''n'', and suppose that ''f''0, ''f''1, ... is a sequence of smooth function, smooth function (mathematics), functions on ''U''. If ''I'' is any open interval in R containing 0 (possibly ''I'' = R), then there exists a smooth function ''F''(''t'', ''x'') defined on ''I''×''U'', such that :\left.\frac\_ = f_k(x), for ''k'' ≥ 0 and ''x'' in ''U''. Proof Proofs of Borel's lemma can be found in many text books on analysis, including and , from which the proof below is taken. Note that it suffices to prove the result for a small interval ''I'' = (−''ε'',''ε''), since if ''ψ''(''t'') is a smooth bump function with compact support in (−''ε'',''ε'') equal identically to 1 near 0, then ''ψ''(''t'') ⋅ ''F''(''t'', ''x'') gives a solution on R × ''U''. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entire Function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. Properties Every entire function can be represented as a power series f(z) = \sum_^\infty a_n z^n that converges everywhere in the complex plane, hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Factorization Theorem
In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root. The theorem, which is named for Karl Weierstrass, is closely related to a second result that every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. A generalization of the theorem extends it to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's zeros and poles, and an associated non-zero holomorphic function. Motivation The consequences of the fundamental theorem of algebra are twofold.. Firstly, any finite sequence \ in the complex plane ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mittag-Leffler Theorem
In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial fractions. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. The theorem is named after the Swedish mathematician Gösta Mittag-Leffler who published versions of the theorem in 1876 and 1884. Theorem Let U be an open set in \mathbb C and E \subset U be a subset whose limit points, if any, occur on the boundary of U. For each a in E, let p_a(z) be a polynomial in 1/(z-a) without constant coefficient, i.e. of the form p_a(z) = \sum_^ \frac. Then there exists a meromorphic function f on U whose poles are precisely the elements of E and such that for each such pole a \in E, the function f(z)-p_a(z) has only a removable singularity at a; in particular, the principal part of f at a is p_a(z). Furthermore, any o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of Unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, and * the sum of all the function values at is 1, i.e., \sum_ \rho(x) = 1. Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They are also important in the interpolation of data, in signal processing, and the theory of spline functions. Existence The existence of partitions of unity assumes two distinct forms: # Given any open cover \_ of a space, there exists a partition \_ indexed ''over the same set'' such that supp \rho_i \subseteq U_i. Such a partition is said to be subordinate to the open cover \_i. # If the space is locally-compact, given any open cover \_ of a space, there exists a partition \_ indexed over a possibly distinct index set such that each has co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirszbraun Theorem
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if is a subset of some Hilbert space , and is another Hilbert space, and : f: U \rightarrow H_2 is a Lipschitz-continuous map, then there is a Lipschitz-continuous map :F: H_1 \rightarrow H_2 that extends and has the same Lipschitz constant as . Note that this result in particular applies to Euclidean spaces and , and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient. The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
