*:* If is unitary then

If is defined as the unitary DFT of the vector then

If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new coordinate system, then the above is just the statement that the dot product of two vectors is preserved under a unitary DFT transformation.

* If the message indicates, Bob would send his qubit through the unitary gate given by the Pauli matrix

If in addition the matrices are all unitary, as are the Dirac set, then S itself is unitary ;

If D is Hermitian the effect is dichroism, while a unitary matrix models birefringence.

If time reversal were implemented as a unitary operator, it would reverse the sign of the energy just as space-reversal reverses the sign of the momentum.

# If the ship was-in fact-a unitary, physical phenomenon which existed separately from the parts, then even when the parts of the ship were organized in a valid spatiotemporal arrangement to be called " Ship of Theseus " it would be possible that the ship could exist in a different spatiotemporal location and that none of the parts would have anything to do with the ship.

If the gauge symmetry is a unitary group or special unitary group then this characteristic class is the second Chern class, which vanishes in the case of the gauge group U ( 1 ).

* If A is skew-Hermitian, then e < sup > A </ sup > is unitary.

If the answer was " yes ", this would have been implemented by further legislation ; and existing two-tier local government areas ( shire counties ) would have become unitary authorities.

If is unitary, then, so.

If is normal, then is unitary, and.

If the coefficient is exactly one, the good is said to be unitary elastic.

If a sufficient number of electors in an area of a proposed new parish ( ranging from 50 % in an area with less than 500 electors to 10 % in one with more than 2, 500 ) sign a petition demanding its creation, then the local district council or unitary authority must consider the proposal.

If V is a unitary representation and the quaternionic structure j is a unitary operator, then V admits an invariant complex symplectic form ω, and hence is a symplectic representation.

If an error is modeled by a unitary transform U, which will act on a qubit, then can be described in the form

If the PED is between zero and 1 demand is said to be inelastic, if PED equals 1, the demand is unitary elastic and if the PED is greater than 1 demand is elastic.

If S is a semigroup or monoid, then a set X on which S acts as above ( on the left, say ) is also known as a ( left ) S-act, S-set, S-action, S-operand, or left act over S. Some authors do not distinguish between semigroup and monoid actions, by regarding the identity axiom ( e • x = x ) as empty when there is no identity element, or by using the term unitary S-act for an S-act with an identity.

If A < sub > 1 </ sub >, ..., A < sub > n </ sub > are n operators each localized in a bounded region and U ( a ) represents the unitary operator actively translating the Hilbert space by the vector a, then if we pick some subset of the n operators to translate,

If experiment shows this 3 × 3 matrix to be not unitary, a sterile neutrino or some other new physics is required.

If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.

If there is an additional magnitude of value or a loss of value after transformation compared with before then the relation between values ( proportional to labor ) and prices ( proportional to total capital advanced ) is incomplete.

If the Sex Pistols ' breakup the previous year had marked the end of the original UK punk scene and its promise of cultural transformation, for many the death of Vicious signified that it had been doomed from the start.

If this is the case, then the transformation of whole lineages should be rare, as the fossil record indicates.

If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory.

If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index.

If every object X < sub > i </ sub > of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φ < sub > i </ sub > then define a natural transformation from 1 < sub > C </ sub > ( the identity functor on C ) to UV.

If an origin is also chosen, this can be decomposed as an affine transformation that sends, namely

If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation.

If the categories C and D have all limits of type J then lim is a functor and the morphisms τ < sub > F </ sub > form the components of a natural transformation

If this object is a class of transformations ( such as " isomorphism " or " permutation "), it implies the equivalence of objects one of which is the image of the other under such a transformation.

If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism between objects of D, called the component of η at X, such that for every morphism we have:

If η is a natural transformation from F to G, we also write or.

If and are natural transformations between functors, then we can compose them to get a natural transformation.

If is a natural transformation between functors and is a natural transformation between functors, then the composition of functors allows a composition of natural transformations.

If is a natural transformation between functors, and is another functor, then we can form the natural transformation by defining

If on the other hand is a functor, the natural transformation is defined by

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.