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List Of Things Named After Peter Gustav Lejeune Dirichlet
The Germany, German mathematician Peter Gustav Lejeune Dirichlet (1805–1859) is the eponym of many things. Mathematics * Theorems named ''Dirichlet's theorem'': **Dirichlet's approximation theorem (diophantine approximation) **Dirichlet's theorem on arithmetic progressions (number theory, specifically prime numbers) **Dirichlet's unit theorem (algebraic number theory and Ring (mathematics), rings) * Dirichlet algebra * Dirichlet beta function * Dirichlet boundary condition (differential equations) **Neumann–Dirichlet method * Dirichlet characters (number theory, specifically Dirichlet series, zeta and Dirichlet L-function, L-functions. 1831) * Dirichlet conditions (Fourier series) * Dirichlet convolution (number theory and arithmetic functions) * Dirichlet density (number theory) **Dirichlet average * Dirichlet distribution (probability theory) **Dirichlet-multinomial distribution ** Dirichlet negative multinomial distribution **Generalized Dirichlet distribution (probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated between the Baltic and North seas to the north, and the Alps to the south; it covers an area of , with a population of almost 84 million within its 16 constituent states. Germany borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west. The nation's capital and most populous city is Berlin and its financial centre is Frankfurt; the largest urban area is the Ruhr. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity. A region named Germania was documented before AD 100. In 962, the Kingdom of Germany formed the bulk of the Holy Roman Empire. During the 16th ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: :1) \chi(ab) = \chi(a)\chi(b); i.e. \chi is completely multiplicative. :2) \chi(a) \begin =0 &\text\; \gcd(a,m)>1\\ \ne 0&\text\;\gcd(a,m)=1. \end (gcd is the greatest common divisor) :3) \chi(a + m) = \chi(a); i.e. \chi is periodic with period m. The simplest possible character, called the principal character, usually denoted \chi_0, (see Notation below) exists for all moduli: : \chi_0(a)= \begin 0 &\text\; \gcd(a,m)>1\\ 1 &\text\;\gcd(a,m)=1. \end The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions. Notation \phi(n) is Euler's totient function. \zeta_n is a complex primitive n-th root of unity: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverted Dirichlet Distribution
In statistics, the inverted Dirichlet distribution is a multivariate generalization of the beta prime distribution, and is related to the Dirichlet distribution. It was first described by Tiao and Cuttman in 1965. The distribution has a density function given by : p\left(x_1,\ldots, x_k\right) = \frac x_1^\cdots x_k^\times\left(1+\sum_^k x_i\right)^,\qquad x_i>0. The distribution has applications in statistical regression and arises naturally when considering the multivariate Student distribution. It can be characterized by its mixed moments: : E\left prod_^kx_i^\right= \frac\prod_^k\frac provided that q_j>-\nu_j, 1\leqslant j\leqslant k and \nu_>q_1+\ldots+q_k. The inverted Dirichlet distribution is conjugate to the negative multinomial distribution if a generalized form of odds ratio is used instead of the categories' probabilities- if the negative multinomial parameter vector is given by p, by changing parameters of the negative multinomial to x_i = \frac, i = 1\ldots k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grouped Dirichlet Distribution
In statistics, the grouped Dirichlet distribution (GDD) is a multivariate generalization of the Dirichlet distribution It was first described by Ng et al. 2008. The Grouped Dirichlet distribution arises in the analysis of categorical data In statistics, a categorical variable (also called qualitative variable) is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or ... where some observations could fall into any of a set of other 'crisp' category. For example, one may have a data set consisting of cases and controls under two different conditions. With complete data, the cross-classification of disease status forms a 2(case/control)-x-(condition/no-condition) table with cell probabilities If, however, the data includes, say, non-respondents which are known to be controls or cases, then the cross-classification of disease status forms a 2-x-3 table. The probabili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Dirichlet Distribution
In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely neutral . The density function of p_1,\ldots,p_ is : \left \prod_^B(a_i,b_i)\right p_k^ \prod_^\left p_i^\left(\sum_^kp_j\right)^\right where we define p_k= 1- \sum_^p_i. Here B(x,y) denotes the Beta function. This reduces to the standard Dirichlet distribution if b_=a_i+b_i for 2\leqslant i\leqslant k-1 (b_0 is arbitrary). For example, if ''k=4'', then the density function of p_1,p_2,p_3 is : \left prod_^B(a_i,b_i)\right p_1^p_2^p_3^p_4^\left(p_2+p_3+p_4\right)^\left(p_3+p_4\right)^ where p_1+p_2+p_3<1 and . Connor and Mosimann define the PDF as they did for the following reason. Define random variables with |
Dirichlet Negative Multinomial Distribution
In probability theory and statistics, the Dirichlet negative multinomial distribution is a multivariate distribution on the non-negative integers. It is a multivariate extension of the beta negative binomial distribution. It is also a generalization of the negative multinomial distribution (NM(''k'', ''p'')) allowing for heterogeneity or overdispersion to the probability vector. It is used in quantitative marketing research to flexibly model the number of household transactions across multiple brands. If parameters of the Dirichlet distribution are \boldsymbol, and if : X \mid p \sim \operatorname(x_0,\mathbf), where : \mathbf \sim \operatorname(\alpha_0,\boldsymbol\alpha), then the marginal distribution of ''X'' is a Dirichlet negative multinomial distribution: : X \sim \operatorname(x_0,\alpha_0,\boldsymbol). In the above, \operatorname(x_0, \mathbf) is the negative multinomial distribution and \operatorname(\alpha_0,\boldsymbol\alpha) is the Dirichlet distributi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet-multinomial Distribution
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution (after George Pólya). It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector \boldsymbol, and an observation drawn from a multinomial distribution with probability vector p and number of trials ''n''. The Dirichlet parameter vector captures the prior belief about the situation and can be seen as a pseudocount: observations of each outcome that occur before the actual data is collected. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, machine learning, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution. It reduces to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \boldsymbol\alpha of positive reals. It is a multivariate generalization of the beta distribution, (Chapter 49: Dirichlet and Inverted Dirichlet Distributions) hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution. The infinite-dimensional generalization of the Dirichlet distribution is the ''Dirichlet process''. Definitions Probability density function The Dirichlet distribution of order ''K'' ≥ 2 with parameters ''α''1, ..., ''α''''K'' > 0 has a probability density function with respect to Lebesgue m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Average
Dirichlet averages are averages of functions under the Dirichlet distribution. An important one are dirichlet averages that have a certain argument structure, namely : F(\mathbf;\mathbf)=\int f( \mathbf \cdot \mathbf) \, d \mu_b(\mathbf), where \mathbf\cdot\mathbf=\sum_i^N u_i \cdot z_i and d \mu_b(\mathbf)=u_1^ \cdots u_N^ d\mathbf is the Dirichlet measure with dimension ''N''. They were introduced by the mathematician Bille C. Carlson in the '70s who noticed that the simple notion of this type of averaging generalizes and unifies many special functions, among them generalized hypergeometric functions or various orthogonal polynomials:. They also play an important role for the solution of elliptic integrals (see Carlson symmetric form) and are connected to statistical applications in various ways, for example in Bayesian analysis Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Density
In mathematics, the Dirichlet density (or analytic density) of a set of primes, named after Peter Gustav Lejeune Dirichlet, is a measure of the size of the set that is easier to use than the natural density. Definition If ''A'' is a subset of the prime numbers, the Dirichlet density of ''A'' is the limit : \lim_ \frac if it exists. Note that since \textstyle as s\rightarrow 1^+ (see Prime zeta function), this is also equal to :\lim_. This expression is usually the order of the "pole" of :\prod_ at ''s'' = 1, (though in general it is not really a pole as it has non-integral order), at least if this function is a holomorphic function times a (real) power of ''s''−1 near ''s'' = 1. For example, if ''A'' is the set of all primes, it is the Riemann zeta function which has a pole of order 1 at ''s'' = 1, so the set of all primes has Dirichlet density 1. More generally, one can define the Dirichlet density of a sequence of primes (or prime powers), possibly with repetition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Functions
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of ''n''". An example of an arithmetic function is the divisor function whose value at a positive integer ''n'' is equal to the number of divisors of ''n''. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions. This article provides links to functions of both classes. Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. Multiplicative and additive functions An arithmetic function ''a'' is * completely additive if ''a''(''mn'') = ''a''(''m'') + ''a''(''n'') for all natural numbers ''m'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet. Definition If f , g : \mathbb\to\mathbb are two arithmetic functions from the positive integers to the complex numbers, the ''Dirichlet convolution'' is a new arithmetic function defined by: : (f*g)(n) \ =\ \sum_ f(d)\,g\!\left(\frac\right) \ =\ \sum_\!f(a)\,g(b) where the sum extends over all positive divisors ''d'' of ''n'', or equivalently over all distinct pairs of positive integers whose product is ''n''. This product occurs naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients: :\left(\sum_\frac\right) \left(\sum_\frac\right) \ = \ \left(\sum_\frac\right). Properties The set of arithmetic functions forms a commutative ring, the , under pointwise addition, where is defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |