The coefficients in the upper figure are computed by the Fourier series integral.

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 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 amplitude coefficients are usually represented by lower case r and t.

Although, for some activities such as dancing or bowling, sliding is desirable, so shoes designed for these activities often have lower coefficients of friction.

Because the coefficients of digital filters are definite, they can be used to achieve much more complex and selective designs – specifically with digital filters, one can achieve a lower passband ripple, faster transition, and higher stopband attenuation than is practical with analog filters.

If a significant amount of the light passed through the sample contains wavelengths that have much lower extinction coefficients than the nominal one, the instrument will report an incorrectly low absorbance.

Modern manufacturing techniques have produced devices with voltages lower than 5. 6 V with negligible temperature coefficients, but as higher voltage devices are encountered, the temperature coefficient rises dramatically.

This effect is particularly pronounced for charged species, which have even lower permeability coefficients than neutral polar molecules.

Since this tends to be a rather large area compared to the projected frontal area, the resulting drag coefficients tend to be low: much lower than for a car with the same drag and frontal area, and at the same speed.

Thus in the common context of polynomials with rational coefficients, a polynomial is irreducible if it cannot be expressed as the product of two or more such polynomials, each of them having a lower degree than the original one.

Although lower complexity than CABAC, CAVLC is more elaborate and more efficient than the methods typically used to code coefficients in other prior designs.

Suppose all the 3 differential dummy coefficients are negative, it means that, female non-white and non-Hispanic workers earn much lower mean wages than female OR non-white and non-Hispanic workers as compared to the base group, which in this case is male, white or Hispanic ( omitted category ).

Both equations are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial, named Cayley resolvent, has a rational root in z, where

Strong bonds result in higher melting temperatures and lower thermal expansion coefficients.

These unit-less activity coefficients are close to 1. 00 for very dilute water solutions, but are usually lower for more concentrated solutions.

Therefore, insulators tend to have very high Seebeck coefficients, while metals have lower values due to their high carrier concentrations.

If the azeotrope can be " jumped over ", distillation can continue, although because the activity coefficients have crossed, the water will boil out of the remaining ethanol, rather than the ethanol out of the water as at lower concentrations.

Rejection, Form Appropriatness, Shading, Pathognomic Verbalization, Barrier, and Penetration had lower stability coefficients while Space, Sex, Abstract and Balance proved to be extreamly infrequent in the children's samples.

The third step is that if the remaining torsion coefficients are not constant in the fibres of the principal bundle PM, it is often possible ( although sometimes difficult ), to normalize them by setting them equal to a convenient constant value and solving these normalization equations, thereby reducing the effective dimension of the Lie group G. If this occurs, one goes back to step one, now having a Lie group of one lower dimension to work with.

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: