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Mathematical operation on arithmetical functions
In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory
Dirichlet_convolution
Integral expressing the amount of overlap of one function as it is shifted over another
matrix Convolution for optical broad-beam responses in scattering media Convolution power Convolution quotient Deconvolution Dirichlet convolution List
Convolution
Function equal to the product of its values on coprime factors
function, so called because it is the multiplicative identity for Dirichlet convolution. Sometimes written as u ( n ) {\displaystyle u(n)} ; not to be confused
Multiplicative_function
Mathematical tool for summing arithmetic functions
first step is to find a pair of functions g and h such that, using Dirichlet convolution, we have f = g ∗ h; the sum then becomes F ( n ) = ∑ k = 1 n ∑ x
Dirichlet_hyperbola_method
Function whose domain is the positive integers
is called the Dirichlet convolution of a and b, and is denoted by a ∗ b {\displaystyle a*b} . A particularly important case is convolution with the constant
Arithmetic_function
Mathematical series
obtained by applying Möbius inversion and Dirichlet convolution to known series. For example, given a Dirichlet character χ(n) one has 1 L ( χ , s ) = ∑
Dirichlet_series
Arithmetic function related to the divisors of an integer
(s-a-b)}{\zeta (2s-a-b)}},} which is a special case of the Rankin–Selberg convolution. A Lambert series involving the divisor function is: ∑ n = 1 ∞ q n σ
Divisor_function
Arithmetic function
f(p)a f(q)b ... While the Dirichlet convolution of two multiplicative functions is multiplicative, the Dirichlet convolution of two completely multiplicative
Completely multiplicative function
Completely_multiplicative_function
Concept in mathematical analysis
{\displaystyle 2\pi } . The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of D n ( x ) {\displaystyle D_{n}(x)}
Dirichlet_kernel
Topics referred to by the same term
mathematics, convolution is a binary operation on functions. Circular convolution Convolution theorem Titchmarsh convolution theorem Dirichlet convolution Infimal
Convolution_(disambiguation)
Probability distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ( α ) {\displaystyle \operatorname
Dirichlet_distribution
Multiplicative function in number theory
(iterated) Dirichlet convolution μ k = μ ∗ ⋯ ∗ μ {\displaystyle \mu _{k}=\mu *\cdots *\mu } to be the k {\displaystyle k} -fold Dirichlet convolution of the
Möbius_function
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
Mathematical operation
an inverse f − 1 ( n ) {\displaystyle f^{-1}(n)} with respect to Dirichlet convolution such that ( f ∗ f − 1 ) ( n ) = δ n , 1 {\displaystyle (f\ast f^{-1})(n)=\delta
Dirichlet_series_inversion
divisors of a natural number n {\displaystyle n} , or equivalently the Dirichlet convolution of an arithmetic function f ( n ) {\displaystyle f(n)} with one:
Divisor_sum_identities
Relation between pairs of arithmetic functions
of Dirichlet convolutions, the first formula may be written as g = 1 ∗ f {\displaystyle g={\mathit {1}}*f} where ∗ denotes the Dirichlet convolution, and
Möbius_inversion_formula
Square (0,1) matrix
j = 1; otherwise, aij = 0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving
Redheffer_matrix
Arithmetical function
also a consequence of the fact that we can write the function as a Dirichlet convolution of ψ = I d ∗ | μ | {\displaystyle \psi =\mathrm {Id} *|\mu |} .
Dedekind_psi_function
On the distribution of prime numbers in arithmetic progressions
and Iwaniec generalized the Elliott-Halberstam conjecture, using Dirichlet convolution of arithmetic functions related to the von Mangoldt function. The
Elliott–Halberstam_conjecture
Number of prime factors of a natural number
number and ε {\displaystyle \varepsilon } is the identity for the Dirichlet convolution, ε ( n ) = ⌊ 1 n ⌋ {\displaystyle \varepsilon (n)=\lfloor {\frac
Prime_omega_function
Associative algebra used in combinatorics
ordered by divisibility The convolution associated to the incidence algebra for intervals [1, n] becomes the Dirichlet convolution, hence the Möbius function
Incidence_algebra
Formal power series
Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require
Generating_function
Mathematical term
}b_{m}q^{m}} where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1: b m = ( a ∗ 1 ) ( m )
Lambert_series
Counting technique in combinatorics
Dirichlet hyperbola method re-expresses a sum of a multiplicative function f ( n ) {\displaystyle f(n)} by selecting a suitable Dirichlet convolution
Inclusion–exclusion_principle
Partial differential equation describing the evolution of temperature in a region
condition u(x, 0) = g(x) for −∞ < x < ∞ and 0 < t < ∞ by applying a convolution: u ( x , t ) = ∫ Φ ( x − y , t ) g ( y ) d y . {\displaystyle u(x,t)=\int
Heat_equation
called the unit function because it is the identity element for Dirichlet convolution. It may be described as the "indicator function of 1" within the
Unit_function
Generalized function whose value is zero everywhere except at zero
series of a function f of period 2π is defined by convolution (on the interval [−π,π]) with the Dirichlet kernel: D N ( x ) = ∑ n = − N N e i n x = sin
Dirac_delta_function
are two polynomial arithmetic functions, one defines f * g, the Dirichlet convolution of f and g, by ( f ∗ g ) ( m ) = ∑ d ∣ m f ( m ) g ( m d ) = ∑ a
Average order of an arithmetic function
Average_order_of_an_arithmetic_function
formula Mod n cryptanalysis Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor
List_of_number_theory_topics
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
Periodic distribution ("function") of "point-mass" Dirac delta sampling
{\displaystyle f(t)} by convolution with Ш T {\displaystyle {\text{Ш}}_{T}} . The Dirac comb identity is a particular case of the Convolution Theorem for tempered
Dirac_comb
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
Arithmetical function
_{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}\,} and
Jordan's_totient_function
Shape made from cubes joined together
Jean-Marc Champarnaud et al, Université de Rouen, France PDF "Dirichlet convolution and enumeration of pyramid polycubes", C. Carré, N. Debroux, M.
Polycube
distribution, a convolution of a normal distribution with an exponential distribution, and the Gaussian minus exponential distribution, a convolution of a normal
List of probability distributions
List_of_probability_distributions
Function in discrete mathematics
partial differential equations, and to perform other operations such as convolutions or multiplying large integers. Since the DFT deals with a finite amount
Discrete_Fourier_transform
{\displaystyle g} , let h = f ∗ g {\displaystyle h=f*g} be their Dirichlet convolution. Then for every prime p {\displaystyle p} , one has: h p ( x ) =
Bell_series
the 2 × 2 complex Hermitian matrices. Redheffer matrix Encodes a Dirichlet convolution. Matrix entries are given by the divisor function; entries of the
List_of_named_matrices
Counts the number of necklaces of n colored beads picked from α available colors
well-defined. The polynomials for M and N are easily related in terms of Dirichlet convolution of arithmetic functions f ( n ) ∗ g ( n ) {\displaystyle f(n)*g(n)}
Necklace_polynomial
transform Discrete Hartley transform List of transforms Dirichlet kernel Fejér kernel Convolution theorem Least-squares spectral analysis List of cycles
List of Fourier analysis topics
List_of_Fourier_analysis_topics
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
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
x_{t}<x_{i}}}(f\star \mu )(d).} Here ⋆ {\displaystyle \star } is the Dirichlet convolution and μ {\displaystyle \mu } is the Möbius function. Further, if f
GCD_matrix
function Trigonometric function Trigonometric polynomial Exponential sum Dirichlet kernel Fejér kernel Gibbs phenomenon Parseval's identity Parseval's theorem
List of harmonic analysis topics
List_of_harmonic_analysis_topics
Formula for the sum of an arithmetic function
_{n=1}^{\infty }{\frac {a(n)}{n^{s}}}} be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for ℜ ( s ) > σ {\displaystyle
Perron's_formula
Function that is discontinuous at rationals and continuous at irrationals
function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused with the integer ruler
Thomae's_function
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
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
Function with a repeating pattern
are periodic but possess properties that make them less intuitive. The Dirichlet function, for example, is periodic, with any nonzero rational number serving
Periodic_function
American mathematician
solution to Landau's problem on the Dirichlet eta function), An introduction to transform theory, and The convolution transform (co-author with I. I. Hirschman)
David_Widder
Discrete Fourier transform algorithm
Multidimensional discrete convolution Multidimensional transform Odlyzko–Schönhage algorithm applies the FFT to finite Dirichlet series Schönhage–Strassen
Fast_Fourier_transform
{\displaystyle \chi ^{2}} . Shimura's proof uses the Rankin-Selberg convolution of f ( z ) {\displaystyle f(z)} with the theta series θ ψ ( z ) = ∑ n
Shimura_correspondence
Construction providing a total order on a free monoid
{\displaystyle \mu } is the classic Möbius function. The sum is a Dirichlet convolution. Some basic definitions are useful. Given a tree t = [ x , y ] {\displaystyle
Hall_word
Green's function for Laplacian
In its general nature, it is a singular integral operator, defined by convolution with a function having a mathematical singularity at the origin, the
Newtonian_potential
and g {\displaystyle g} can be written in an elegant way using a Dirichlet convolution: f = 1 ∗ g {\displaystyle f=\mathbf {1} *g} , i.e. f ( n , k ) =
Root_of_unity_modulo_n
Family of functions
(k_{n})} be a summability kernel, and ∗ {\displaystyle *} denote the convolution operation. If ( k n ) , f ∈ C ( T ) {\displaystyle (k_{n}),f\in {\mathcal
Summability_kernel
Mathematical concept
kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as
Poisson_kernel
Type of differential equation
source P ( D ) u = δ {\displaystyle P(D)u=\delta } ), then taking the convolution with the boundary conditions to get the solution. This is analogous in
Partial_differential_equation
Mathematical problem in classical harmonic analysis
S_{N}(f)=f*D_{N}} where ∗ stands for the periodic convolution and D N {\displaystyle D_{N}} is the Dirichlet kernel, which has an explicit formula, D n ( t
Convergence_of_Fourier_series
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
Type of probability distribution
variables is still subgaussian, the convolution of subgaussian distributions is still subgaussian. In particular, any convolution of the normal distribution with
Sub-Gaussian_distribution
Mathematical concept
representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They
Automorphic_L-function
Method of solution to differential equations
solution of the inhomogeneous problem L y = f {\displaystyle Ly=f} is the convolution, y = ( G ∗ f ) . {\displaystyle y=(G\ast f).} By the superposition principle
Green's_function
Function used in signal processing
polynomial function of degree k − 1 that is obtained by k-fold self-convolution of the rectangular function. Triangular windows are given by w [ n ]
Window_function
Mathematical function
may be seen as a particular case of the identity for the integral of a convolution. Taking f ( u ) := e − u u z 1 − 1 1 R + g ( u ) := e − u u z 2 − 1 1
Beta_function
Image classification model
different themes. Probabilistic latent semantic analysis (pLSA) and latent Dirichlet allocation (LDA) are two popular topic models from text domains to tackle
Bag-of-words model in computer vision
Bag-of-words_model_in_computer_vision
many properties with the usual Laplace transform. For instance, the convolution theorem holds: { L ∗ ( g ∗ h ) } ( s ) = { L ∗ g } ( s ) { L ∗ h } (
Laplace–Stieltjes_transform
Type of mathematical function
is log-concave (see Prékopa–Leindler inequality). This implies that convolution preserves log-concavity, since h(x,y) = f(x-y) g(y) is log-concave if
Logarithmically concave function
Logarithmically_concave_function
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
Robert Alexander Rankin and Atle Selberg independently constructed their convolution L-functions, now thought of as the Langlands L-function associated to
Rankin–Selberg_method
American computer scientist
Itemset Mining, PLDA for Latent Dirichlet Allocation, PSC for Spectral Clustering, and SPeeDO for Parallel Convolutional Neural Networks. Through his research
Edward_Y._Chang
Generating pseudo-random numbers that follow a probability distribution
decreasing density functions as well as symmetric unimodal distributions Convolution random number generator, not a sampling method in itself: it describes
Non-uniform random variate generation
Non-uniform_random_variate_generation
Integral transform useful in probability theory, physics, and engineering
integral equations with algebraic polynomial equations, and by replacing convolution with multiplication. For example, through the Laplace transform, the
Laplace_transform
Mapping involving integration between function spaces
integration kernels are then biperiodic functions; convolution by functions on the circle yields circular convolution. If one uses functions on the cyclic group
Integral_transform
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
were first used by Hilbert (1909) in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later Weyl
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Automated recognition of patterns and regularities in data
empirical observations – using e.g., the Beta- (conjugate prior) and Dirichlet-distributions. The Bayesian approach facilitates a seamless intermixing
Pattern_recognition
Singularities of holomorphic functions extend infinitely outward
identify the property of solutions of systems of partial differential or convolution equations satisfying Hartogs-type theorems. The original proof was given
Hartogs's_extension_theorem
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
American mathematician
generalizations of Vandermonde's convolution, Amer. Math. Monthly, 63(1956), 84–91. Final analysis of Vandermonde's convolution, Amer. Math. Monthly, 64(1957)
Henry_W._Gould
summations Cesàro mean Abel's summation formula Convolution Cauchy product –is the discrete convolution of two sequences Farey sequence – the sequence
List_of_real_analysis_topics
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
Numerical technique
equation (ADE) approach (equivalently, i/ω appears as an integral or convolution in time domain). Perfectly matched layers, in their original form, only
Perfectly_matched_layer
Fundamental principle of physics
Fourier. Additive state decomposition Beat (acoustics) Coherence (physics) Convolution Green's function Impulse response Interference Quantum superposition
Superposition_principle
Inputs for which a function's value is non-zero
smooth functions approximating nonsmooth (generalized) functions, via convolution. In good cases, functions with compact support are dense in the space
Support_(mathematics)
approximations) Dirichlet's approximation theorem (Diophantine approximations) Dirichlet's theorem on arithmetic progressions (number theory) Dirichlet's unit theorem
List_of_theorems
Unsolved problem in mathematics
L-function, nowadays called the Ramanujan L-function. It can be defined as a Dirichlet series for Ramanujan tau function: L ( s , τ ) = ∑ n = 1 ∞ τ ( n ) n s
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Concept in mathematical analysis
volume, the ball has the smallest first eigenvalue for the Laplacian with Dirichlet boundary conditions. The proof goes by restating the problem as a minimization
Pólya–Szegő_inequality
Machine learning methods using multiple input modalities
Shibo; Zhang, Zhengdong; Wu, Yonghui; Pang, Ruoming (2020). "Conformer: Convolution-augmented Transformer for Speech Recognition". arXiv:2005.08100 [eess
Multimodal_learning
Pair of polynomial sequences
These identities can be proven using generating functions and discrete convolution. From their definition by recurrence it follows that the Chebyshev polynomials
Chebyshev_polynomials
Transform in mathematics
derived from the DCT by replacing the Neumann condition at x=0 with a Dirichlet condition. Both the DCT and the DST were described by Nasir Ahmed, T.
Discrete_sine_transform
Type of operator in Fourier analysis
operators and convolution operators; every multiplier T can also be expressed in the form Tf = f∗K for some distribution K, known as the convolution kernel of
Multiplier_(Fourier_analysis)
Statistical model used in machine learning
Jacobian ∏ c s c H W {\displaystyle \prod _{c}s_{c}^{HW}} . invertible 1x1 convolution z c i j = ∑ c ′ K c c ′ y c i j {\displaystyle z_{cij}=\sum _{c'}K_{cc'}y_{cij}}
Flow-based_generative_model
American mathematician
papers together, Hirschman and Widder published a book entitled The Convolution Transform. Hirschman spent most of his career (1949–1978) at Washington
Isidore_Isaac_Hirschman_Jr.
Classification of Artificial Neural Networks (ANNs)
general public via chatbots, code generators and many other forms. Convolutional neural networks (CNN): a FNN that uses kernels and regularization to
Types of artificial neural networks
Types_of_artificial_neural_networks
Oscillatory error in Fourier series
value of the original function at the discontinuity) as a consequence of Dirichlet's theorem. The quantity ∫ 0 π sin t t d t = ( 1.851937051982 … ) =
Gibbs_phenomenon
Certain vector fields are the sum of an irrotational and a solenoidal vector field
kernel K ( r , r ′ ) {\displaystyle K(\mathbf {r} ,\mathbf {r} ')} in the convolution integrals has to be replaced by K ′ ( r , r ′ ) = K ( r , r ′ ) − K (
Helmholtz_decomposition
Infinite series with alternating signs
central idea is that 1 − 2 + 3 − 4 + ... is the Cauchy product (discrete convolution) of 1 − 1 + 1 − 1 + ... with 1 − 1 + 1 − 1 + .... The Cauchy product
1_−_2_+_3_−_4_+_⋯
Overview of and topical guide to machine learning
neural network Feedforward neural network Extreme learning machine Convolutional neural network Recurrent neural network Long short-term memory (LSTM)
Outline_of_machine_learning
\left({\frac {1}{2}}x\right)}}.} The convolution of any integrable function of period 2 π {\displaystyle 2\pi } with the Dirichlet kernel coincides with the function's
List of trigonometric identities
List_of_trigonometric_identities
Direct relationship Directional statistics Dirichlet distribution Dirichlet-multinomial distribution Dirichlet process Disattenuation Discrepancy function
List_of_statistics_articles
DIRICHLET CONVOLUTION
DIRICHLET CONVOLUTION
DIRICHLET CONVOLUTION
DIRICHLET CONVOLUTION
Girl/Female
Hindu
Feather, Line, Saintly
Boy/Male
English
noble.
Male
Arthurian
, a giant.
Boy/Male
Hindu, Indian
Glorious
Girl/Female
Arabic, Muslim
Princess
Boy/Male
African
one who is born in the wintertime.
Girl/Female
Muslim/Islamic
Desiring desirous
Boy/Male
Indian, Telugu
Lord Siva
Girl/Female
Tamil
Prayuta | பà¯à®°à®¯à¯à®¤à®¾
Mingled with
Boy/Male
Irish
Valour; high honor.
DIRICHLET CONVOLUTION
DIRICHLET CONVOLUTION
DIRICHLET CONVOLUTION
DIRICHLET CONVOLUTION
DIRICHLET CONVOLUTION
a.
Pertaining to a gyrus, or convolution.
v. t.
To turn completely, or with repeated turns; especially, to turn about something fixed; to cause to form convolutions about anything; to coil; to twine; to twist; to wreathe; as, to wind thread on a spool or into a ball.
n.
A sinuous depression or sulcus like those separating the convolutions of the brain.
a.
Consisting of many folds, coils, or convolutions.
a.
Below the margin; submarginal; as, an inframarginal convolution of the brain.
v. t.
To unite by winding one thread, strand, or other flexible substance, round another; to form by convolution, or winding separate things round each other; as, to twist yarn or thread.
n. pl.
A general name for all those placental mammals that have a brain with few or no cerebral convolutions, as Rodentia, Insectivora, etc.
n.
The state of being intervolved or coiled up; a convolution; as, the intervolutions of a snake.
n.
The state of being rolled upon itself, or rolled or doubled together; a tortuous or sinuous winding or fold, as of something rolled or folded upon itself.
n.
The act of twisting; a contortion; a flexure; a convolution; a bending.
a.
Having convolutions.
n.
Anything of a rounded or swelling form resembling a roll; a turn; a convolution; a coil.
n.
A twist; a convolution.
n.
An oblong vermiform mass on the dorsal side of the testicle, composed of numerous convolutions of the excretory duct of that organ.
n.
A twist; a convolution.
n.
A band of gray matter bordering the fimbria in the brain; the dentate convolution.
n.
A nonplane curve whose tangents are all equally inclined to a given plane. The common helix is the curve formed by the thread of the ordinary screw. It is distinguished from the spiral, all the convolutions of which are in the plane.
n.
The act of rolling anything upon itself, or one thing upon another; a winding motion.
n.
An irregular, tortuous folding of an organ or part; as, the convolutions of the intestines; the cerebral convolutions. See Brain.
n.
A convoluted ridge between grooves; a convolution; as, the gyri of the brain; the gyri of brain coral. See Brain.