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 λ < sub > 1 </ sub >, ..., λ < sub > ν </ sub > are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others.

If Carol has a hand like < font color = red > K ♥</ font > J ♠, she may choose to fold.

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

If J is the empty category there is only one diagram of type J: the empty one ( similar to the empty function in set theory ).

If J is a discrete category then a diagram F is essentially nothing but a family of objects of C, indexed by J.

If J is a category with two objects and two parallel morphisms from object 1 to object 2 then a diagram of type J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms.

* If J = 1, the category with a single object and morphism, then a diagram of type J is essentially just an object of C. A cone to an object X is just a morphism with codomain X.

which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected ( and nonempty ) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ.

The unit of this adjunction is the universal cocone from F to colim F. If J is connected ( and nonempty ) then the counit is an isomorphism, so that colim is a left inverse of Δ.

If a diagram F: J → C has a limit in C, denoted by lim F, there is a canonical isomorphism

If F: J → C is a diagram in C and G: C → D is a functor then by composition ( recall that a diagram is just a functor ) one obtains a diagram GF: J → D. A natural question is then:

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 G lifts limits of all type J and D has all limits of type J, then C also has all limits of type J and G preserves these limits.

" Later in the film, when the brothers are driving rapidly through Chicago, Elwood ( Dan Aykroyd ) comments " If my estimations are correct, we should be very close to the Honorable Richard J. Daley Plaza.

Since a one sided maximal ideal A is not necessarily two-sided, the quotient R / A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J ( R ).

If the current J is alternating because the applied voltage or E-field varies in time, then reactance must be added to resistance to account for self-inductance, see electrical impedance.

If he had married her, he'd have been asking for trouble.

He said: `` If it's all right with you, Mr. Morgan, I'll sleep out here on the couch.

If we let them go, they won't stay away, they'll find men to ride with them and they'll be back.

He added, `` If this doesn't work out, the three of you barricade yourself in the house and talk terms with them ''.

If you've got any ideas ''.

If she did, he could stand it better in the light.

If you ever try anything without my orders I'll kill you ''.

`` If you hadn't I'd have killed you ''.

If she sensed any unusual preoccupation on the part of her mother, she did not comment upon it.

If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.

If, as she walked, her steps fumbled from time to time, she chose to ignore that omen.

If the slope grew steeper and the groves more dim, she tried not to heed.

If, when this was all over, she found the words to tell him about it, she wondered if he would ever understand.

If you take the one, you'd better take both ''.

If the bluff failed and they ran into trouble, Brannon had told the others, they would withdraw -- and he would come after his son another time.

If you don't leave this country within 3 days, your life will be taken the same as Powell's was.

If someone were to drop a match in here, this place would go up like a haystack ''.

If we was both armed, you wouldn't talk so tough ''.

`` If you spot Carmer give a yell before you move in ''.

If he wondered whether the attackers would allow him to pull away unmolested, he had his answer a moment later.

If the other pilots were worried, they did not show it.

If it were the enemy, tactically his position was correct.

If they're Japs.

If any of us miss, they can pick up the pieces.

If the turn was too tight, a barrel roll would bring them out.