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Equidistribution Theorem
In mathematics, the equidistribution theorem is the statement that the sequence :''a'', 2''a'', 3''a'', ... mod 1 is uniformly distributed on the circle \mathbb/\mathbb, when ''a'' is an irrational number. It is a special case of the ergodic theorem where one takes the normalized angle measure \mu=\frac. History While this theorem was proved in 1909 and 1910 separately by Hermann Weyl, Wacław Sierpiński and Piers Bohl, variants of this theorem continue to be studied to this day. In 1916, Weyl proved that the sequence ''a'', 22''a'', 32''a'', ... mod 1 is uniformly distributed on the unit interval. In 1937, Ivan Vinogradov proved that the sequence ''p''''n'' ''a'' mod 1 is uniformly distributed, where ''p''''n'' is the ''n''th prime. Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every sufficiently large odd number is the sum of three primes. George Birkhoff, in 1931, and Aleksandr Khinchin, in 1933, proved that the generalization ''x''&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equidistribution Theorem
In mathematics, the equidistribution theorem is the statement that the sequence :''a'', 2''a'', 3''a'', ... mod 1 is uniformly distributed on the circle \mathbb/\mathbb, when ''a'' is an irrational number. It is a special case of the ergodic theorem where one takes the normalized angle measure \mu=\frac. History While this theorem was proved in 1909 and 1910 separately by Hermann Weyl, Wacław Sierpiński and Piers Bohl, variants of this theorem continue to be studied to this day. In 1916, Weyl proved that the sequence ''a'', 22''a'', 32''a'', ... mod 1 is uniformly distributed on the unit interval. In 1937, Ivan Vinogradov proved that the sequence ''p''''n'' ''a'' mod 1 is uniformly distributed, where ''p''''n'' is the ''n''th prime. Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every sufficiently large odd number is the sum of three primes. George Birkhoff, in 1931, and Aleksandr Khinchin, in 1933, proved that the generalization ''x''&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Bourgain
Jean, Baron Bourgain (; – ) was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic theory and nonlinear partial differential equations from mathematical physics. Biography Bourgain received his PhD from the Vrije Universiteit Brussel in 1977. He was a faculty member at the University of Illinois, Urbana-Champaign and, from 1985 until 1995, professor at Institut des Hautes Études Scientifiques at Bures-sur-Yvette in France, at the Institute for Advanced Study in Princeton, New Jersey from 1994 until 2018. He was an editor for the ''Annals of Mathematics''. From 2012 to 2014, he was a visiting scholar at UC Berkeley. His research work included several areas of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, analytic number theory, combinatorics, ergodic theory, partial differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias M
Elias is the Greek equivalent of Elijah ( he, אֵלִיָּהוּ ''ʾĒlīyyāhū''; Syriac: ܐܠܝܐ ''Eliyā''; Arabic: الیاس Ilyās/Elyās), a prophet in the Northern Kingdom of Israel in the 9th century BC, mentioned in several holy books. Due to Elias' role in the scriptures and to many later associated traditions, the name is used as a personal name in numerous languages. Variants * Éilias Irish * Elia Italian, English * Elias Norwegian * Elías Icelandic * Éliás Hungarian * Elías Spanish * Eliáš, Elijáš Czech * Elias, Eelis, Eljas Finnish * Elias Danish, German, Swedish * Elias Portuguese * Elias, Iliya () Persian * Elias, Elis Swedish * Elias, Elyas Ethiopian * Elias, Elyas Philippines * Eliasz Polish * Élie French * Elija Slovene * Elijah English, Hebrew * Elis Welsh * Elisedd Welsh * Eliya (එලියා) Sinhala * Eliyas (Ілияс) Kazakh * Eliyahu, Eliya (אֵלִיָּהוּ, אליה) Biblical Hebrew, Hebrew * Elyās, Ilyās, Eliya ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Map
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. Definition Functions of a real variable The shift operator (where ) takes a function on R to its translation , : T^t f(x) = f_t(x) = f(x+t)~. A practical operational calculus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rendiconti Del Circolo Matematico Di Palermo
The Circolo Matematico di Palermo (Mathematical Circle of Palermo) is an Italian mathematical society, founded in Palermo by Sicilian geometer Giovanni B. Guccia in 1884.The Mathematical Circle of Palermo . Retrieved 2011-06-19. It began accepting foreign members in 1888, and by the time of Guccia's death in 1914 it had become the foremost international mathematical society, with approximately one thousand members. However, subsequently to that time it declined in influence. Publications ''Rendiconti del Circolo Matemat ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-gap Theorem
In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places points on a circle, at angles of , , , ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two. Unless is a rational multiple of , there will also be at least two distinct distances. This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, , and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square. Statement The three-gap theorem can be stated geometrically in terms of points on a circle. In this form, it states that if one places n points on a circle, at angles of \theta, 2\theta, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet's Approximation Theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and : \left , q \alpha -p \right , \leq \frac < \frac. Here represents the integer part of . This is a fundamental result in , showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality : is satisfied by infinitely many integers ''p'' and ''q''. This shows that any irratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Low-discrepancy Sequence
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. 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 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, 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 numbers. The "quasi" modifier is used to denote more clearly that the values of a low-discrepancy sequence are neither random nor pseudorandom, but such sequences share so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''a''/''b'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''a''/''b'' and ''α'' may not decrease if ''a''/''b'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric seri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential 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 absolute va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
