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Of a Kronecker product (combinatorics)
In mathematics, Kronecker coefficients gλ μν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations
Kronecker_coefficient
Area of mathematics
possible to recover the Kronecker coefficients as linear combinations of reduced Kronecker coefficients. Reduced Kronecker coefficients are implemented in
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Mathematical operation on matrices
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a
Kronecker_product
Mathematical rule
_{p+q-i+1}=a+b,\quad {i=1,\dots ,q}.} For example, . The reduced Kronecker coefficient of the symmetric group C ¯ λ , μ , ν {\displaystyle {\bar {C}}_{\lambda
Littlewood–Richardson_rule
Coefficients in angular momentum eigenstates of quantum systems
Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular
Clebsch–Gordan_coefficients
Theorem about Diophantine approximations
In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by Leopold Kronecker (1884). Kronecker's approximation theorem
Kronecker's_theorem
Number of solutions of linear systems in terms of matrix ranks
and coefficient matrix. The theorem is variously known as the: Rouché–Capelli theorem in English speaking countries, Italy and Brazil; Kronecker–Capelli
Rouché–Capelli_theorem
Kronecker substitution is a technique named after Leopold Kronecker for determining the coefficients of an unknown polynomial by evaluating it at a single
Kronecker_substitution
Problem about mathematical number fields
ganzzahligen Abel’schen Gleichungen durch die Kreisteilungsgleichungen. — Kronecker in a letter to Dedekind in 1880 reproduced in volume V of his collected
Hilbert's_twelfth_problem
Coefficient used in numerical approximation
difference can be central, forward or backward. This table contains the coefficients of the central differences, for several orders of accuracy and with uniform
Finite_difference_coefficient
Decomposition of periodic functions
functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied
Fourier_series
Every finite abelian extension of Q is contained within some cyclotomic field
n Z ) × {\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }} . The Kronecker–Weber theorem provides a partial converse: every finite abelian extension
Kronecker–Weber_theorem
or more of the sets Give a combinatorial interpretation of the Kronecker coefficients The size m ( n ) {\displaystyle m(n)} of the smallest collection
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Measure of spatial correlation
similar to rank correlation coefficients like Spearman's rank correlation coefficient and the Kendall rank correlation coefficient but also explicitly considers
Tjøstheim's_coefficient
Array of numbers describing a metric connection
be the "flat-space" metric tensor. For Riemannian manifolds, it is the Kronecker delta η a b = δ a b {\displaystyle \eta _{ab}=\delta _{ab}} . For pseudo-Riemannian
Christoffel_symbols
Computational method
Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension
Factorization_of_polynomials
Mathematician (1845–1918)
encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while
Georg_Cantor
Mathematical form
while others have very different names (outer product, tensor product, Kronecker product) and yet convey essentially the same idea. A brief overview of
Product_(mathematics)
Known channel properties of a communication link
^{-1}{\mbox{vec}}(\mathbf {Y} )} where ⊗ {\displaystyle \otimes } denotes the Kronecker product and the identity matrix I {\displaystyle \scriptstyle \mathbf
Channel_state_information
Vectors whose components are all 0 except one that is 1
I}=((\delta _{ij})_{j\in I})_{i\in I}} where I is any set and δij is the Kronecker delta, equal to zero whenever i ≠ j and equal to 1 if i = j. This family
Standard_basis
Fourth letter in the Greek alphabet
δ)-definition of limits, in mathematics and more specifically in calculus. The Kronecker delta in mathematics. The central difference for a function. The degree
Delta_(letter)
Determinant of a product of rectangular matrices
g(k)})_{j\in [n],k\in [m]}{\bigr )}} where " δ {\displaystyle \delta } " is the Kronecker delta, and the Cauchy−Binet formula to prove has been rewritten as ∑ f
Cauchy–Binet_formula
Coefficients coupled with angular momentum
symbols, also called 3-jm symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta. While the two approaches address
3-j_symbol
Type of filter in signal processing
[citation needed] The impulse response (that is, the output in response to a Kronecker delta input) of an Nth-order discrete-time FIR filter lasts exactly N
Finite_impulse_response
Commutative group where every element is the sum of elements from one finite subset
finite case was proven by Gauss in 1801, the finite case was proven by Kronecker in 1870, and stated in group-theoretic terms by Frobenius and Stickelberger
Finitely generated abelian group
Finitely_generated_abelian_group
Sufficient condition for polynomial irreducibility
criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not
Eisenstein's_criterion
Symbols for constants, special functions
this coefficient quite often, we will give it a special name, the Lorentz factor, and stick to our symbol γ(V),... Weisstein, Eric W. "Kronecker Delta"
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Generalized function whose value is zero everywhere except at zero
called the delta function because it is a continuous analogue of the Kronecker delta function. The mathematical rigor of the delta function was disputed
Dirac_delta_function
First article on transfinite set theory
facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekind's contributions to the article
Cantor's first set theory article
Cantor's_first_set_theory_article
Tensor index notation for tensor-based calculations
_{2}^{2}+\delta _{3}^{3}=4.} The Kronecker delta is one of the family of generalized Kronecker deltas. The generalized Kronecker delta of degree 2p may be defined
Ricci_calculus
Theory of a class of elliptic curves
\lambda } in K {\displaystyle K} . Conversely, Kronecker conjectured – in what became known as the Kronecker Jugendtraum – that every abelian extension of
Complex_multiplication
Material property in strain-stress relationship
In continuum mechanics, Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) are two material-dependent quantities denoted
Lamé_parameters
German mathematician (1810–1893)
of high school, where he inspired the mathematical career of Leopold Kronecker. Kummer was born in Sorau, Brandenburg (then part of Prussia). His father
Ernst_Kummer
Concept in machine learning
data tensors can be expressed in terms of matrix multiplication and the Kronecker product. The computation of gradients, a crucial aspect of backpropagation
Tensor_(machine_learning)
Statistical method
between lasso coefficient estimates and so-called soft thresholding. It also reveals that (like standard linear regression) the coefficient estimates do
Lasso_(statistics)
Mathematical technique used in data compression and analysis
_{km},\end{aligned}}} where δ j l {\displaystyle \delta _{jl}\,} is the Kronecker delta. Completeness is satisfied if every function f ∈ L 2 ( R ) {\displaystyle
Wavelet_transform
Function in discrete mathematics
understood more precisely as converting between sample values and the coefficients of a trigonometric polynomial that interpolates those values. It is therefore
Discrete_Fourier_transform
Theorem used in quantum mechanics for angular momentum calculations
independent of angular momentum orientation, and the other a Clebsch–Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart, who
Wigner–Eckart_theorem
Mathematical model of a system in control engineering
systems theory into an algebraic framework, making it possible to use Kronecker structures for efficient analysis. State-space models are applied in fields
State-space_representation
Boolean polynomials as sums of monomials
polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because
Algebraic_normal_form
Number with an integer power equal to 1
cyclotomic field – this is the content of a theorem of Kronecker, usually called the Kronecker–Weber theorem on the grounds that Weber completed the proof
Root_of_unity
Formula in number theory
( d m ) {\displaystyle \chi =\left(\!{\frac {d}{m}}\!\right)} be the Kronecker symbol. Then χ {\displaystyle \chi } is a Dirichlet character. Write L
Class_number_formula
Irish American mathematician
294–318. doi:10.1090/S0002-9904-1929-04734-7. Acoustoelastic effect Kronecker coefficient Murnaghan–Nakayama rule Murnaghan–Tait equation of state Lewis,
Francis Dominic Murnaghan (mathematician)
Francis_Dominic_Murnaghan_(mathematician)
Mathematical operation on vector spaces
describing the tensor product f ⊗ g {\displaystyle f\otimes g} is the Kronecker product of the two matrices. For example, if V, X, W, and U above are
Tensor_product
Type of pseudorandom binary sequence
of length 3. ... etc. ... The circular autocorrelation of an MLS is a Kronecker delta function (with DC offset and time delay, depending on implementation)
Maximum_length_sequence
Random motion of particles suspended in a fluid
respect to P (and its own natural filtration), where δij denotes the Kronecker delta. The spectral content of a stochastic process X t {\displaystyle
Brownian_motion
Rational number sequence
where m = 0 , 1 , 2... {\displaystyle m=0,1,2...} and δ denotes the Kronecker delta. The first of these is sometimes written as the formula (for m >
Bernoulli_number
Polynomial function of degree 5
Kronecker per la Risoluzione delle Equazioni di Quinto Grado". Atti Dell'i. R. Istituto Lombardo di Scienze, Lettere ed Arti. I: 275–282. Kronecker,
Quintic_function
solution of a polynomial equation with integer (or equivalently rational) coefficients. Polynomials: Can be generated solely by addition, multiplication, and
List of mathematical functions
List_of_mathematical_functions
Infinite series of Bessel functions
n})]^{2},} (where: δ m n {\displaystyle \delta _{mn}} is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the
Fourier–Bessel_series
Set of vectors used to define coordinates
in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates
Basis_(linear_algebra)
Regression for more than two discrete outcomes
y_{i}}={\begin{cases}1,{\text{ for }}j=y_{i}\\0,{\text{ otherwise}}\end{cases}}} is the Kronecker delta. The negative log-likelihood function is therefore the well-known
Multinomial logistic regression
Multinomial_logistic_regression
Algebraic object with geometric applications
\mathbf {e} _{i}} , where δ j k {\displaystyle \delta _{j}^{k}} is the Kronecker delta, which functions similarly to the identity matrix, and has the effect
Tensor
Computational model of cells and tissues
cell type of cell σ, J is the coefficient determining the adhesion between two cells of types τ(σ),τ(σ'), δ is the Kronecker delta, v(σ) is the volume of
Cellular_Potts_model
Shorthand notation for tensor operations
^{i}(\mathbf {e} _{j})=\delta _{j}^{i}.} where δ {\displaystyle \delta } is the Kronecker delta. As Hom ( V , W ) = V ∗ ⊗ W {\displaystyle \operatorname {Hom}
Einstein_notation
Concept in statistical mathematics
where IR is the R-dimensional identity matrix and ⊗ denotes the matrix Kronecker product. The SUR model is usually estimated using the feasible generalized
Seemingly unrelated regressions
Seemingly_unrelated_regressions
Irreducible representation of the rotation group SO
{\displaystyle \sum _{j}\chi ^{j}(\beta )(2j+1)=\delta (\beta ).} The set of Kronecker product matrices D j ( α , β , γ ) ⊗ D j ′ ( α , β , γ ) {\displaystyle
Wigner_D-matrix
Algebra associated to any vector space
avoids this confusion. This axiomatization of areas is due to Leopold Kronecker and Karl Weierstrass; see Bourbaki (1989b, Historical Note). For a modern
Exterior_algebra
System of complete and orthogonal polynomials
_{-1}^{1}P_{m}(x)P_{n}(x)\,dx={\frac {2}{2n+1}}\delta _{mn},} where δmn denotes the Kronecker delta. That the polynomials are complete means the following. Given any
Legendre_polynomials
Sum of elements on the main diagonal
is not true in general for more than three factors. The trace of the Kronecker product of two matrices is the product of their traces: tr ( A ⊗ B )
Trace_(linear_algebra)
German mathematician (1849–1917)
semester before returning to Berlin, where he attended lectures by Leopold Kronecker, Ernst Kummer and Karl Weierstrass. He received his doctorate (awarded
Ferdinand_Georg_Frobenius
Mathematical method
_{a}^{b}\phi _{i}^{*}(x)\phi _{j}(x)\,dx=\delta _{ij},} where δij is the Kronecker delta. Substituting function fn into these equations then leads to the
Least-squares function approximation
Least-squares_function_approximation
Sum type in number theory
linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains
Quadratic_Gauss_sum
Array of numbers
310–312, doi:10.2307/2691101, JSTOR 2691101 Kronecker, Leopold (1897), Hensel, Kurt (ed.), Leopold Kronecker's Werke, Teubner Miller, G. A. (May 1930), "On
Matrix_(mathematics)
Polynomial sequence
}}\,n!\,\delta _{nm},} where δ n m {\displaystyle \delta _{nm}} is the Kronecker delta. The probabilist polynomials are thus orthogonal with respect to
Hermite_polynomials
Stress-strain relation in a linear elastic material
_{i}^{l}\delta _{j}^{k}\right)} where δ n m {\displaystyle \delta _{n}^{m}} is the Kronecker delta. Unless otherwise noted, this article assumes C {\displaystyle \mathbf
Elasticity_tensor
0 if p divides a. The same notation is used for the Jacobi symbol and Kronecker symbol, which are generalizations where p is respectively any odd positive
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Approach to general relativity
a {\displaystyle \delta _{b}^{a}} is the Kronecker delta. A vielbein is usually specified by its coefficients e μ a {\displaystyle e^{\mu }{}_{a}} with
Tetrad_formalism
Theorem in probability theory
{\displaystyle \{1,\ldots ,2k\}} , δ i , j {\displaystyle \delta _{i,j}} is the Kronecker delta. Since there are | P 2 k 2 | = ( 2 k − 1 ) ! ! {\displaystyle |P_{2k}^{2}|=(2k-1)
Isserlis's_theorem
Matrices important in quantum mechanics and the study of spin
_{k}\ ,\end{aligned}}} where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta, which equals + 1 {\displaystyle +1} if i = j {\displaystyle i=j}
Pauli_matrices
Artificial neural network node function
previous layer or layers: ^ Here, δ i j {\displaystyle \delta _{ij}} is the Kronecker delta. ^ For instance, j {\displaystyle j} could be iterating through
Activation_function
Algebraic integer which represents an ideal in a ring of integers
Kummer's ideas to the general case was accomplished independently by Kronecker and Dedekind during the next forty years. A direct generalization encountered
Ideal_number
Matrix equation in control theory
solution X exactly when the spectra of A and −B are disjoint. Using the Kronecker product notation and the vectorization operator vec {\displaystyle \operatorname
Sylvester_equation
Concept in algebraic geometry
and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects
Quantum_cohomology
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
in lattice reduction and in the Hermite normal form of matrices. The Kronecker product of two unimodular matrices is also unimodular. This follows since
Unimodular_matrix
Mathematical sequences in combinatorics
between them. A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise
Stirling_number
Tensor operator generalizes the notion of operators which are scalars and vectors
computing or looking up Clebsch–Gordan coefficients. The selection rule m′ = q + m in the Clebsch–Gordan coefficient means that many of the integrals vanish
Tensor_operator
Integral expressing the amount of overlap of one function as it is shifted over another
\bullet } is face-splitting product, ⊗ {\displaystyle \otimes } denotes Kronecker product, ∘ {\displaystyle \circ } denotes Hadamard product (this result
Convolution
Type of signal in signal processing
\sigma ^{2}} is the variance and δ ( n ) {\displaystyle \delta (n)} is the Kronecker delta. The variables W ( n ) {\displaystyle W(n)} are only required to
White_noise
Measure of polynomial height
He pointed out that the classical Kronecker's theorem, which characterizes monic polynomials with integer coefficients all of whose roots are inside the
Mahler_measure
Special coordinate system in differential geometry
manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric
Normal_coordinates
Pair of polynomial sequences
Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1, 1] is bounded by 1. They are
Chebyshev_polynomials
Function that "converges" to periodicity
vector that is not proportional to a vector of integers). A theorem of Kronecker from diophantine approximation can be used to show that any particular
Almost_periodic_function
Analog of the continuous Laplace operator
very special properties, e.g., they are Kronecker sums of one-dimensional discrete Laplacians, see Kronecker sum of discrete Laplacians, in which case
Discrete_Laplace_operator
Multiplicative function in number theory
(n)\Omega (n)}\lambda (n),} where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta, λ ( n ) {\displaystyle \lambda (n)} is the Liouville function,
Möbius_function
Device for suppressing part of a discretely-sampled signal
{\displaystyle h_{k}} , is a measurement of how a filter will respond to the Kronecker delta function. For example, given a difference equation, one would set
Digital_filter
Multiplication algorithm
practical applications such as Lenstra elliptic curve factorization via Kronecker substitution, which reduces polynomial multiplication to integer multiplication
Schönhage–Strassen_algorithm
Branch of algebraic geometry
equations with rational coefficients exist if non-zero rational solutions exist. In the 1850s, Leopold Kronecker formulated the Kronecker–Weber theorem, introduced
Arithmetic_geometry
\ ]} is the commutator and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta. These operators change the eigenvalues of the number operator,
Jordan_map
Real root of the polynomial x^5+x+a
{\displaystyle a=d_{0}(-d_{1})^{-5/4}} . This form is required by the Hermite–Kronecker–Brioschi method, Glasser's method, and the Cockle–Harley method of differential
Bring_radical
Tensor that describes the 4D geometry of spacetime
contravariant index of a tensor with one of a covariant metric tensor coefficient has the effect of lowering the index g μ ν A ν = A μ {\displaystyle g_{\mu
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Mathematical series
\beta }r^{2}\right)v_{\alpha \beta }(\mathbf {R} ),} where δαβ is the Kronecker delta and r2 ≡ |r|2. Removing the trace is common, because it takes the
Multipole_expansion
Representation of a type of random process
the time-varying autoregressive (TVAR) model, where the autoregressive coefficients are allowed to change over time to model evolving or non-stationary processes
Autoregressive_model
Operation in graph theory
_{n_{1}}\otimes \mathbf {A} _{2}} , where ⊗ {\displaystyle \otimes } denotes the Kronecker product of matrices and I n {\displaystyle \mathbf {I} _{n}} denotes the
Cartesian_product_of_graphs
Mathematical operation on vectors in 3D space
values 1 to 3. In a positively-oriented orthonormal basis ηmi = δmi (the Kronecker delta) and E i j k = ε i j k {\displaystyle E_{ijk}=\varepsilon _{ijk}}
Cross_product
Operation that pairs a left and a right R-module into an abelian group
_{\mathbb {Z} }G} is the homology group of C with coefficients in G (see also: universal coefficient theorem). The tensor product of sheaves of modules
Tensor_product_of_modules
Algorithm to multiply two numbers
algorithms can also be used to multiply polynomials by means of the method of Kronecker substitution. If a positional numeral system is used, a natural way of
Multiplication_algorithm
Algorithm for computing polynomial coefficients
{\displaystyle (x_{0},y_{0}),\ldots ,(x_{n},y_{n})} , the method calculates the coefficients of the interpolation polynomial of these points in the Newton form. It
Divided_differences
Mathematical function of a linear operator
_{ij}={\begin{cases}1&i=j\\0&i\neq j\end{cases}},} where δij is the Kronecker delta and can be thought of as the elements of the identity matrix. Functions
Eigenfunction
Mathematical connection between field theory and group theory
Galois' theory until well after the turn of the century. In Germany, Kronecker's writings focused more on Abel's result. Dedekind wrote little about Galois'
Galois_theory
KRONECKER COEFFICIENT
KRONECKER COEFFICIENT
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from any of the various places in Normandy, France, called Crèvecoeur (‘heartbreak’), from Old French creve(r) ‘to break or destroy’, ‘to die’ + ceur ‘heart’, a reference to the infertility and unproductiveness of the land.English : occupational name for a potter, Middle English crockere, an agent derivative of Middle English crock ‘pot’ (Old English croc(ca)).Americanized spelling of German Krocker.
KRONECKER COEFFICIENT
KRONECKER COEFFICIENT
Boy/Male
Arabic, Farsi, French, Iranian, Malaysian, Muslim, Parsi, Pashtun
Prophet Name; Zachary
Boy/Male
Indian
Favor
Boy/Male
Hindu
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sindhi, Telugu
Immortal; Lord Shiva
Boy/Male
Muslim
Silence, Peace, Calm
Girl/Female
Indian
Pampered girl
Female
Hebrew
(× ×„×’Ö·×”) Unisex form of Hebrew Nogahh, NOGA means "shining splendor," as of the fire or the sun.Â
Boy/Male
Hindu, Indian
Winner
Boy/Male
Indian, Sikh
Devoted to God
Boy/Male
French American
Destiny; fate.
KRONECKER COEFFICIENT
KRONECKER COEFFICIENT
KRONECKER COEFFICIENT
KRONECKER COEFFICIENT
KRONECKER COEFFICIENT
n.
One of the variables of a quantic as distinguished from a coefficient.
n.
An invariable quantity; specifically, a function of the coefficients of one or more forms, which remains unaltered, when these undergo suitable linear transformations.
n.
A number, commonly used in computation as a factor, expressing the amount of some change or effect under certain fixed conditions as to temperature, length, volume, etc.; as, the coefficient of expansion; the coefficient of friction.
n.
That which unites in action with something else to produce the same effect.
n.
A number or letter put before a letter or quantity, known or unknown, to show how many times the latter is to be taken; as, 6x; bx; here 6 and b are coefficients of x.
n.
The operation of finding the primitive function which has a given function for its differential coefficient. See Integral.
a.
Cooperating; acting together to produce an effect.
n.
A numerical coefficient in any particular case of the binomial theorem.
n.
A function involving the coefficients and the variables of a quantic, and such that when the quantic is lineally transformed the same function of the new variables and coefficients shall be equal to the old function multiplied by a factor. An invariant is a like function involving only the coefficients of the quantic.
n.
A quantity or coefficient, or constant, which expresses the measure of some specified force, property, or quality, as of elasticity, strength, efficiency, etc.; a parameter.
v. t.
To obtain the differential, or differential coefficient, of; as, to differentiate an algebraic expression, or an equation.
n.
In the theory of gravitation, or of other forces acting in space, a function of the rectangular coordinates which determine the position of a point, such that its differential coefficients with respect to the coordinates are equal to the components of the force at the point considered; -- also called potential function, or force function. It is called also Newtonian potential when the force is directed to a fixed center and is inversely as the square of the distance from the center.
n.
A curve whose contact with a given curve, at a given point, is of a higher order (or involves the equality of a greater number of successive differential coefficients of the ordinates of the curves taken at that point) than that of any other curve of the same kind.