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If and denotes
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 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 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 and quantum
If a particle and antiparticle are in the appropriate quantum states, then they can annihilate each other and produce other particles.
If a suitably sized quantum computer capable of running Grover's algorithm reliably becomes available, it would reduce a 128-bit key down to 64-bit security, roughly a DES equivalent.
If the quantum state is initially symmetric ( antisymmetric ), it will remain symmetric ( antisymmetric ) as the system evolves.
If S contains two elements that are not pairwise orthogonal ( in particular, the set of all quantum states includes such pairs ) then an argument like that given above shows that the answer is no.
If the error rate is small enough, it is thought to be possible to use quantum error correction, which corrects errors due to decoherence, thereby allowing the total calculation time to be longer than the decoherence time.
If the theory of quantum gravity also achieves a grand unification of the other known interactions, it is referred to as a theory of everything ( TOE ).
If one used Planck's energy quanta, and demanded that electromagnetic radiation at a given frequency could only transfer energy to matter in integer multiples of an energy quantum, then the photoelectric effect could be explained very simply.
If certain quantum inequalities conjectured by Ford and Roman hold, then the energy requirements for some warp drives may be absurdly gigantic as well as negative, for example the energy equivalent of − 10 < sup > 64 </ sup > kg might be required to transport a small spaceship across the Milky Way galaxy.
If the universe is described by an effective local quantum field theory down to the Planck scale, then we would expect a cosmological constant of the order of.
:" If the essential frequency range is limited to B cycles per second, 2B was given by Nyquist as the maximum number of code elements per second that could be unambiguously resolved, assuming the peak interference is less half a quantum step.
If a quantum computer with a sufficient number of qubits were to be constructed, Shor's algorithm could be used to break public-key cryptography schemes such as the widely used RSA scheme.
If true, sonoluminescence may be the first observable example of quantum vacuum radiation.
If quantum electrodynamics were an exact theory, the fine-structure constant would actually diverge at an energy known as the Landau pole.
If more than one quantum mechanical state is at the same energy, the energy levels are " degenerate ".
If quantum mechanics were to be applicable to macroscopic objects, there must be some limit in which quantum mechanics reduces to classical mechanics.
If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be in nuclear coordinates.
If a more general measurement is made that detects the system in a state then the system makes a " jump " or quantum leap from the original state to the final state with probability of.
If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.
If the middle layer is made thin enough, it acts as a quantum well.
If parity transforms a pair of quantum states into each other, then the sum and difference of these two basis states are states of good parity.
If we allow the elements of the system to use quantum computation, the system is called a quantum interactive proof system, and the corresponding complexity class is called QIP.
If we introduce a vector then each quantum state corresponds to a point in ' n-space ' with energy

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