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 denotes the conjugate transpose of ( i. e., the adjoint of ), then

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 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 the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

If X is a real Banach space, then the polarization identity is

If the applied voltage is large enough, the liquid crystal molecules in the center of the layer are almost completely untwisted and the polarization of the incident light is not rotated as it passes through the liquid crystal layer.

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ < sup >( 3 )</ sup >, this produces a nonlinear polarization field:

If the waves have the correct frequency ( or wavelength ) and polarization, their energy can be transferred to the charged particles in the plasma, which in turn collide with other plasma particles, thus increasing the temperature of the bulk plasma.

If this wave interacts with a material at which the horizontal component ( green sinusoid ) travels at a different speed than the vertical component ( blue sinusoid ), the two components will fall out of the 90 degree phase difference ( required for circular polarization ) changing the Kerr Ellipticity.

If this depends on the polarization states of the waves, then their polarization may also be altered.

If the polarization of the input light is not aligned with the stress direction in the fiber, the output will vary between linear and circular polarization ( and generally will be elliptically polarized ).

If the axis of polarization of the incident wave is chosen so that it makes a 45 ° with the fast and slow axes of the waveplate, then E < sub > f </ sub > = E < sub > s </ sub > ≡ E, and the resulting wave upon exiting the waveplate is

If a capacitor is driven with a time-varying voltage that changes rapidly enough, then the polarization of the dielectric cannot follow the signal.

If the solar wind has energetic bursts, contracting and expanding the magnetosphere, or if the solar wind takes a southward polarization, geomagnetic storms can be expected.

Prior to any measurements being made, the polarizations of the photons are indeterminate ; If a measurement is made on one of the photons using a two-channel polarizer aligned with the axes of the coordinate system, the photon assumes either horizontal or vertical polarization, with 50 % probability for each orientation, and the other two photons immediately assume the identical polarization.

If the Faraday medium is of such thickness as to cause a 45 degree rotation on the way in, the back-reflected beam will have polarization perpendicular to the incident beam, allowing it to be cleanly blocked by the light source's polarizer.

If polarization and phase are known in the waveform, and are accurate enough, then these values might be added to a more complex voxel that holds such measurements separately.

If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics.

If the initial polarization filter is inserted between the lamp and the image generation element, the light intensity striking the image element is not any higher than normal without the polarizing filter, and overall image contrast transmitted to the screen is not affected.

If we consider a polarization of a shear ( transverse ) elastic wave propagating along an acoustic axis in a solid, a polarization rotation effect due to rotation of the body as a whole ( so-called, " polarization inertia effect ") can be observed too ( it was noted by Ukrainian scientist Sergii A. Sarapuloff in early 80th, as well as a corresponding modification of Green-Christoffel's tensors in Acoustics ).

) If the photons share the same polarization it is deemed Type I correlation ; if they have perpendicular polarisations it is deemed Type II.

If a jammer and receiver do not have the same polarization, the jamming signal will incur a loss that reduces its effectiveness.