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Weyl Sequence
In mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...: The sequence of all multiples of an irrational ''α'', ::0, ''α'', 2''α'', 3''α'', 4''α'', ... :is equidistributed modulo 1. In other words, the sequence of the fractional parts of each term will be uniformly distributed in the interval , 1). In computing In computing, an integer version of this sequence is often used to generate a discrete uniform distribution rather than a continuous one. Instead of using an irrational number, which cannot be calculated on a digital computer, the ratio of two integers is used in its place. An integer ''k'' is chosen, relatively prime to an integer modulus ''m''. In the com ... [...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] |
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'' + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone bore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Uniform Distribution
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, ''a'' and ''b'', which are the minimum and maximum values. The interval can either be closed (e.g. , b or open (e.g. (a, b)). Therefore, the distribution is often abbreviated ''U'' (''a'', ''b''), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable ''X'' under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is: : f(x)=\begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, engineering, mathematical, technological and social aspects. Major computing disciplines include computer engineering, computer science, cybersecurity, data science, information systems, information technology and software engineering. The term "computing" is also synonymous with counting and calculating. In earlier times, it was used in reference to the action performed by mechanical computing machines, and before that, to human computers. History The history of computing is longer than the history of computing hardware and includes the history of methods intended for pen and paper (or for chalk and slate) with or without the aid of tables. Computing is intimately tied to the representation of numbers, though mathematical conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Uniform Distribution
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair dice. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a unif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Prime
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Marsaglia
George Marsaglia (March 12, 1924 – February 15, 2011) was an American mathematician and computer scientist. He is best known for creating the diehard tests, a suite of software for measuring statistical randomness. Research on random numbers George Marsaglia established the lattice structure of linear congruential generators in the paper "Random numbers fall mainly in the planes", later termed the Marsaglia's theorem. This phenomenon means that ''n''-tuples with coordinates obtained from consecutive use of the generator will lie on a small number of equally spaced hyperplanes in ''n''-dimensional space. He also developed the diehard tests, a series of tests to determine whether or not a sequence of numbers have the statistical properties that could be expected from a random sequence. In 1995 he published a CD-ROM of random numbers, which included the diehard tests. His diehard paper came with the quotation "Nothing is random, only uncertain" attributed to ''Gail Gasram'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Statistical Software
The ''Journal of Statistical Software'' is a peer-reviewed open-access scientific journal that publishes papers related to statistical software. The ''Journal of Statistical Software'' was founded in 1996 by Jan de Leeuw of the Department of Statistics at the University of California, Los Angeles. Its current editors-in-chief are Achim Zeileis, Bettina Grün, Edzer Pebesma, and Torsten Hothorn. It is published by the Foundation for Open Access Statistics. The journal charges no author fees or subscription fees. The journal publishes peer-reviewed articles about statistical software, together with the source code. It also publishes reviews of statistical software and books (by invitation only). Articles are licensed under the Creative Commons Attribution License, while the source codes distributed with articles are licensed under the GNU General Public License. Articles are often about free statistical software and coverage includes packages for the R programming language. Abs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Things Named After Hermann Weyl
This is a list of topics named after Hermann Weyl, the influential German mathematician from the 20th century. Mathematics and physics * Cartan–Weyl theory ** Cartan–Weyl basis * Courant–Fischer–Weyl min-max principle * De Donder–Weyl theory * Hodge−Weyl decomposition * Majorana–Weyl spinor * Peter–Weyl theorem * Schur–Weyl duality * Weyl–Berry conjecture * Weyl–Groenewold product * Wigner–Weyl transform * Weyl algebra * Weyl almost periodic functions * Weyl anomaly * Weyl basis of the gamma matrices * Weyl chamber * Weyl character formula ** Weyl denominator formula ** Weyl dimension formula ** Weyl–Kac character formula * Weyl curvature: see Weyl tensor * Weyl curvature hypothesis * Weyl dimension formula, a specialization of the character formula * Weyl distance function * Weyl equation, a relativistic wave equation * Weyl expansion * Weyl fermion * Weyl gauge * Weyl gravity * Weyl group ** Length of a Weyl group element ** Restricte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
