If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.

If both are accepted, one has to additionally signal a preference.

If the aging for one of these types of whiskey reaches two years or beyond, the whiskey is then additionally designated as " straight " e. g., " straight rye whiskey ".

If we use NC reductions, that is, reductions which can operate in polylogarithmic time on a parallel computer with a polynomial number of processors, then all P-complete problems lie outside NC and so cannot be effectively parallelized, under the unproven assumption that NC ≠ P. If we use the weaker log-space reduction, this remains true, but additionally we learn that all P-complete problems lie outside L under the weaker unproven assumption that L ≠ P. In this latter case the set P-complete may be smaller.

If UV / Vis spectrophotometry is used in quantitative chemical analysis then the results are additionally affected by uncertainty sources arising from the nature of the compounds and / or solutions that are measured.

If, additionally, the recipient of the criticism is not the intended target, that is also a factor to consider.

If we additionally assume that the image plane is perpendicular to the ground plane ( an imaginary plane Tangential to the surface of the Earth ), and that the waterfall flows on a plane that is parallel to the image plane, we can estimate the height by measuring the vertical separation of waterfall head and base in

* If the operation additionally has an identity element, we have a commutative monoid

If additionally

If, additionally, the real part of s is positive, then the limit γ ( s, u ) → 0 for u → 0 applies, finally arriving at the complex integral definition of γ

If not, then the judge must additionally assess the application under the " centralized mode of existence " approach, guided by the non-exhaustive factors set out in Koo ( Re ).

If the poset P additionally is a join-semilattice ( i. e., if it has binary suprema ) then these conditions are equivalent to the following statement:

If the surface is additionally known to be axisymmetric with,

If a player cannot follow suit, he must play a trump card if able, and additionally, the player is compelled to play a higher trump than any existing trump in the trick if he is able ( The " Petit " or 1 is valued lowest, and the " Monde " or 21 is valued highest ).

If it were a warship, and repairable, usually the Crown bought it at a fair price ; additionally, the Crown added " head money " of 5 pounds per enemy sailor aboard the captured warship.

Their sets included songs spanning Strummer's career to that point, including works from The 101ers and The Clash, additionally material co written with Mick Jones for Big Audio Dynamite's No. 10, Upping St. album as well as a cover of The Pogues ' famous " If I Should Fall From Grace With God.

If the sample is depolarizing the Mueller matrix formalism should be used, because it gives additionally access to the amount of depolarization.

If additionally then is called Turing complete for.

If X was in good faith, and additionally was in legal possession of the house, then any vindicatio brought by Y could be defeated by X using the defence of fraud ( exceptio dolus malus ) until Y paid X for the materials.

If the two vertices are additionally connected by a path of length 1, i. e. by a single edge, the vertices are called adjacent.

If police additionally have reasonable suspicion that a person so detained may be armed, they may " frisk " the person for weapons, but not for contraband like drugs.

If the officer additionally has reasonable suspicion that the person is armed, the officer may perform a search of the person's outer garments for weapons.

If abstract index notation is used also for spinors then these will carry a spinorial index and the Dirac gamma will carry one Lorentzian and two spinorian indices, but it is more common to regard spinors as column matrices and the Dirac gamma as a matrix which additionally carries a Lorentzian index.

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 f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

If a function that is not injective is applied to both sides of a true equation, then the resulting equation will still be true, but it may be less useful.

If the domain consists of the non-negative numbers, then the function is injective and invertible.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

If M is simple, then f is either the zero homomorphism or injective because the kernel of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image of f is a submodule of N. If M = N, then f is an endomorphism of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism.

If K is a field, then for every vector space V over K we have a " natural " injective linear map from the vector space into its double dual.

* If the domain X = ∅ or X has only one element, the function is always injective.

If any horizontal line intersects the graph more than once, the function fails the horizontal line test and is not injective.

If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring.

If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.

No notation is used to describe the set of all λ, which satisfy this condition, but for a subset: If λ-T does not have dense range but is injective, λ is said to be in the residual spectrum of T, denoted by σ < sub > r </ sub >( T ).

If an operator is not injective ( so there is some nonzero x with T ( x ) = 0 ), then it is clearly not invertible.

: If U is an open subset of R < sup > n </ sup > and f: U → R < sup > n </ sup > is an injective continuous map, then V = f ( U ) is open and f is a homeomorphism between U and V.

If is a covering and and are such that, then is injective at the level of fundamental groups, and the induced homomorphisms are isomorphisms for all.

If S is infinite, the function can be chosen to be injective.

If a curve is not injective, then one can find two intersecting subcurves of the curve, each obtained by considering the images of two disjoint segments from the curve's domain ( the unit line segment ).

If there is an injective many-one reduction function then we say A is 1 reducible or one-one reducible to B and write

If the object A in the above short exact sequence is injective, then the sequence splits.

If X is itself injective, then we can choose the injective resolution 0 → X → X → 0, and we obtain that R < sup > i </ sup > F ( X )

* Invariance of domain: If U is an open subset of and is an injective continuous map, then is open and is a homeomorphism between and.

If is a divisible group, then is injective in the category of-modules.