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 x < sub > 0 </ sub > is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum ( HWHM ).

Note that if the operations are performed in a colour space where γ is not equal to 1 then the operation will lead to non-linear effects which can potentially be seen as aliasing artifacts ( or ' jaggies ') along sharp edges in the matte.

* 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 shifts in origin have been ignored, the relative velocity is assumed to be in the-direction and the Lorentz factor γ is defined by:

where γ is the Lorentz factor

The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ '( 0 ), then the corresponding derivation is D ( ƒ ) = ( ƒ ∘ γ )'( 0 ) ( where the derivative is taken in the ordinary sense, since ƒ ∘ γ is a function from (- 1, 1 ) to R ).

where α, β, and γ are coordinates on the unit sphere S, and ω is the area element on S. This result has the interpretation that u ( t, x ) is t times the mean value of φ on a sphere of radius ct centered at x:

where d denotes the transition dipole moment, γ the natural linewidth of the transition, ν the frequency, N the number of atoms, and A the cross-section of the beam.

where c-speed of light in vacuum, v-speed of light in medium, γ-wavelength in vacuum, γ °- wavelength in medium

where γ ≈ 0. 57721 56649 01532 ... is the Euler – Mascheroni gamma constant.

where p, q ∈ C and γ ∈ R are constants and i is the imaginary unit.

where is the solubility constant for the solute particles with the molar surface area A, is the solubility constant for substance with molar surface area tending to zero ( i. e., when the particles are large ), γ is the surface tension of the solute particle in the solvent, A < sub > m </ sub > is the molar surface area of the solute ( in m < sup > 2 </ sup >/ mol ), R is the universal gas constant, and T is the absolute temperature.

In some cases, it is possible to define abstractions using Galois connections ( α, γ ) where α is from L to L ′ and γ is from L ′ to L. This supposes the existence of best abstractions, which is not necessarily the case.

where the numerical value was found by taking T < sub > i </ sub >= T < sub > e </ sub > and γ < sub > i </ sub >= γ < sub > e </ sub >.

a < sub > X </ sub > = γ < sub > X </ sub > c < sub > X </ sub >, where γ < sub > X </ sub > is the activity coefficient of species X.

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 >

These nodes are areas where the action potential is amplified using a high density of sodium ( Na < big >+</ big >) ions and is subsequently passed along the axon.

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.

where n < sup > c </ sup > denotes the charge conjugate state, i. e., the antiparticle.

where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E ( k ), and a < sub > k </ sub > denotes the corresponding annihilation operators.

where E < sub > 0 </ sub > is an infinite negative constant.

The Battle of Qarqar is mentioned in extra-biblical records, and was perhaps at Apamea where Shalmaneser III of Assyria fought a great confederation of princes from Cilicia, Northern Syria, Israel, Ammon, and the tribes of the Syrian desert ( 853 BC ), including Ahab ( A-ha-ab-bu < sup > mat </ sup >) ( Adad -' idri ).

where < big ></ big > is the number of degrees of freedom divided by two, R is the universal gas constant and n is the number of moles in the system ( a constant ).

We could, alternatively, choose an encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring < M >.

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

auxiliary regression is TR < sup > 2 </ sup >, where T is the sample size and R < sup > 2 </ sup > is the coefficient of determination.

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

where n > 1 is an integer and p, q, and r are prime numbers, then 2 < sup > n </ sup >× p × q and 2 < sup > n </ sup >× r are a pair of amicable numbers.

where θ ∈ T < sub > f ( x )</ sub >< sup >*</ sup > N and X < sub > x </ sub > ∈ T < sub > x </ sub > M.

: where M is any Hermitian positive-definite matrix, and x < sup >*</ sup > the conjugate transpose of x.

rk ( A < sup >*</ sup >), where A < sup >*</ sup > is the conjugate transpose or hermitian transpose of A.

A subbase for the weak topology is the collection of sets of the form φ < sup >- 1 </ sup >( U ) where φ ∈ X < sup >*</ sup > and U is an open subset of the base field K. In other words, a subset of X is open in the weak topology if and only if it can be written as a union of ( possibly infinitely many ) sets, each of which is an intersection of finitely many sets of the form φ < sup >- 1 </ sup >( U ).

More generally, an complex matrix M is said to be positive definite if z < sup >*</ sup > Mz is real and positive for all non-zero complex vectors z ; where z < sup >*</ sup > denotes the conjugate transpose of z.

where x < sub >*</ sub > can be any point in interval j.

Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be several local minima, where a local minimum x < sup >*</ sup > is defined as a point for which there exists some δ > 0 so that for all x such that

This can be seen by looking at the diagonal entries of A < sup >*</ sup > A and AA < sup >*</ sup >, where A is a normal, triangular matrix.

where M < sup >*</ sup > denotes the conjugate transpose of M. Other authors retain the definition () for complex matrices and call matrices satisfying () conjugate symplectic.

Thus MR pulse sequences sensitive to T < sub > 2 </ sub >< sup >*</ sup > show more MR signal where blood is highly oxygenated and less where it is not.

The hard part to deal with is more or less a group H < sup > 1 </ sup >( P < sup > 1 </ sup >, j < sub >*</ sub > E ) = H ( U, E ), where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibers the spaces E < sub > x </ sub > of vanishing cycles.

If we specifically choose the Euclidean norm on both R < sup > n </ sup > and R < sup > m </ sup >, then we obtain the matrix norm which to a given matrix A assigns the square root of the largest eigenvalue of the matrix A < sup >*</ sup > A ( where A < sup >*</ sup > denotes the conjugate transpose of A ).

where z and b are complex numbers, z < sup >*</ sup > and b < sup >*</ sup > are the complex conjugates of z and b, respectively, and c is a real number.

where M < sub >*</ sub >< sup > n-1 </ sup > is the ( n-1 )- dimensional Minkowski content, L < sup > n </ sup > is the n-dimensional Lebesgue measure, and ω < sub > n </ sub > is the volume of the unit ball in R < sup > n </ sup >.

where GL ( V, F ) is the general linear group of invertible linear transformations of V over F and F < sup >*</ sup > here is the normal subgroup consisting of multiplications of vectors in V by nonzero elements of F ( that is, scalar multiples of the identity ; scalar transformations ).

where ƒ < sup >*</ sup >( x ) ƒ ( x ) = ƒ ( x ) ƒ < sup >*</ sup >( x ) = x.

This can be seen by looking at the diagonal entries of A < sup >*</ sup > A and AA < sup >*</ sup >, where A is a normal, triangular matrix.