Here, the coefficients of the configurations and the coefficients of the basis functions are optimized together.

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.

( Here primitive is used in the sense that the highest common factor of the set of coefficients of P is 1 ; this is weaker than requiring the coefficients to be pairwise relatively prime.

Here are the unknowns, are the coefficients of the system, and are the constant terms.

Here the user specifies a desired frequency response, a weighting function for errors from this response, and a filter order N. The algorithm then finds the set of N coefficients that minimize the maximum deviation from the ideal.

Here the vectors are elements of a given vector space V over a field K, and the coefficients are scalars in K.

Here is an image showing the Chebyshev filters next to other common kind of filters obtained with the same number of coefficients ( all filters are fifth order ):

Here, as in the earlier example of, the coefficients 1 prevent Eisenstein's criterion from applying directly.

Here Z < sub > l </ sub > denotes the ℓ-adic integers, but the definition is by means of the system of ' constant ' sheaves with the finite coefficients Z / l < sup > k </ sup > Z.

Here the user specifies a desired frequency response, a weighting function for errors from this response, and a filter order N. The algorithm then finds the set of coefficients that minimize the maximum deviation from the ideal.

( Here, homology and cohomology is taken with coefficients in the ring of integers, but the isomorphism holds for any coefficient ring.

Here the ratio of coefficients of y is same as that of the constant terms.

Here it is easy to see that if some linear combination of the x < sub > i </ sub > and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a character χ of the group T other than the trivial character takes the value 1 on P. By Pontryagin duality we have T ′ contained in the kernel of χ, and therefore not equal to T.

Here, the for are coefficients fully determined by the data, being defined as

Here, are the standard spherical harmonics, and are constant coefficients which depend on the function.

Here is an image showing the elliptic filter next to other common kind of filters obtained with the same number of coefficients:

( Here, are symbols of components or states, are coefficients ).

Here χ is not a number as before but a tensor of rank 2, the electric susceptibility tensor.

Here χ is a Dirichlet character and s a complex variable with real part greater than 1.

Here we wrote τ ( χ ) for the Gauss sum

Here is the Legendre symbol, which is a quadratic character mod p. An analogous formula with a general character χ in place of the Legendre symbol defines the Gauss sum G ( χ ).

Here χ is the quadratic residue symbol modulo D, where − D is the discriminant of an imaginary quadratic field.

Here, χ is the character.

Here χ ″( G ) is the total chromatic number ; Δ ( G ), maximum degree ; and δ ( G ), minimum degree.

Here χ < sup >′</ sup >( G ) is the chromatic index of G ; and K < sub > n, n </ sub >, the complete bipartite graph with equal partite sets.

Here n ( G ) is the order of G, α ( G ) is the independence number, ω ( G ) is the clique number, and χ ( G ) is the chromatic number.

Here, C, χ, M and N are

Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.

Therefore, given any positive integer n, it produces a string with Kolmogorov complexity at least as great as n. The program itself has a fixed length U. The input to the program GenerateComplexString is an integer n. Here, the size of n is measured by the number of bits required to represent n, which is log < sub > 2 </ sub >( n ).

Here K denotes the field of real numbers or complex numbers, I is a closed and bounded interval b and p, q are real numbers with 1 < p, q < ∞ so that

Here k is first-order rate constant having dimension 1 / time, ( t ) is concentration at a time t and < sub > 0 </ sub > is the initial concentration.

Here I < sub > 2 </ sub > is reduced to I < sup >–</ sup > and S < sub > 2 </ sub > O < sub > 3 </ sub >< sup > 2 –</ sup > ( thiosulfate anion ) is oxidized to S < sub > 4 </ sub > O < sub > 6 </ sub >< sup > 2 –</ sup >.

The simplest, and original, implementation of the protocol uses the multiplicative group of integers modulo p, where p is prime and g is primitive root mod p. Here is an example of the protocol, with non-secret values in < span style =" color: blue "> blue </ span >, and secret values in < span style =" color: red "> boldface red </ span >:

Here is the center of the ellipse, and is the angle between the < math > X </ Math >- axis and the major axis of the ellipse.

Here C < sub > p </ sub > is the heat capacity at constant pressure and α is the coefficient of ( cubic ) thermal expansion

Here ( Z / 2Z ) is the polynomial ring of Z / 2Z and ( Z / 2Z )/( T < sup > 2 </ sup >+ T + 1 ) are the equivalence classes of these polynomials modulo T < sup > 2 </ sup >+ T + 1.

Here the kets and columns are identified with the vectors of V and the bras and rows with the dual vectors or linear functionals of the dual space V < sup >*</ sup >, with conjugacy associated with duality.

Here, R < sub > ij </ sub > is the Ricci tensor.

# If A is a cartesian product of intervals I < sub > 1 </ sub > × I < sub > 2 </ sub > × ... × I < sub > n </ sub >, then A is Lebesgue measurable and Here, | I | denotes the length of the interval I.

Here the notion of convergence corresponds to the norm on L < sup > 2 </ sup >.

Here e < sub > a </ sup >< sup > μ </ sup > is the vierbein and D < sub > μ </ sub > is the covariant derivative for fermion fields, defined as follows

" Here, take her, sir " ( Sir Joseph, Josephine, Ralph, Cousin Hebe and Chorus )< sup > 1 </ sup >