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Averages of functions under the Dirichlet distribution
Dirichlet averages are averages of functions under the Dirichlet distribution. An important one are dirichlet averages that have a certain argument structure
Dirichlet_average
Probability distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ( α ) {\displaystyle \operatorname
Dirichlet_distribution
Family of stochastic processes
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes
Dirichlet_process
If there are more items than boxes holding them, one box must contain at least two items
commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the
Pigeonhole_principle
and rings) Dirichlet algebra Dirichlet beta function Dirichlet boundary condition (differential equations) Neumann–Dirichlet method Dirichlet characters
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
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 via
B-spline
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
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
Formula in number theory
is the average of the coefficients of the Dirichlet series representation of the Dedekind zeta function. The n-th coefficient of the Dirichlet series
Class_number_formula
Mathematical form
functions) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar fields). Dirichlet forms can be defined on any
Dirichlet_form
{\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
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
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
Second-order partial differential equation
probabilistic solution of the Dirichlet problem: the harmonic extension of a boundary function is obtained by averaging the boundary values against the
Laplace's_equation
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
Topics referred to by the same term
individuals in the dataset Dirichlet process, a stochastic process corresponding to an infinite generalization of the Dirichlet distribution. Dynamic programming
DP
Multiplicative function in number theory
Larger values can be checked in: Wolframalpha the b-file of OEIS The Dirichlet series that generates the Möbius function is the (multiplicative) inverse
Möbius_function
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
divisors of a natural number n {\displaystyle n} , or equivalently the Dirichlet convolution of an arithmetic function f ( n ) {\displaystyle f(n)} with
Divisor_sum_identities
Discrete probability distribution
and multinomial distributions can lead to problems. For example, in a Dirichlet-multinomial distribution, which arises commonly in natural language processing
Categorical_distribution
Function whose domain is the positive integers
has the average order log(n). Given an arithmetic function a(n), let Fa(s), for complex s, be the function defined by the corresponding Dirichlet series
Arithmetic_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
Infinite series summing alternating 1 and -1 terms
limits of the Dirichlet, Fejér, and Poisson kernels, respectively. Multiplying the terms of Grandi's series by 1/nz yields the Dirichlet series η ( z )
Grandi's_series
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
Mathematical conjecture
Ali Erhan Özlük, proved the pair correlation conjecture for some of Dirichlet's L-functions (A. E. Ozluk (1982)). The connection with random unitary
Montgomery's pair correlation conjecture
Montgomery's_pair_correlation_conjecture
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
Boundary-value problem in differential equations
derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. It is named after the prolific 19th-century
Cauchy_boundary_condition
Arithmetic function related to the divisors of an integer
\infty }d(n)=2.} In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality:
Divisor_function
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
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
Averages of repeated trials converge to the expected value
but using conditional convergence and interpreting the integral as a Dirichlet integral, which is an improper Riemann integral, we can say: E ( sin
Law_of_large_numbers
Probability distribution
prime distribution. The generalization to multiple variables is called a Dirichlet distribution. The probability density function (PDF) of the beta distribution
Beta_distribution
American mathematician (born 1947)
Goldfeld has introduced the theory of multiple Dirichlet series, objects that extend the fundamental Dirichlet series in one variable. He has also made contributions
Dorian_M._Goldfeld
Mathematical operation
transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and
Mellin_transform
Family of functions in mathematics
expresses the Fejér kernel F n ( x ) {\displaystyle F_{n}(x)} in terms of the Dirichlet kernel F n ( x ) = 1 n ∑ k = 0 n − 1 D k ( x ) {\displaystyle F_{n}(x)={\frac
Fejér_kernel
part of a ratio. In 1838, Peter Gustav Lejeune Dirichlet proved an approximate formula for the average number of divisors of all the numbers from 1 to
Poussin_proof
Arithmetical function
\sum _{d|n}J_{k}(d)=n^{k}.\,} which may be written in the language of Dirichlet convolutions as J k ( n ) ⋆ 1 = n k {\displaystyle J_{k}(n)\star 1=n^{k}\
Jordan's_totient_function
Theorem in analysis
Consider the first Wirtinger inequality given above. Take L to be 2π. Since Dirichlet's conditions are met, we can write y ( x ) = 1 2 a 0 + ∑ n ≥ 1 ( a n sin
Wirtinger's inequality for functions
Wirtinger's_inequality_for_functions
Statistics and machine learning technique
space of possible ensembles (with model weights drawn randomly from a Dirichlet distribution having uniform parameters). This modification overcomes the
Ensemble_learning
On the distribution of prime numbers in arithmetic progressions
which are equal to a {\displaystyle a} modulo q {\displaystyle q} . Dirichlet's theorem on primes in arithmetic progressions then tells us that π ( x
Elliott–Halberstam_conjecture
Summability method in physics
In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the
Zeta_function_regularization
Mathematical problem in classical harmonic analysis
series converges to the average of the left and right limits (but see Gibbs phenomenon). The Dirichlet–Dini Criterion (see Dirichlet conditions and Dini test)
Convergence_of_Fourier_series
The Dirichlet divisor problem that estimates the average order of the divisor function d(n) and Gauss's circle problem that estimates the average order
Multiplicative_number_theory
Representation of a type of random process
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive
Autoregressive_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
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
Theory of subatomic structure
refers to a certain mathematical condition on the system known as the Dirichlet boundary condition. The study of D-branes in string theory has led to
String_theory
Divergent sum of positive unit fractions
this average to differ from the harmonic numbers by a small constant, and Peter Gustav Lejeune Dirichlet showed more precisely that the average number
Harmonic_series_(mathematics)
Mathematical theorem
publisher (link) Vinogradov, A. I. (1965). "The density hypothesis for Dirichlet L-series". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). 29 (4): 903–934
Bombieri–Vinogradov_theorem
Infinite series that is not convergent
series by the limit above. A series of this type is known as a generalized Dirichlet series; in applications to physics, this is known as the method of heat-kernel
Divergent_series
Used in the summation of divergent series
applications of this kind of result in number theory, in particular in handling Dirichlet series. The development of the field of Tauberian theorems received a
Abelian and Tauberian theorems
Abelian_and_Tauberian_theorems
Math method
major application of large sieves using estimations of mean values of Dirichlet characters. In the late 1960s and early 1970s, many of the key ingredients
Large_sieve
American meteorologist (1872–1956)
a geometric method for dividing land areas, that although known from Dirichlet Tessellation (1850) and the Voronoi Diagram (1908), apparently had never
Alfred_H._Thiessen
Smooth approximation of one-hot arg max
in a differentiable manner. Softplus Multinomial logistic regression Dirichlet distribution – an alternative way to sample categorical distributions
Softmax_function
Theorem in mathematics
is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to the instantaneous
Mean_value_theorem
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
Concept in fluid mechanics
boundaries may also be regions exposed to constant pressure in which case a Dirichlet boundary condition for p {\displaystyle p} is appropriate. Similarly,
Hele-Shaw_flow
System of complete and orthogonal polynomials
is the complete elliptic integral of the first kind. The formulas of Dirichlet-Mehler: P n ( cos θ ) = 2 π ∫ 0 θ cos ( n + 1 2 ) ϕ ( 2 cos ϕ −
Legendre_polynomials
Algorithm for computing greatest common divisors
Lejeune Dirichlet seems to have been the first to describe the Euclidean algorithm as the basis for much of number theory. Lejeune Dirichlet noted that
Euclidean_algorithm
"The Dirichlet: A Comprehensive Model of Buying behaviour" and was read to the Royal Statistical Society. Finally published in 1984 the NBD-Dirichlet model
Andrew_S._C._Ehrenberg
Australian and American mathematician (born 1975)
Szemerédi theorem. In 2010, Green and Tao gave a multilinear extension of Dirichlet's celebrated theorem on arithmetic progressions. Given a k × n matrix A
Terence_Tao
Instantaneous rate of change (mathematics)
of distributions and only require that a function is differentiable "on average". Properties of the derivative have inspired the introduction and study
Derivative
Numerical method for solving physical or engineering problems
with respect to x {\displaystyle x} . P2 is a two-dimensional problem (Dirichlet problem) P2 : { u x x ( x , y ) + u y y ( x , y ) = f ( x , y ) in
Finite_element_method
Concept in fluid dynamics
flow experiments. The form of this boundary condition is an example of a Dirichlet boundary condition. In the majority of fluid flows relevant to fluids
No-slip_condition
distribution is the product of their individual density functions. The Dirichlet distribution, a generalization of the beta distribution. The Ewens's sampling
List of probability distributions
List_of_probability_distributions
Function defined on integers in number theory
Jorma K.; Tossavainen, Timo (2020). "Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative". Mathematical Communications. 25
Arithmetic_derivative
Type of functional equation (mathematics)
dynamical systems analyzes the qualitative aspects of solutions, such as their average behavior over a long time interval. Differential equations came into existence
Differential_equation
Characterization of how many integers are prime
Dirichlet's formulas imply the same conjectured asymptotic equivalence of π(x) and x / log(x) stated above, although it turned out that Dirichlet's approximation
Prime_number_theorem
Difference between logarithm and harmonic series
constants. Values of the derivative of the Riemann zeta function and Dirichlet beta function. In connection to the Laplace and Mellin transform. In the
Euler's_constant
Concept in probability theory
corresponds to 0 successes and 0 failures. The same issues apply to the Dirichlet distribution. β is rate or inverse scale. In parameterization of gamma
Conjugate_prior
Function used in signal processing
The rectangular window (sometimes known as the boxcar or uniform or Dirichlet window or misleadingly as "no window" in some programs) is the simplest
Window_function
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
1966 result in mathematical analysis
L2 function converges to it in L2 norm. After Dirichlet's result, several experts, including Dirichlet, Riemann, Weierstrass and Dedekind, stated their
Carleson's_theorem
Way to write a number as a product of other numbers
is a n {\displaystyle a_{n}} , McMahon and Oppenheim observed that its Dirichlet series generating function f ( s ) {\displaystyle f(s)} has the product
Multiplicative_partition
French mathematician (1875–1941)
historical context, addressing Augustin-Louis Cauchy, Peter Gustav Lejeune Dirichlet, and Bernhard Riemann. Lebesgue presents six conditions which it is desirable
Henri_Lebesgue
and the only prime which is the sum of 4 consecutive primes. 24, all Dirichlet characters mod n are real if and only if n is a divisor of 24. 25, the
List_of_numbers
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
Statistical technique for smoothing categorical data
to the expected value of the posterior distribution, using a symmetric Dirichlet distribution with parameter α as a prior distribution. In the special
Additive_smoothing
Unproved conjecture in mathematics
{\displaystyle L} -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. It is a special
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Vector calculus formulas relating the bulk with the boundary of a region
is chosen to be Green's function that vanishes on the boundary of U (Dirichlet boundary condition), ∮ ∂ U ψ ( y ) ∂ G ( y , η ) ∂ n d S y = { ψ ( η )
Green's_identities
Linear operator acting on modular forms
possesses an Euler product. More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime p
Hecke_operator
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
System where changes of output are not proportional to changes of input
principle. 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
Nonlinear_system
Algorithm used by recommender systems
probabilistic latent semantic analysis, multiple multiplicative factor, latent Dirichlet allocation and Markov decision process-based models. Through this approach
Collaborative_filtering
Chinese-American mathematician (born 1949)
their new "conformal invariant", which is a min-max quantity based on the Dirichlet energy. The main work of their article is devoted to relating their conformal
Shing-Tung_Yau
Counterintuitive mathematical object
1]} onto [ 0 , 1 ] × [ 0 , 1 ] {\displaystyle [0,1]\times [0,1]} . The Dirichlet function, which is the indicator function for rationals, is a bounded
Pathological_(mathematics)
Variable capable of taking on a limited number of possible values
such cases, more advanced techniques must be used. An example is the Dirichlet process, which falls in the realm of nonparametric statistics. In such
Categorical_variable
Probability distribution
then the vector (X1/S, ..., Xn/S), where S = X1 + ... + Xn, follows a Dirichlet distribution with parameters α1, ..., αn. For large α the gamma distribution
Gamma_distribution
System to predict users' preferences
latent semantic analysis (LSA), singular value decomposition (SVD), latent Dirichlet allocation (LDA), etc. Their uses have consistently aimed to provide customers
Recommender_system
Number divisible only by 1 and itself
the prime number theorem. Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions
Prime_number
How many integer lattice points there are in a circle
is no reason not to consider other shapes, for example conics; indeed Dirichlet's divisor problem is the equivalent problem where the circle is replaced
Gauss_circle_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
2D conformal field theories
state, while Dirichlet boundary states are parametrized by a real parameter. The corresponding one-point functions are ⟨ V α ( z ) ⟩ Dirichlet , θ = e α
Massless free scalar bosons in two dimensions
Massless_free_scalar_bosons_in_two_dimensions
Infinite series with alternating signs
leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function. The series' terms (1, −2,
1_−_2_+_3_−_4_+_⋯
Branch of mathematics
speed h) traveled in that segment. Associated with each segment is the average value of the function above it, f(x) = h. The sum of all such rectangles
Calculus
Mathematical formula in harmonic analysis
estimates the size of d(n), the number of positive divisors of an integer n. Dirichlet proved D ( X ) = ∑ n = 1 X d ( n ) − X log X − ( 2 γ − 1 ) X = O ( X
Voronoi_formula
Four basic unsolved problems about prime numbers
6\cdot 10^{3321634}} assuming the Generalized Riemann hypothesis (GRH) for Dirichlet L-functions. Johnston and Starichkova give a version working for all n
Landau's_problems
function: is not continuous at any point of its domain; for example, the Dirichlet function. Locally constant function: a continuous function into a discrete
List_of_types_of_functions
Method of analysis
More robust parameter-learning methods involve placing hierarchical Dirichlet process priors over the HMM transition matrix. Step detection Keogh, Eamonn
Time-series_segmentation
Newspaper in digital format
published. With new methods of Natural Language Processing such as Latent Dirichlet allocation it is possible to gain insights into the core characteristics
Online_newspaper
DIRICHLET AVERAGE
DIRICHLET AVERAGE
Boy/Male
Tamil
Moderate, Average
Boy/Male
Hindu
Moderate, Average
Boy/Male
Muslim/Islamic
Moderate average
Boy/Male
Buddhist, Indian
High Above Average
Boy/Male
Muslim
Moderate, Average
Boy/Male
Arabic, Muslim, Sindhi
Moderate; Average
Surname or Lastname
English
English : nickname from Middle English gode ‘good’ + enoh ‘enough’ (Old English genÅh). Reaney suggests that it was bestowed on one who was easily satisfied; it may also have been used with reference to one whose achievements were average, ‘good enough’ though not outstanding.English : possibly a nickname meaning ‘good lad’ or ‘good servant’, from Middle English gode knave, from Old English gÅd ‘good’ + cnafa ‘boy’, ‘servant’.
DIRICHLET AVERAGE
DIRICHLET AVERAGE
Girl/Female
Irish
Irish form of Jane “God is gracious.â€
Boy/Male
Tamil
Intelligent, Wise, Prudent, Learned
Boy/Male
Indian, Punjabi, Sikh
All Victorious
Girl/Female
Muslim
Helper, Supporter, Protector, Blossoming flower, Brilliant, Visible, Dawn
Girl/Female
Muslim
Peaceful
Boy/Male
Gujarati, Hindu, Indian
Brightness; Son of Sun Like Karna
Boy/Male
Indian
Part of Body
Boy/Male
Indian, Sanskrit
Friendly to the Needy
Boy/Male
American, Australian, British, Christian, English, Scandinavian
Divine Warrior; Born from a Bear
Boy/Male
Hindu, Indian, Traditional
King of Surya Dynasty
DIRICHLET AVERAGE
DIRICHLET AVERAGE
DIRICHLET AVERAGE
DIRICHLET AVERAGE
DIRICHLET AVERAGE
a.
Forming an exception; not ordinary; uncommon; rare; hence, better than the average; superior.
a.
Average; having an intermediate value between two extremes, or between the several successive values of a variable quantity during one cycle of variation; as, mean distance; mean motion; mean solar day.
n.
A chart or graphic representation of the average distribution of rain over the surface of the earth.
n.
A mean proportion, medial sum or quantity, made out of unequal sums or quantities; an arithmetical mean. Thus, if A loses 5 dollars, B 9, and C 16, the sum is 30, and the average 10.
a.
Pertaining to an average or mean; medial; containing a mean proportion; of a mean size, quality, ability, etc.; ordinary; usual; as, an average rate of profit; an average amount of rain; the average Englishman; beings of the average stamp.
n.
The unit of the English system of weights; -- so called because considered equal to the average of grains taken from the middle of the ears of wheat. 7,000 grains constitute the pound avoirdupois, and 5,760 grains the pound troy. A grain is equal to .0648 gram. See Gram.
v. i.
To form, or exist in, a mean or medial sum or quantity; to amount to, or to be, on an average; as, the losses of the owners will average twenty five dollars each; these spars average ten feet in length.
n.
The period of a synodic revolution of the moon, or the time from one new moon to the next; varying in length, at different times, from about 29/ to 29/ days, the average length being 29 d., 12h., 44m., 2.9s.
v. t.
To divide among a number, according to a given proportion; as, to average a loss.
imp. & p. p.
of Average
v. t.
To do, accomplish, get, etc., on an average.
n.
A quantity having an intermediate value between several others, from which it is derived, and of which it expresses the resultant value; usually, unless otherwise specified, it is the simple average, formed by adding the quantities together and dividing by their number, which is called an arithmetical mean. A geometrical mean is the square root of the product of the quantities.
n.
An average.
v. t.
To make equal; to reduce to an average; to make such an allowance or correction in as will reduce to a common standard of comparison; to reduce to mean time or motion; as, to equate payments; to equate lines of railroad for grades or curves; equated distances.
a.
According to the laws of averages; as, the loss must be made good by average contribution.
a.
Of or pertaining to a mean or average; mean; as, medial alligation.
n.
A money of account in Persia, whose value varies greatly at different times and places. Its average value may be reckoned at about two and a half dollars.
n.
A uniform or average height; a normal plane or altitude; a condition conformable to natural law or which will secure a level surface; as, moving fluids seek a level.
superl.
Free from any marked characteristic; average; middling; as, a fair specimen.
n.
A fall or descent of rain; the water, or amount of water, that falls in rain; as, the average annual rainfall of a region.