Here variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds.

The coefficients in the upper and lower figures are respectively computed by the Fourier series integral and the discrete Fourier transform | DFT summation.

The matrix expressing with respect to is almost upper triangular, in the sense that the coefficients are the only nonzero subdiagonal coefficients.

A holomorphic function in the upper half plane which is invariant under linear fractional transformations with integer coefficients and determinant 1 is called a modular form.

When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations.

where is as above and the mean of the non-redundant correlation coefficients ( i. e., the mean of an upper triangular, or lower triangular, correlation matrix ).

The subgroup that maps the upper half-plane, H, onto itself is PSL ( 2, R ), the transforms with real coefficients, and these act transitively and isometrically on the upper half-plane, making it a homogeneous space.

where z is in the upper half-plane and the normalization constant m < sup > k − 1 </ sup > assures that the image of a form with integer Fourier coefficients has integer Fourier coefficients.

The DFT computes the coefficients in the lower figure.

The Smith chart can be used to represent many parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis.

This figure is much smaller than the modern value: but it is close to the difference between the modern coefficients of the equation of the center ( 1st term ) and that of the evection: the difference is accounted for by the fact that the ancient measurements were taken at times of eclipses, and the effect of the evection ( which subtracts under those conditions from the equation of the center ) was at that time unknown and overlooked.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

* The quadratic surds ( irrational roots of a quadratic polynomial with integer coefficients,, and ) are algebraic numbers.

Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

The coefficients are displayed in their hexadecimal equivalent of the binary representation of bit polynomials from GF ( 2 ).

For a two-dimensional array, the element with indices i, j would have address B + c · i + d · j, where the coefficients c and d are the row and column address increments, respectively.

Often the coefficients are chosen so that the elements occupy a contiguous area of memory.

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

These Zernike coefficients are linearly independent, thus individual aberration contributions to an overall wavefront may be isolated and quantified separately.

where are the binomial coefficients.

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.

The properties of binomial coefficients have led to extending the meaning of the symbol beyond the basic case where n and k are nonnegative integers with ; such expressions are then still called binomial coefficients.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him.

The coefficients that appear in the binomial expansion are called binomial coefficients.

is a multiple of d. x and y are called Bézout coefficients for ( a, b ); they are not unique.

A pair of Bézout coefficients ( in fact the ones that are minimal in absolute value ) can be computed by the extended Euclidean algorithm.

The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:

In the following chemical equation with arrows pointing both ways to indicate equilibrium, A and B are reactant chemical species, S and T are product species, and α, β, σ, and τ are the stoichiometric coefficients of the respective reactants and products:

The results of the study, based on collision integrals computed from the latest critically evaluated data on intermolecular forces in air, will be reported in the form of a table of viscosity, thermal conductivity, thermal diffusion, and diffusion coefficients at temperatures of 1,000 to 10,000 Af and of logarithm of pressure in atmospheres from Af to Af times normal density.

Sedimentation coefficients were computed as Af values and relative amounts of the various components were calculated from the Schlieren patterns.

After one pair of Bézout coefficients ( x, y ) has been computed ( using extended Euclid or some other algorithm ), all pairs may be found using the formula

Binomial coefficients can be computed explicitly in various ways.

* Distance-weighted estimator – the measure uses weighting coefficients for x < sub > i </ sub > that are computed as the inverse mean distance between x < sub > i </ sub > and the other data points.

where f is a known power series with f ( 0 ) = 0, then the coefficients of g can be explicitly computed using the Lagrange inversion formula.

* Linear algebra: matrix operations Multiply or add matrices of a type and with coefficients you choose and see how the result was computed.

The probability that 20 flips of a fair coin would result in 14 or more heads can be computed from binomial coefficients as

The coefficients b < sub > α, β </ sub > at a given point in the parametric ( u < sup > 1 </ sup >, u < sup > 2 </ sup >)- plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed with the aid of the dot product as follows:

More significantly, the value of the coefficients of each term can be computed directly from the exponents.

Now the total spin and its projection onto the previously defined axis can be computed using the rules for adding angular momentum in quantum mechanics using the Clebsch – Gordan coefficients.

where the difference or detail signals are computed from the detail coefficients as

The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer.

Provided algebraic equations can be solved algorithmically in the base field K, then the coefficients of the Puiseux series solutions can be computed to any given order.