where C < sub > α </ sub > denotes I < sub > α </ sub > or e < sup > απi </ sup > K < sub > α </ sub >.

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

for α > − 1 / 2 and where δ is the Dirac delta function.

where A < sub > α </ sub > and B < sub > α </ sub > are any two solutions of Bessel's equation, and C < sub > α </ sub > is a constant independent of x ( which depends on α and on the particular Bessel functions considered ).

where A ∈ N ( i. e., A is a single nonterminal ), α, β ∈ ( N U Σ )* ( i. e., α and β are strings of nonterminals and terminals ) and γ ∈ ( N U Σ )< sup >+</ sup > ( i. e., γ is a nonempty string of nonterminals and terminals ).

where it already is established that α = ρ.

where, and are nonterminal symbols, α is a terminal symbol ( a symbol that represents a constant value ), is the start symbol, and ε is the empty string.

where, and are nonterminal symbols, and α is a terminal symbol.

The basic operation used is known as ' alpha blending ', where an opacity value, ' α ' is used to control the proportions of two input pixel values that end up a single output pixel.

* Prediction: For every state in S ( k ) of the form ( X → α • Y β, j ) ( where j is the origin position as above ), add ( Y → • γ, k ) to S ( k ) for every production in the grammar with Y on the left-hand side ( Y → γ ).

where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the α < sub > i </ sub > are positive integers ; 1 is represented by the empty product.

For anisotropic multicomponent diffusion coefficients one needs 4-index quantities, for example,, where i, j are related to the components and α, β = 1, 2, 3 correspond to the space coordinates.

where ω is implicitly a function of k. We assume that the wave packet α is almost monochromatic, so that A ( k ) is nonzero only in the vicinity of a central wavenumber k < sub > 0 </ sub >.

where α is the fine-structure constant and j is a number which is the total angular momentum eigenvalue ; that is, depending on the direction of the electron spin.

He also offers a conversion table ( see Cohen, 1988, p. 283 ) for eta squared ( η < sup > 2 </ sup >) where 0. 0099 constitutes a small effect, 0. 0588 a medium effect and 0. 1379 a large effect.

Reactions of acids are often generalized in the form HA H < sup >+</ sup > + A < sup >−</ sup >, where HA represents the acid and A < sup >−</ sup > is the conjugate base.

The numerical value of K < sub > a </ sub > is equal to the concentration of the products divided by the concentration of the reactants, where the reactant is the acid ( HA ) and the products are the conjugate base and H < sup >+</ sup >.

Because the range of possible values for K < sub > a </ sub > spans many orders of magnitude, a more manageable constant, pK < sub > a </ sub > is more frequently used, where pK < sub > a </ sub >

functions as real combinations of spherical harmonics Y < sub > lm </ sub >( θ, φ ) ( where l and m are quantum numbers ).

Within a shell where n is some integer n < sub > 0 </ sub >, ranges across all ( integer ) values satisfying the relation.

It lacks the NH < sub > 2 </ sub > group because of the cyclization of the side-chain and is known as an imino acid ; it falls under the category of special structured amino acids .</ ref > where R is an organic substituent known as a " side-chain "); often the term " amino acid " is used to refer specifically to these.

½ ( pKa < sub > 1 </ sub > + pKa < sub > R </ sub >), where pKa < sub > R </ sub > is the side-chain pKa.

## < tt > SubBytes </ tt >— a non-linear substitution step where each byte is replaced with another according to a lookup table.

## < tt > ShiftRows </ tt >— a transposition step where each row of the state is shifted cyclically a certain number of steps.

The < tt > MixColumns </ tt > function takes four bytes as input and outputs four bytes, where each input byte affects all four output bytes.

Detailed analysis shows that the complexity class is unchanged by allowing error as high as 1 / 2 − n < sup >− c </ sup > on the one hand, or requiring error as small as 2 < sup >− n < sup > c </ sup ></ sup > on the other hand, where c is any positive constant, and n is the length of input.

where ν > − 1 / 2 and z is a complex number.

In some applications it is useful to be able to compute the Bernoulli numbers B < sub > 0 </ sub > through B < sub > p − 3 </ sub > modulo p, where p is a prime ; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime.

Finally, weight of the chain is represented by ( 0, − λgs ) where λ is the mass per unit length, g is the acceleration of gravity and s is the length of chain between c and r.

The equation of a circle is ( x − a )< sup > 2 </ sup > + ( y − b )< sup > 2 </ sup > = r < sup > 2 </ sup > where a and b are the coordinates of the center ( a, b ) and r is the radius.

Innova Discs rate stability as " turn " on a scale of + 1 to − 4, where + 1 is the most overstable and − 4 is the most understable.

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

As seen from locations in the Earth's Northern Hemisphere, celestial objects with declinations greater than 90 ° − ( where = observer's latitude ) appear to circle daily around the celestial pole without dipping below the horizon, and are therefore called circumpolar stars.

This similarly occurs in the Southern Hemisphere for objects with declinations less ( i. e. more negative ) than-90 ° − ( where is always a negative number for southern latitudes ).

Generally, if a star whose declination is is circumpolar for some observer ( where is either positive or negative ), then a star whose declination is − never rises above the horizon, as seen by the same observer.

− 26 "/ cy < sup > 2 </ sup >, where n is the mean sidereal angular motion of the Moon.

Conversely, if the measured potential is negative, the electrode is more oxidizing than the SHE ( such as Zn in ZnSO < sub > 4 </ sub > where the standard electrode potential is − 0. 76 V ).

where r < sub > k </ sub > < r < sub > k − 1 </ sub >.

It follows from Euler's polyhedron formula, V − E + F = 2 ( where V, E, F are the numbers of vertices, edges, and faces ), that there are exactly 12 pentagons in a fullerene and V / 2 − 10 hexagons.

There is in fact no complex number z for which, and hence the reciprocal gamma function is an entire function, with zeros at z = 0, − 1, − 2 ,.... We see that the gamma function has a local minimum at where it attains the value.

; Laplacian matrix or " Kirchhoff matrix " or " Admittance matrix ": This is defined as D − A, where D is the diagonal degree matrix.

radius ) a and flattening f. The quantity f = ( a − b )/ a, where b is the semi-minor axis ( polar radius ), is a purely geometrical one.

Then one need only check the records in each bucket T against those in buckets T where k ranges between − m and m.

For instance, suppose that each input is an integer z in the range 0 to N − 1, and the output must be an integer h in the range 0 to n − 1, where N is much larger than n. Then the hash function could be h