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Concept in number theory
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
Dirichlet_density
Probability distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ( α ) {\displaystyle \operatorname
Dirichlet_distribution
Concept in number theory
Davenport–Erdős theorem states that the natural density, when it exists, is equal to the logarithmic density. Dirichlet density Erdős conjecture on arithmetic progressions
Natural_density
Topics referred to by the same term
Optical density, the absorbance of a material Natural density, also called asymptotic density Dirichlet density Schnirelmann density Density (polytope)
Density_(disambiguation)
1831) Dirichlet conditions (Fourier series) Dirichlet convolution (number theory and arithmetic functions) Dirichlet density (number theory) Dirichlet average
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Distributions in probability theory
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite
Dirichlet-multinomial distribution
Dirichlet-multinomial_distribution
Theorem on the number of primes in arithmetic sequences
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
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Type of plane partition
Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons
Voronoi_diagram
Probability distribution
The logistic normal distribution is a more flexible alternative to the Dirichlet distribution in that it can capture correlations between components of
Logit-normal_distribution
Mathematical function
\left({\frac {1}{s-1}}\right)} . This is used in the definition of Dirichlet density. This gives the continuation of P ( s ) {\displaystyle P(s)} to
Prime_zeta_function
Second-order partial differential equation
kernels are densities of the harmonic measure with respect to boundary measure in these model domains. A classical approach to the Dirichlet problem for
Laplace's_equation
Modes of vibration in mathematics
In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can
Dirichlet_eigenvalue
has density 1/|E/F|. On the other hand, by studying Dirichlet L-series of characters of the group H0(E/F), one shows that the Dirichlet density of primes
Class_formation
Probability distribution
In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and
Generalized Dirichlet distribution
Generalized_Dirichlet_distribution
Describes statistically the splitting of primes in a given Galois extension of Q
Chebotarev density theorem may be viewed as a generalisation of Dirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem
Chebotarev_density_theorem
Exploring properties of the integers with complex analysis
begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions
Analytic_number_theory
Monte Carlo algorithm
as latent Dirichlet allocation and various other models used in natural language processing, it is quite common to collapse out the Dirichlet distributions
Gibbs_sampling
In statistics, the inverse Dirichlet distribution is a derivation of the matrix variate Dirichlet distribution. It is related to the inverse Wishart distribution
Inverse Dirichlet distribution
Inverse_Dirichlet_distribution
Mathematics
is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed
Neumann_boundary_condition
Mathematical conjecture on the Riemann zeta function
2024-07-16. "Density hypothesis - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-07-16. "New Bounds for Large Values of Dirichlet Polynomials
Lindelöf_hypothesis
Russian mathematician (1929–2005
theorem is partially named after him. A.I. Vinogradov, The density hypothesis for Dirichlet L-series. Izv. Akad. Nauk SSSR Ser. Mat., 29 (1965), pages
Askold_Vinogradov
Conjecture on zeros of the zeta function
this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation ( 1 − 2 2 s ) ζ ( s ) = η ( s ) = ∑
Riemann_hypothesis
Analytic function in mathematics
Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. The Riemann zeta function
Riemann_zeta_function
Probability distribution
In statistics, the grouped Dirichlet distribution (GDD) is a multivariate generalization of the Dirichlet distribution It was first described by Ng et
Grouped Dirichlet distribution
Grouped_Dirichlet_distribution
Type of problem involving ODEs or PDEs
studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle
Boundary_value_problem
Distribution of new data marginalized over the posterior
three-parameter Student's t distribution, beta-binomial distribution and Dirichlet-multinomial distribution are all predictive distributions of exponential-family
Posterior predictive distribution
Posterior_predictive_distribution
Gives the rank of the group of units in the ring of algebraic integers of a number field
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of
Dirichlet's_unit_theorem
Mathematical conjecture about zeros of L-functions
Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Statistical Markov model
two-level prior Dirichlet distribution, in which one Dirichlet distribution (the upper distribution) governs the parameters of another Dirichlet distribution
Hidden_Markov_model
Solution method for linear differential equations
at the associated turning point. One can then compute the probability density associated to the approximate wave function. The probability that the quantum
WKB_approximation
{\displaystyle \delta } is multiplicative, and since it is bounded by 1, its Dirichlet series converges absolutely in the half-plane Re ( s ) > 1 {\displaystyle
Average order of an arithmetic function
Average_order_of_an_arithmetic_function
the inverted Dirichlet distribution is a multivariate generalization of the beta prime distribution, and is related to the Dirichlet distribution. It
Inverted Dirichlet distribution
Inverted_Dirichlet_distribution
Concept in mathematics
also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory
Harmonic_map
Probability distribution
The generalization to multiple variables is called a Dirichlet distribution. The probability density function (PDF) of the beta distribution, for 0 ≤ x
Beta_distribution
Topics referred to by the same term
enzyme Hierarchical decision process Hierarchical Dirichlet process, a stochastic process High-density plasma, a type of plasma (physics) Hurricane Destruction
HDP
statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution. Suppose U
Matrix variate Dirichlet distribution
Matrix_variate_Dirichlet_distribution
trial data with a Dirichlet prior requires only adding the outcome frequencies to the Dirichlet prior alpha values, resulting in a Dirichlet posterior distribution
Expected value of sample information
Expected_value_of_sample_information
used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Type of probabilistic logic
(Probability Density Function). A multinomial opinion applies to a state variable of multiple possible values, and can be represented as a Dirichlet PDF (Probability
Subjective_logic
Partial differential equation describing the evolution of temperature in a region
,0)=g(\mathbf {x} )&\mathbf {x} \in \Omega \end{cases}}} with either Dirichlet or Neumann boundary data. A Green's function always exists, but unless
Heat_equation
Statistical concept
weights are typically viewed as a K-dimensional random vector drawn from a Dirichlet distribution (the conjugate prior of the categorical distribution), and
Mixture_model
Differential operator in mathematics
Laplacian can be defined wherever the Dirichlet energy functional makes sense, which is the theory of Dirichlet forms. For spaces with additional structure
Laplace_operator
Calculation technique for classical electrostatics
charges, which replicates the boundary conditions of the problem (see Dirichlet boundary conditions or Neumann boundary conditions). The validity of the
Method_of_image_charges
formula Mod n cryptanalysis Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula
List_of_number_theory_topics
Equations of motion for viscous fluids
{\textstyle \Gamma _{N}} portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied ( Γ D ∩ Γ N = ∅ {\textstyle
Navier–Stokes_equations
Mathematical theorem
location missing publisher (link) Vinogradov, A. I. (1965). "The density hypothesis for Dirichlet L-series". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). 29
Bombieri–Vinogradov_theorem
Description in spectral theory
he proved that the number, N ( λ ) {\displaystyle N(\lambda )} , of Dirichlet eigenvalues (counting their multiplicities) less than or equal to λ {\displaystyle
Weyl_law
On the distribution of prime numbers in arithmetic progressions
1112/s0025579300005313. MR 0197425. Vinogradov, Askold Ivanovich (1965). "The density hypothesis for Dirichlet L-series". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). 29
Elliott–Halberstam_conjecture
Canadian computer scientist and statistician (born 1956)
S2CID 1890561. Neal, Radford M. (2000). "Markov Chain Sampling Methods for Dirichlet Process Mixture Models". Journal of Computational and Graphical Statistics
Radford_M._Neal
Circulation density in a vector field
respectively. The curl of a field is formally defined as the circulation density at each point of the field. A vector field whose curl is zero is called
Curl_(mathematics)
Type of statistical analysis
nonparametric hierarchical Bayesian models, such as models based on the Dirichlet process, which allow the number of latent variables to grow as necessary
Nonparametric_statistics
Topics referred to by the same term
Sabbatarian organization Laser Doppler anemometry, to measure velocity Latent Dirichlet allocation, in natural language processing Left-displaced abomasum, a
LDA
classical result on primes in arithmetic progressions of Dirichlet generalises to Chebotaryov's density theorem; what is asked for is a generalisation, of the
Non-abelian class field theory
Non-abelian_class_field_theory
Mathematical function for the probability a given outcome occurs in an experiment
distribution, the precision (inverse variance) of a normal distribution, etc. Dirichlet distribution, for a vector of probabilities that must sum to 1; conjugate
Probability_distribution
Generalized function whose value is zero everywhere except at zero
integrals only converge in the sense of distributions (an example is the Dirichlet kernel below), rather than in the sense of measures. Another example is
Dirac_delta_function
Russian mathematician (1937–2008)
mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series. For most of his student and professional life he was associated
Anatoly_Karatsuba
Differential calculus on function spaces
Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area
Calculus_of_variations
Method of solution to differential equations
0 or Poisson's equation ∇2φ(x) = −ρ(x), subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for φ(x) everywhere
Green's_function
Integral expressing the amount of overlap of one function as it is shifted over another
scattering media Convolution power Convolution quotient Deconvolution Dirichlet convolution List of convolutions of probability distributions LTI system
Convolution
Number without repeated prime factors
is, |μ(n)| is equal to 1 if n is square-free, and 0 if it is not. The Dirichlet series of this indicator function is ∑ n = 1 ∞ | μ ( n ) | n s = ζ ( s
Square-free_integer
variables the probability density function of their joint distribution is the product of their individual density functions. The Dirichlet distribution, a generalization
List of probability distributions
List_of_probability_distributions
Number of prime factors of a natural number
moments of the function ω ( n ) {\displaystyle \omega (n)} . A known Dirichlet series involving ω ( n ) {\displaystyle \omega (n)} and the Riemann zeta
Prime_omega_function
Generalization of beta distribution
approximated by a transform of the Tracy–Widom distribution. Matrix variate Dirichlet distribution (Potters & Bouchaud 2020) Johnstone, Iain M. (2008-12-01)
Matrix variate beta distribution
Matrix_variate_beta_distribution
Process which assigns captioning to a digital image
from the original on 2007-09-28. Latent Dirichlet Allocation model D Blei; A Ng & M Jordan (2003). "Latent Dirichlet allocation" (PDF). Journal of Machine
Automatic_image_annotation
Theorem in measure theory
not be readily apparent, as can be demonstrated by example. Consider Dirichlet function, that is the indicator function 1 Q : [ 0 , 1 ] → { 0 , 1 } {\displaystyle
Lusin's_theorem
considering partial sums, which Dirichlet transformed into a particular Dirichlet integral involving what is now called the Dirichlet kernel. This paper introduced
List of publications in mathematics
List_of_publications_in_mathematics
Technique to solve partial differential equations
the reasons behind the failure of regular PINNs is soft-constraining of Dirichlet and Neumann boundary conditions which pose a multi-objective optimization
Physics-informed neural networks
Physics-informed_neural_networks
Probability distribution
S2CID 120066454. Penny, W. D. "KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities". "LogGammaDistribution—Wolfram Language Documentation".
Gamma_distribution
Integral transform useful in probability theory, physics, and engineering
to compute using elementary methods of real calculus. For example, the Dirichlet integral can be evaluated using the Laplace transform: ∫ 0 ∞ sin x x
Laplace_transform
Fourier transform of the probability density function
probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Model for generating observable data in probability and statistics
Generative artificial intelligence Averaged one-dependence estimators Latent Dirichlet allocation Boltzmann machine (e.g. Restricted Boltzmann machine, Deep
Generative_model
Generating pseudo-random numbers that follow a probability distribution
Poisson-distributed random variables Beta distribution#Random variate generation Dirichlet distribution#Random variate generation Exponential distribution#Random
Non-uniform random variate generation
Non-uniform_random_variate_generation
Statistical model used in machine learning
can also be obtained by factoring the density of the SGB distribution, which is obtained by sending Dirichlet variates through f cal {\displaystyle f_{\text{cal}}}
Flow-based_generative_model
Type of mathematical function
hyperbolic secant distribution, the Wishart distribution, if n ≥ p + 1, the Dirichlet distribution, if all parameters are ≥ 1, the gamma distribution if the
Logarithmically concave function
Logarithmically_concave_function
Functions in mathematics
are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions). On regions without
Harmonic_function
Cosmological paradox involving gravity
some special infinite nonuniforms systems. The Dirichlet problem cannot be solved in uniform density systems and general relativity is required. Under
Bentley's_paradox
Model for representing text documents
document frequency, latent semantic indexing, random projections and latent Dirichlet allocation. Weka. Weka is a popular data mining package for Java including
Vector_space_model
Operation in differential calculus
derivative is finite at 0, i.e. this is an essential discontinuity. The Dirichlet function, defined as: f ( x ) = { 1 , if x is rational 0 , if x is
Symmetric_derivative
Decomposition of periodic functions
integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fourier's results with greater precision
Fourier_series
Spline function
{\displaystyle B_{i,n,{\textbf {norm}}}} can be written as Carlson's Dirichlet average R k {\displaystyle R_{k}} , which in turn can be solved exactly
B-spline
Extended physical object in string theory
required to lie on a D-brane. The letter "D" in D-brane refers to the Dirichlet boundary condition, which the D-brane satisfies. One crucial point about
Brane
Find the value of the De Bruijn–Newman constant. Is Selberg class of Dirichlet series equal to class of automorphic L-functions? Hardy–Littlewood zeta
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Statement relating differentiable symmetries to conserved quantities
{S}}} is left invariant. This will certainly be true if the Lagrangian density L {\displaystyle {\mathcal {L}}} is left invariant, but it will also be
Noether's_theorem
Results about asymptotic posterior normality
Bernstein–von Mises theorem usually fails to hold with a notable exception of the Dirichlet process. A remarkable result was found by Freedman in 1965: the Bernstein–von
Bernstein–von_Mises_theorem
Calculation of complex statistical distributions
sampling over nonparametric Bayesian models such as those involving the Dirichlet process or Chinese restaurant process, where the number of mixing
Markov_chain_Monte_Carlo
Symbols for constants, special functions
of a linear response function a character in mathematics; especially a Dirichlet character in number theory sometimes the mole fraction a characteristic
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Prime number congruent to 1 mod 4
61, 73, 89, 97, 101, 109, 113, ... (sequence A002144 in the OEIS). By Dirichlet's theorem on arithmetic progressions, this sequence is infinite. More strongly
Pythagorean_prime
French mathematician and physicist (1781–1840)
to the works of Dirichlet and Hermann Schwarz, the Poisson kernel is now typically presented in the context of solving the Dirichlet problem for harmonic
Siméon_Denis_Poisson
Auxiliary functions used to probe equations, distributions, and weak formulations
space H 0 1 ( U ) {\displaystyle H_{0}^{1}(U)} corresponds to homogeneous Dirichlet boundary conditions in the trace sense on sufficiently regular domains
Test_function
Shape taken by a self-gravitating fluid body rotating at constant velocity
3160200203. Lagrange, J. L. (1811). Mécanique Analytique sect. IV 2 vol. Dirichlet, G. L. (1856). "Gedächtnisrede auf Carl Gustav Jacob Jacobi". Journal
Jacobi_ellipsoid
Generalization of the one-dimensional normal distribution to higher dimensions
the covariance matrix is singular, the corresponding distribution has no density; see the section below for details. This case arises frequently in statistics;
Multivariate normal distribution
Multivariate_normal_distribution
Generalization of the binomial distribution
(x_{i}+1)}}\prod _{i=1}^{k}p_{i}^{x_{i}}.} This form shows its resemblance to the Dirichlet distribution, which is its conjugate prior. Suppose that in a three-way
Multinomial_distribution
Multivariate derivative (mathematics)
Wikimedia Commons has media related to Gradient fields. Curl – Circulation density in a vector field Divergence – Vector operator in vector calculus Four-gradient –
Gradient
Vector operator in vector calculus
However any closed surface not enclosing the point will have a constant density of gas inside, so just as many fluid particles are entering as leaving
Divergence
Hungarian mathematician
ISSN 0001-5954. Montgomery, H. L. (1969). "Mean and large values of Dirichlet polynomials". Inventiones Mathematicae. 8 (4): 334–345. doi:10.1007/BF01404637
Gábor_Halász
Technique in integral evaluation
with probability density pX and another random variable Y such that Y= ϕ(X) for injective (one-to-one) ϕ, what is the probability density for Y? It is easiest
Integration_by_substitution
British mathematician
1112/plms/s3-47.2.225. Heath-Brown, D. R. (1992). "Zero-Free Regions for Dirichlet L-Functions, and the Least Prime in an Arithmetic Progression". Proceedings
Roger_Heath-Brown
for Dirichlet (mem)brane, a submanifold (of dimension p+1) on which the ends of strings are constrained to lie, so that the strings satisfy Dirichlet boundary
Glossary_of_string_theory
Concept in statistical mechanics
meanings of the word "distribution"). Given a domain Ω⊆Rn, consider the Dirichlet inner product ⟨ f , g ⟩ := ∫ Ω ( D f ( x ) , D g ( x ) ) d x {\displaystyle
Gaussian_free_field
Concept in statistics
distribution with probability vector distributed according to a Dirichlet distribution yields a Dirichlet-multinomial distribution. Compounding a Poisson distribution
Compound probability distribution
Compound_probability_distribution
DIRICHLET DENSITY
DIRICHLET DENSITY
DIRICHLET DENSITY
DIRICHLET DENSITY
Boy/Male
Hindu
Eternal, Constant, Perpetually
Girl/Female
Bengali, Hindu, Indian, Marathi
Flower
Boy/Male
Australian, Scottish
River
Girl/Female
Australian, Hungarian, Latin
Violet
Girl/Female
Arabic, Australian, French, Lebanese
Smiling
Girl/Female
Australian, Greek
Wretchedness
Boy/Male
Arabic, Muslim
Friend of God
Surname or Lastname
English
English : perhaps a variant of Bedworth, a habitational name from a place in Warwickshire, so named with an Old English personal name Bē(a)da + worð ‘enclosure’.
Boy/Male
Tamil
Mahendran | மஹேநà¯à®¤à¯€à®°à®£Â
The great God Indra the God of the Sky), Lord Indra, Lord of the Sky
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Marathi, Tamil, Telugu
Born
DIRICHLET DENSITY
DIRICHLET DENSITY
DIRICHLET DENSITY
DIRICHLET DENSITY
DIRICHLET DENSITY
n.
Depth of shade.
n.
The quality or state of being porous; -- opposed to density.
n.
An instrument for ascertaining the specific gravity or density of a substance.
n.
The quality of being dense, close, or thick; compactness; -- opposed to rarity.
n.
To break the natural course of, as rays of light orr heat, when passing from one transparent medium to another of different density; to cause to deviate from a direct course by an action distinct from reflection; as, a dense medium refrcts the rays of light as they pass into it from a rare medium.
n.
The science of the determination of the density of vapors and gases.
n.
A measure of weight, being a thousand grams, equal to 2.2046 pounds avoirdupois (15,432.34 grains). It is equal to the weight of a cubic decimeter of distilled water at the temperature of maximum density, or 39¡ Fahrenheit.
n.
The change in the direction of ray of light, heat, or the like, when it enters obliquely a medium of a different density from that through which it has previously moved.
n.
The quality of being dense; density.
a.
Having equal density, as different regions of a medium; passing through points at which the density is equal; as, an isopycnic line or surface.
n.
The unit of weight in the metric system. It was intended to be exactly, and is very nearly, equivalent to the weight in a vacuum of one cubic centimeter of pure water at its maximum density. It is equal to 15.432 grains. See Grain, n., 4.
n.
The ratio of mass, or quantity of matter, to bulk or volume, esp. as compared with the mass and volume of a portion of some substance used as a standard.
n.
The curve formed by a rope or chain of uniform density and perfect flexibility, hanging freely between two points of suspension, not in the same vertical line.
n.
An instrument for determining the strength or purity of wine by measuring its density.
n.
The quality or state of being rare; rareness; thinness; as, the rarity (contrasted with the density) of gases.
n.
A form of hydrometer, specially graduated, for finding the density of milk, and thus discovering whether it has been mixed with water or some of the cream has been removed.
n.
A line or surface passing through those points in a medium, at which the density is the same.
n.
The state or quality of being solid; density; consistency, -- opposed to fluidity; compactness; fullness of matter, -- opposed to openness or hollowness; strength; soundness, -- opposed to weakness or instability; the primary quality or affection of matter by which its particles exclude or resist all others; hardness; massiveness.