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 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 φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

If this output is then used as the clock signal for a similarly arranged D flip-flop ( remembering to invert the output to the input ), you will get another 1 bit counter that counts half as fast.

If v < sub > n </ sub > is the number of vertices of degree n and D is the maximum degree of any vertex,

If you work as a No. 2 in a New York City D. A.

If one chooses a critical value of the test statistic D < sub > α </ sub > such that P ( D < sub > n </ sub > > D < sub > α </ sub >)

In 2001, David Wiley criticized learning object theory in his paper, The Reusability Paradox which is summarized by D ' Arcy Norman as, If a learning object is useful in a particular context, by definition it is not reusable in a different context.

The " diamond problem " ( sometimes referred to as the " deadly diamond of death ") is an ambiguity that arises when two classes B and C inherit from A, and class D inherits from both B and C. If D calls a method defined in A ( and does not override the method ), and B and C have overridden that method differently, then from which class does it inherit: B, or C?

If ( D ) is the wheel diameter, the torque on the runner is: T = F ( D / 2 ) = ρQD ( V < sub > i </ sub > − u ).

* Matter, William D. If It Takes All Summer: The Battle of Spotsylvania ( 1988 )

If this type of contact utilizes a " make before break " functionality, then it is called a Form D contact.

If chiral pairs ( D and S ) are considered as identical, the remaining seven pieces can fill a 7 × 2 × 2 box.

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.

Similar statements apply to the dual situation of terminal morphisms from U. If such morphisms exist for every X in C one obtains a functor V: C → D which is right-adjoint to U ( so U is left-adjoint to V ).

If the ratio of the two resistances in the known leg is equal to the ratio of the two in the unknown leg, then the voltage between the two midpoints ( B and D ) will be zero and no current will flow through the galvanometer.

If the opponent places a stone at B or D, the remaining hex can be filled to join the original two stones into a single group.

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.

If a loan word is too foreign sounding inflection or derivation rules may not be able to transform it.

If the rotational speed is measured in revolutions per unit of time, the linear speed and distance are increased proportionately by 2π in the above derivation to give:

If A ∧ B is true, then B ∧ A is true ; This derivation can be drawn by composing inference rules in such a fashion that premises of a lower inference match the conclusion of the next higher inference.

Gilbert Murray summarizes the derivation from pelas gē, (" neighboring land "):" If Pelasgoi is connected with πέλας, ' near ', the word would mean ' neighbor ' and would denote the nearest strange people to the invading Greeks ..."

If the Pelasgians were not Indo-Europeans, the name in this derivation must have been assigned by the Hellenes.

If premises are left unsatisfied in the derivation, then the derivation is a proof of a hypothetical statement: " if the premises hold, then the conclusion holds.

If ASCII characters are used, the 256 bit key is calculated by applying the PBKDF2 key derivation function to the passphrase, using the SSID as the salt and 4096 iterations of HMAC-SHA1.

If ambiguity can be resolved by applying rightmost derivation the grammar is context-free, otherwise it is context-sensitive.

If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. An algebra A equipped with a distinguished derivation d forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

If in the algebraic derivation above more than two polynomials are used, it results in the generalized Vandermonde's identity.

# If Δ is a derivation ending in an expression of the form α ( Iβ ) ι, then Δ followed by the term αβι is a derivation.

# If Δ is a derivation ending in an expression of the form α (( Kβ ) γ ) ι, then Δ followed by the term αβι is a derivation.

# If Δ is a derivation ending in an expression of the form α ((( Sβ ) γ ) δ ) ι, then Δ followed by the term α (( βδ )( γδ )) ι is a derivation.

If the images consist of a flat background, with all detail situated away from the edges, then a linear shift will be equivalent to a circular shift, and the above derivation will hold exactly.

Mommsen wrote :" In contrast to these irrefutable conditioning factors, Nolte ’ s derivation based on personalities and the history of ideas seems artificial, even for the explanation of Hitler ’ s anti-semitism … If one emphasizes the indisputably important connection in isolation, one should not then force a connection with Hitler's weltanschauung, which was in no ways original itself, in order to deprive from it the existence of Auschwitz.

If quantization were homogeneous ( e. g., if it were linear ), the following would be a sufficient derivation of the equivalence of DM and SDM:

If the origin is moved to ( a, 0 ) then a derivation similar to that given above shows that the equation of the curve in polar coordinates becomes