Denjoy–Koksma Inequality
In mathematics, the Denjoy–Koksma inequality, introduced by as a combination of work of Arnaud Denjoy and the Koksma–Hlawka inequality of Jurjen Ferdinand Koksma, is a bound for Weyl sums \sum_^f(x+k\omega) of functions ''f'' of bounded variation. Statement Suppose that a map ''f'' from the circle ''T'' to itself has irrational rotation number ''α'', and ''p''/''q'' is a rational approximation to ''α'' with ''p'' and ''q'' coprime, , ''α'' – ''p''/''q'',  < 1/''q''². Suppose that ''φ'' is a function of bounded variation, and ''μ'' a

Arnaud Denjoy
Arnaud Denjoy (; 5 January 1884 – 21 January 1974) was a French mathematician. Biography Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able to integrate all derivatives. Among his students is Gustave Choquet. He is also known for the more general broad Denjoy integral, or Khinchin integral. Denjoy was an Invited Speaker of the ICM with talk ''Sur une classe d'ensembles parfaits en relation avec les fonctions admettant une dérivée seconde généralisée'' in 1920 at Strasbourg and with talk ''Les equations differentielles periodiques'' in 1950 at Cambridge, Massachusetts. In 1931 he was the president of the Société Mathématique de France. In 1942 he was elected a member of the Académie des sciences and was its president in 1962. Denjoy married in 1923 and was the father of three sons. He died in Paris in 1974. He was an atheist with a strong interest in philosophy, psy ...
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Koksma–Hlawka Inequality
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of ''N'', its subsequence ''x''1, ..., ''x''''N'' has a low discrepancy of a sequence, discrepancy. Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set ''B'' is close to proportional to the Measure (mathematics), measure of ''B'', as would happen on average (but not for particular samples) in the case of an equidistributed sequence. Specific definitions of discrepancy differ regarding the choice of ''B'' (hyperspheres, Hypercube, hypercubes, etc.) and how the discrepancy for every B is computed (usually normalized) and combined (usually by taking the worst value). Low-discrepancy sequences are also called quasirandom sequences, due to their common use as a replacement of uniformly distributed random sequence, random numbers. The "quasi" modifier is used to denote more clearly that the values of a low-discrepancy s ...
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Jurjen Ferdinand Koksma
Jurjen Ferdinand Koksma (21 April 1904, Schoterland – 17 December 1964, Amsterdam) was a Dutch mathematician who specialized in analytic number theory. Koksma received his Ph.D. degree (''cum laude'') in 1930 at the University of Groningen under supervision of Johannes van der Corput, with a thesis on ''Systems of Diophantine Inequalities''. Around the same time, aged 26, he was invited to become full professor at the ''Vrije Universiteit Amsterdam''. He accepted and in 1930 became the first professor in mathematics at this university. Koksma is also one of the founders of the Dutch ''Mathematisch Centrum'' (today ''Centrum Wiskunde & Informatica''). One of Koksma's main works was the book ''Diophantische Approximationen'', published in 1936 by Springer. He also wrote several papers with Paul Erdős. In 1950 he became member of the Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie ...
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Weyl Sum
In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typical exponential sum may take the form :\sum_n e(x_n), summed over a finite sequence of real numbers ''x''''n''. Formulation If we allow some real coefficients ''a''''n'', to get the form :\sum_n a_n e(x_n) it is the same as allowing exponents that are complex numbers. Both forms are certainly useful in applications. A large part of twentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started by basic work of Hermann Weyl in diophantine approximation. Estimates The main thrust of the subject is that a sum :S=\sum_n e(x_n) is ''trivially'' estimated by the number ''N'' of terms. That is, the absolute value :, S, \le N\, by the triangle inequality, since each summand has absolu ...
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Bounded Variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the -axis, neglecting the contribution of motion along -axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed -axis and to the -axis. Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltj ...
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Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ...
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Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the unit interval , 1 returning 0 for the empty set and 1 for t ...
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Publications Mathématiques De L'IHÉS
''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche Scientifique. The journal was established in 1959 and was published at irregular intervals, from one to five volumes a year. It is now biannual. The editor-in-chief is Claire Voisin (Collège de France). See also *''Annals of Mathematics'' *'' Journal of the American Mathematical Society'' *''Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...'' External links * Back issues from 1959 to 2010 Mathematics journals Publications established in 1959 Springer Science+Business Media academic journals Biannual journal ...
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Theorems In Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ...
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