:* non-parametric hierarchical Bayesian models, such as models based on the Dirichlet process, which allow the number of latent variables to grow as necessary to fit the data, but where individual variables still follow parametric distributions and even the process controlling the rate of growth of latent variables follows a parametric distribution.

The exponential families include many of the most common distributions, including the normal, exponential, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, Wishart, Inverse Wishart and many others.

The normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, geometric, inverse Gaussian, von Mises and von Mises-Fisher distributions are all exponential families.

In hierarchical Bayesian models with categorical variables, such as latent Dirichlet allocation and various other models used in natural language processing, it is quite common to collapse out the Dirichlet distributions that are typically used as prior distributions over the categorical variables.

In probability and statistics, the Dirichlet distribution ( after Johann Peter Gustav Lejeune Dirichlet ), often denoted, is a family of continuous multivariate probability distributions parametrized by a vector of positive reals.

Symmetric Dirichlet distributions are often used when a Dirichlet prior is called for, since there typically is no prior knowledge favoring one component over another.

In Bayesian mixture models and other hierarchical Bayesian models with mixture components, Dirichlet distributions are commonly used as the prior distributions for the categorical variables appearing in the models.

This distribution plays an important role in hierarchical Bayesian models, because when doing inference over such models using methods such as Gibbs sampling or variational Bayes, Dirichlet prior distributions are often marginalized out.

Dirichlet distributions are most commonly used as the prior distribution of categorical variables or multinomial variables in Bayesian mixture models and other hierarchical Bayesian models.

* All Dirichlet characters are completely multiplicative functions.

If f and g are two multiplicative functions, one defines a new multiplicative function f * g, the Dirichlet convolution of f and g, by

More examples are shown in the article on Dirichlet series.

Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.

Their Dirichlet generating functions are

However, there are other applications where there is a need to describe the uncertainty with which a function is known and where the state of knowledge about the true function can be expressed by saying that it is an unknown realisation of a random function, for example in the Dirichlet process.

According to the definition above, some exotic functions, for example the Dirichlet function, are also periodic ; in the case of Dirichlet function, any nonzero rational number is a period.

In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0.

Global L-functions can be associated to elliptic curves, number fields ( in which case they are called Dedekind zeta-functions ), Maass forms, and Dirichlet characters ( in which case they are called Dirichlet L-functions ).

There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series ; definitions and examples are given below.

Formal Dirichlet series are often classified as generating functions, although they are not strictly formal power series.

The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group ; and states that these L-functions are identical to certain Dirichlet L-series or more general series ( that is, certain analogues of the Riemann zeta function ) constructed from Hecke characters.

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of.

Dirichlet characters are used to define Dirichlet L-functions, which are meromorphic functions with a variety of interesting analytic properties.

1912 Plemelj published a very simple proof for the Fermat's last theorem for exponent n = 5, which was first given almost simultaneously by Dirichlet in 1828 and Legendre in 1830.

A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies ; this makes solutions very flexible.

In fact his direction was very different, as his publications show, with an interest in Dirichlet series, lacunary series, entire functions and other major topics in complex analysis and harmonic analysis.

A very common special case is the symmetric Dirichlet distribution, where all of the elements making up the parameter vector have the same value.

It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions.

It is widely believed that Kummer was led to his " ideal complex numbers " by his interest in Fermat's Last Theorem ; there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Dirichlet told him his argument relied on unique factorization ; but the story was first told by Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources.

This construction ties in with concept of a base measure when discussing Dirichlet processes and is often used in the topic modelling literature.

Inference over hierarchical Bayesian models is often done using Gibbs sampling, and in such a case, instances of the Dirichlet distribution are typically marginalized out of the model by integrating out the Dirichlet random variable.

Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.

In the proof he notably used the principle that the solution is the function that minimizes the so-called Dirichlet energy.

Riemann later named this approach the Dirichlet principle, although he knew it had also been used by Gauss and by Lord Kelvin.

Gradient estimates were also used crucially in Yau's joint work with S. Y. Cheng to give a complete proof of the higher dimensional Hermann Minkowski problem and the Dirichlet problem for the real Monge – Ampère equation, and other results on the Kähler – Einstein metric of bounded pseudoconvex domains.

Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products.

He also used this technique on the Dirichlet divisor problem, allowing him to estimate the number of integer points under an arbitrary curve.

This form is used to construct solutions to Dirichlet boundary condition problems.

This fact can be used to prove existence and uniqueness of solutions to the Dirichlet problem for the Poisson equation in suitably regular domains, and for suitably well-behaved functions ƒ: one first applies a Newtonian potential to obtain a solution, and then adjusts by adding a harmonic function to get the correct boundary data.

The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters.

Other types of boundary conditions, e. g., the homogeneous Dirichlet boundary condition, where f ( x, y )= 0 on the boundary of the grid, are rarely used for graph Laplacians, but are common in other applications.

In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc.