If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.

If D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.

If Af denotes the net profit from stage R and Af, then the principle of optimality gives Af.

If denotes the quantum state of a particle ( n ) with momentum p, spin J whose component in the z-direction is σ, then one has

* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

Frege, however, did not conceive of objects as forming parts of senses: If a proper name denotes a non-existent object, it does not have a reference, hence concepts with no objects have no truth value in arguments.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

# 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.

If denotes the state of the system at any one time t, the following Schrödinger equation holds:

* If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

* If the base field is C, then for all complex numbers λ, where denotes the complex conjugation of λ.

If the sender has nothing more to send, the line simply remains in the marking state ( as if a continuing series of stop bits ) until a later space denotes the start of the next character.

If the position was found to be r < sub > 0 </ sub > then in an interpretation satisfying CFD, the statistical population describing position and momentum would contain all pairs ( r < sub > 0 </ sub >, p ) for every possible momentum value p, whereas an interpretation that rejects counterfactual values completely would only have the pair ( r < sub > 0 </ sub >,⊥) where ⊥ denotes an undefined value.

If an origin is chosen, and denotes its image, then this means that for any vector:

If denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between and is − θ.

If the string is stretched between two points where x = 0 and x = L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0 < x < L and t is unlimited.

If the heuristic h satisfies the additional condition for every edge x, y of the graph ( where d denotes the length of that edge ), then h is called monotone, or consistent.

If A is n-by-n, B is m-by-m and denotes the k-by-k identity matrix then the Kronecker sum is defined by:

If Sym < sub > n </ sub > denotes the space of symmetric matrices and Skew < sub > n </ sub > the space of skew-symmetric matrices then since and

If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables X < sub > i </ sub > / σ ( X < sub > i </ sub >) for i = 1, ..., n. This applies to both the matrix of population correlations ( in which case " σ " is the population standard deviation ), and to the matrix of sample correlations ( in which case " σ " denotes the sample standard deviation ).

If denotes the total energy of a system, one may write

If Skew < sub > n </ sub > denotes the space of skew-symmetric matrices and Sym < sub > n </ sub > denotes the space of symmetric matrices and then since and

The conjecture is stated in terms of three positive integers, a, b and c ( whence comes the name ), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d cannot be much smaller than c.

If A is expressed as an N × N matrix, then A < sup >†</ sup > is its conjugate transpose.

If < math > c ^ 2 < 4km </ math > there are two complex conjugate roots a ± ib, and the solution ( with the above boundary conditions ) will look like this:

If is a complex-valued random variable, with values in, then its variance is, where is the conjugate transpose of.

If the respective impedances of the branches of the hybrid that are connected to the conjugate sides of the hybrid are known, hybrid balance may be computed by the formula for return loss.

is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U. If f is antiholomorphic ( that is, the conjugate to a holomorphic function ), it still preserves angles, but it reverses their orientation.

If z is replaced by the negative reciprocal of its complex conjugate, then the functions g < sub > 1 </ sub >, g < sub > 2 </ sub >, and g < sub > 3 </ sub > of z are left unchanged.

For a particle which has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is everywhere constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right.

If pH is above the pKa value, the concentration of the conjugate base is greater than the concentration of the acid, and the color associated with the conjugate base dominates.

If the conjugate transpose of a matrix is denoted by, then the Hermitian property can be written concisely as

As above, let f and g denote measurable real-or complex-valued functions defined on S. If || fg ||< sub > 1 </ sup > is finite, then the products of f with g and its complex conjugate function, respectively, are μ-integrable, the estimates

The complex conjugate of is written If the scalar field is taken to be real-valued, then

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 ).

If c < sub > 1 </ sub > ∈ C is another pre-image of x in C then the subgroups p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c )) and p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c < sub > 1 </ sub >)) are conjugate in π < sub > 1 </ sub >( X, x ) by p-image of a curve in C connecting c to c < sub > 1 </ sub >.

If either L or H are independent of a generalized coordinate q, meaning the L and H so not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved ( this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry ).

If a conjugate of HS in Co < sub > 0 </ sub > fixes a particular point of type 3, this point is found in 276 triangles of type 2-2-3, which this copy of HS permutes in orbits of 176 and 100.

* If the Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are constants of the motion, i. e. are conserved, this immediately follows from Lagrange's equations:

If is a creation operator, its hermitian conjugate ( destruction or annihilation operator ) acts on the vacuum as follows:

If one thinks of operators on a Hilbert space as " generalized complex numbers ", then the adjoint of an operator plays the role of the complex conjugate of a complex number.

If unrecognized, the condition leads to liver failure -- but not kernicterus, as the liver is still able to conjugate bilirubin, and conjugated bilirubin is unable to cross the blood – brain barrier.

If we are to show that X = AB ∩ DE, Y = BC ∩ EF, Z = CD ∩ FA are collinear for conconical ABCDEF, then notice that ADY and CYF are similar, and that X and Z will correspond to the isogonal conjugate if we overlap the similar triangles.

If the underlying field has an involution, one can instead ask φ to be conjugate-linear, as in conjugate transpose, below.

If the likelihood and its prior take on simple parametric forms ( such as 1-or 2-dimensional likelihood functions with simple conjugate priors ), then the empirical Bayes problem is only to estimate the marginal and the hyperparameters using the complete set of empirical measurements.

If the likelihood function belongs to the exponential family, then a conjugate prior exists, often also in the exponential family ; see Exponential family: Conjugate distributions.