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 is injective we say is 1-reducible to and write

If additionally

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 competition beats you to it, this exciting new product era can have real headaches in store.

If the photographically realistic continuity of dreams, however bizarre their combinations, denies that it is purely a composition of the brain, it must be compounded from views of diverse realities, although some of them may never be encountered in what we are pleased to call the real life.

I kept saying, `` If I could just build up a reputation for myself, make some real money, get to be well known as an illustrator -- like Peter Askington, for instance -- then I could take some time off and paint ''.

If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.

Existence is the principle that gives reality to an essence not the same in any way as the existence: " If things having essences are real, and it is not of their essence to be, then the reality of these things must be found in some principle other than ( really distinct from ) their essence.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If X is a real Banach space, then the polarization identity is

If A is a self-adjoint operator, then is always a real number ( not complex ).

If the Czech ethnic origin was to be stressed, combinations like " Bohemian of Bohemian language "( Čech českého jazyka ), " a real Bohemian " ( pravý Čech ) etc.

If the exponent r is even, then the inequality is valid for all real numbers x.

If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.

If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.

If the eigenvalues are all positive, then the frequencies are all real and the stationary point is a local minimum.

If zeal for orthodoxy caused him to overstep the limits of discretion, his real attitude towards Rome is sufficiently clear.

If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ).

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.

If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

If a DBMS system responses users ' request in a given time period, it can be regarded as a real time database.

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

If this operation has to be done in real time video games there is an easy trick to boost performance.

* If we think of as the set of real numbers, then the direct product is precisely just the cartesian product,.

* If we think of as the group of real numbers under addition, then the direct product still consists of.

* If we think of as the ring of real numbers, then the direct product again consists of.

The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring.