mathematical analysis

Series mathematics - Convergence criteria. The investigation of the validity of infinite series is considered to begin with Gauss. Euler had already considered the hypergeometric series on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence. Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with ...

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Series mathematics, Series mathematics - Infinite series, Series mathematics - Formal definition, Series mathematics - History of the theory of infinite series, Series mathematics - Convergence criteria, Series mathematics - Uniform convergence, Series mathematics - Semi-convergence, Series mathematics - Fourier series, Series mathematics - Some types of infinite series, Series mathematics - Absolute convergence, Series mathematics - Convergence tests, Series mathematics - Power series, Series mathematics - Generalizations

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The most basic of improper integrals are integrals such as: As stated above, this need not be defined as an improper integral, since it can be construed as a Lebesgue integral instead. Nonetheless, for purposes of actually computing this integral, it is more convenient to treat it as an improper integral, i.e., to evaluate it when the upper bound of integration is finite and then take the limit as that bound approaches ∞. The antiderivative of the function being inte ...

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Improper integral, Improper integral - Infinite bounds of integration, Improper integral - Vertical asymptotes at bounds of integration, Improper integral - Cauchy principal values

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Extended real number line - Limits. We often wish to describe the behavior of a function f(x), as either the argument x or the function value f(x) get "very big" in some sense. For example, consider the function The graph of this function has a horizontal asymptote of y = 0. Geometrically, as we move farther and farther to the right down the x-axis, the value of 1/x gets closer and closer to 0. This limiting behavior is similar to the limit of a function at a real number, except tha ...

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Extended real number line, Extended real number line - Motivation, Extended real number line - Limits, Extended real number line - Measure and integration, Extended real number line - Order and topological properties, Extended real number line - Arithmetic operations, Extended real number line - Algebraic properties, Extended real number line - Miscellaneous

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Fractal - Contributions from classical analysis. Objects that are now called fractals were discovered and explored long before the word was coined. As Mandelbrot himself pointed out the idea of "recursive self similarity" was originally developed by the philosopher Leibniz and he even worked many details. In 1872, Karl Weierstrass found an example of a function with the non-intuitive property that it is everywhere continuous but nowhere differentiable — the graph of this function would now be called a fractal. ...

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Fractal, Fractal - History, Fractal - Contributions from classical analysis, Fractal - Aspects of set description, Fractal - Mandelbrot's contributions, Fractal - The fractional dimension of the boundary of the Koch snowflake, Fractal - Definitions, Fractal - Categories of fractals, Fractal - Examples, Fractal - Fractals in nature, Fractal - Applications, Fractal - Fractal generation

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1. Mentioned in Of Human Bondage, Chapter 44 by W. Somerset Maugham, 1915. 2. Extract from Noctambule, Ballads of a Bohemian by Robert Service, 1921. Zut! it's two o'clock. See! the lights are jumping. Finish up your bock, Time we all were humping. Waiters stack the chairs, Pile them on the tables; Let us to our lairs Underneath the gables. Up the old Boul ...

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Boulevard Saint-Michel, Boulevard Saint-Michel - History, Boulevard Saint-Michel - Composition, Boulevard Saint-Michel - Literature, Boulevard Saint-Michel - Extension to the sea

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A set S of real numbers is called bounded above if there is a real number k such that k ≥ s for all s in S. The number k is called an upper bound of S. The terms bounded below and lower bound are similarly defined. A set S is bounded if it is bounded both above and below. Therefore, a set of real numbers is bounde ...

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Bounded set, Bounded set - Definition, Bounded set - Metric space, Bounded set - Boundedness in topological vector spaces, Bounded set - Boundedness in order theory

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Functions with compact support in X are those with support that is a compact subset of X. For example, if X is the real line, they are examples of functions that vanish at infinity. In good cases, functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. Note that every function on a compact space has compact support s ...

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Support mathematics, Support mathematics - Compact support, Support mathematics - Singular support, Support mathematics - Family of supports

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If S is a real interval (or indeed any topological space), we can talk about the continuity of the functions fn and f. The following is the more important result about uniform continuity: Uniform convergence theorem. If (fn) is a sequence of continuous functions which converges uniformly towards the function f, then f is continuous as well. The former theorem is important, since pointwise convergence of continuous functions is not enough to guarantee continuity ...

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Uniform convergence, Uniform convergence - Definition and comparison with pointwise convergence, Uniform convergence - Topological reformulation, Uniform convergence - Theorems, Uniform convergence - Generalizations, Uniform convergence - History, Uniform convergence - Reference

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In a normed vector space V, the triangle inequality is ||x + y|| ≤ ||x|| + ||y||     for all x, y in V that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. The real line is a normed vector space with the absolute value as the norm, and so the triangle inequality states that for any real numbers < ...

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Triangle inequality, Triangle inequality - Normed vector space, Triangle inequality - Metric space, Triangle inequality - Consequences, Triangle inequality - Reversal in Minkowski space

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The following examples illustrate how the squeeze theorem is applied in practice. Squeeze theorem - Example 1. Consider f(x) = x2 sin(1/x), which is defined on any interval containing x = 0, but is not defined at x = 0 itself. Computing the limit of f(x) as x → 0 is difficult by conventional means. Direct substitution fails because the function is not defined at x = 0, (let alone continuous). We cannot use L'Hopital's ...

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Squeeze theorem, Squeeze theorem - Statement, Squeeze theorem - Examples and applications, Squeeze theorem - Example 1, Squeeze theorem - Example 2, Squeeze theorem - Example 3, Squeeze theorem - Proof

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The O-notation means that the function is bounded from above (also known as dominated) by a function which varies like f, i.e., Thus, f = O(1) means that f is bounded and f = O(x) means that f does not grow faster than linear (sub-linear growth). It corresponds for functions in asymptotic limit to the r ...

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Asymptotic notation, Asymptotic notation - Definitions, Asymptotic notation - Properties

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Suppose X is a topological space, x0 is a point in X and f : X → R is a real-valued function. We say that f is upper semi-continuous at x0 if for every ε > 0 there exists a neighborhood U of x0 such that f(x) < f(x0) + ε for all x in U. Equivalently, this can be expressed as lim sup x → x0fSee also:

Semi-continuity, Semi-continuity - Examples, Semi-continuity - Formal definition, Semi-continuity - Properties

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If X is any set, then the family consisting only of the empty set and X is a σ-algebra over X, the so-called trivial σ-algebra. Another σ-algebra over X is given by the full power set of X. The collection of subsets of X which are countable or whose complements are countable is a σ-algebra, which is distinct from the powerset of X iff X is uncountable. If {Σa} is a family of σ-algebras over X, then the intersection of all Σa ...

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Sigma-algebra, Sigma-algebra - Notation, Sigma-algebra - Examples

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Related article: Criticism of Noam Chomsky. Chomsky is one of the best known figures of radical American politics. He defines himself as being in the tradition of anarchism, a political philosophy he summarizes as challenging all forms of hierarchy and attempting to eliminate them if they are unjustified. He especially identifies with the labor-oriented anarcho-syndicalist current of anarchism. Unlike many anarchists, Chomsky does not totally object to electoral politics; his stance on U.S. elections is that citizens should vot ...

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Noam Chomsky, Noam Chomsky - Biography, Noam Chomsky - Chomsky's name, Noam Chomsky - Contributions to linguistics, Noam Chomsky - Generative grammar, Noam Chomsky - Chomsky hierarchy, Noam Chomsky - Contributions to psychology, Noam Chomsky - Opinion on criticism of science culture, Noam Chomsky - Chomsky's influence in other fields, Noam Chomsky - Political views, Noam Chomsky - Chomsky on terrorism, Noam Chomsky - Criticism of United States government, Noam Chomsky - Views on globalization, Noam Chomsky - Views on socialism, Noam Chomsky - Mass media analysis, Noam Chomsky - Chomsky and the Middle East, Noam Chomsky - Criticism of intellectual communities, Noam Chomsky - Chomsky's influence as a political activist, Noam Chomsky - Opposition to the Vietnam War, Noam Chomsky - Alleged marginalization in the mainstream media, Noam Chomsky - Worldwide audience, Noam Chomsky - Criticisms, Noam Chomsky - Academic Achievements Awards and Honors, Noam Chomsky - Bibliography, Noam Chomsky - Linguistics, Noam Chomsky - Political works, Noam Chomsky - About Chomsky, Noam Chomsky - Filmography, Noam Chomsky - Political contemporaries

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The term theory is used informally within mathematics to mean a self-consistent body of definitions, axioms, theorems, examples, and so on. (Examples include group theory, Galois theory, control theory, and K-theory.) In particular there is no connotation of hypothetical. Thus the term unifying theory is more like sociological term used to study the actions of mathematicians. It may assume nothing conjectural, that would be analogous to an undiscovered scientific link. There is really no cognate within mathematics to ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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Aiming at a completely self-contained treatment of most of modern mathematics based on set theory, the group produced the following volumes (with the original French titles in brackets): I Set theory (Théorie des ensembles) II Algebra (Algèbre) III Topology (Topologie générale) IV Functions of one real variable (Fonctions d'une variable réelle) V Topological vector spaces (Espaces vectoriels topologiques ...

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Nicolas Bourbaki, Nicolas Bourbaki - Books by Bourbaki, Nicolas Bourbaki - Influence on mathematics in general, Nicolas Bourbaki - The group, Nicolas Bourbaki - The Bourbaki perspective and its limitations, Nicolas Bourbaki - Dieudonné as speaker for Bourbaki, Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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Feller held a docent position at the University of Kiel beginning in 1928. He fled the Nazis and went to Denmark, (Copenhagen) in 1933. He also lectured in Sweden, (Stockholm and Lund). Finally, in 1939 he arrived in the USA where he became a citizen in 1944 and was on the faculty at Brown and Cornell. In 1950 he became a professor at Princeton University. The works of Feller are contained in 104 papers and two books on a variety of topics such as mathematical analysis, theory of measurement, functional analysis ...

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William Feller, William Feller - Early life and education, William Feller - Work, William Feller - Results

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For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. A measure that takes values in a Banach space is called a spectral measure; these are used mainly in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values fro ...

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Measure mathematics, Measure mathematics - Formal definition, Measure mathematics - Properties, Measure mathematics - Monotonicity, Measure mathematics - Measures of infinite unions of measurable sets, Measure mathematics - Measures of infinite intersections of measurable sets, Measure mathematics - Sigma-finite measures, Measure mathematics - Completeness, Measure mathematics - Examples, Measure mathematics - Counterexamples, Measure mathematics - Generalizations

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The space Q of rational numbers, with the standard metric given by the absolute value, is not complete. Consider for instance the sequence defined by x1 := 1 and xn+1 := xn/2 + 1/xn. This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit; in fact, it converges towards the irra ...

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Complete space, Complete space - Examples, Complete space - Some theorems, Complete space - Completion, Complete space - Topologically complete spaces, Complete space - Generalisations

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Cauchy sequence - Formal definition. Formally, a Cauchy sequence is a sequence in a metric space (M, d) such that for every positive real number r > 0, there is an integer N such that for all integers m,n > N, the distance d(xm,xn) is less than r. Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ough ...

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Cauchy sequence, Cauchy sequence - Cauchy sequence in a metric space, Cauchy sequence - Formal definition, Cauchy sequence - Completeness, Cauchy sequence - Other properties, Cauchy sequence - Cauchy sequences in topological vector spaces, Cauchy sequence - Cauchy sequences in groups

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The institute is highly regarded throughout the world as a leader in applied mathematics, mathematical analysis, and scientific computation. There is emphasis on partial differential equations and their applications. Within the field of computer science, CIMS is regarded as a leader in theory, programming languages, computer graphics, and parallel computing. The Institute offers Master of Science and Ph.D. programs in both mathematics and computer science. There are currently about 230 full-time graduate students and another 370 part-ti ...

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Courant Institute of Mathematical Sciences, Courant Institute of Mathematical Sciences - Academics, Courant Institute of Mathematical Sciences - History, Courant Institute of Mathematical Sciences - Notable Courant Alumni

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Take the periodic Dirac delta function, which is not really a function, in the sense of mapping one set into another, but is rather a "generalized function", also called a "distribution", and multiply by 2π. We get the identity element for convolution on functions of period 2π. In other words, we have for every function f of period 2π. The Fourier series representation of this "function" is Therefore the Dirichlet kernel, which is just the sequence of partial sums of this seri ...

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Dirichlet kernel, Dirichlet kernel - Relation to the delta function, Dirichlet kernel - Proof of the trigonometric identity

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