If the considered poset has binary suprema ( a. k. a. joins ), as do the posets within this article, then this is equivalently characterized as a lower set I which is closed for binary suprema ( i. e. x, y in I imply x ' y in I ).

If all directed subsets of a poset have a supremum, then the order is a directed complete partial order ( dcpo ).

If every subset that has some upper bounds has also a least upper bound, then the respective poset is called bounded complete.

If it exists, the least element of a poset is always compact.

If the poset X is finite then the statement of the theorem has a clear interpretation that leads to the proof.

* If P is a non-empty upwards ccc poset and Y is a family of cofinal subsets of P with then there is an upwards directed set A such that A meets every element of Y.

If B is a cofinal subset of a poset A and C is a cofinal subset of B with the partial ordering of A restricted to B, then C is also a cofinal subset of A.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

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 P, then Q.

If P is a program which outputs a string x, then P is a description of x.

* If the balance factor of P is-2 then the right subtree outweighs the left subtree of the given node, and the balance factor of the right child ( R ) must be checked.

* If the balance factor of R is-1, a single left rotation ( with P as the root ) is needed ( Right-Right case ).

* If the balance factor of P is 2, then the left subtree outweighs the right subtree of the given node, and the balance factor of the left child ( L ) must be checked.

* If the balance factor of L is + 1, a single right rotation ( with P as the root ) is needed ( Left-Left case ).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

If either P or Q is true and P is false, then Q is true.

If the partial pressure of A at x < sub > 1 </ sub > is P < sub > A < sub > 1 </ sub ></ sub > and x < sub > 2 </ sub > is P < sub > A < sub > 2 </ sub ></ sub >, integration of above equation,

If ~ is an equivalence relation on X, and P ( x ) is a property of elements of X, such that whenever x ~ y, P ( x ) is true if P ( y ) is true, then the property P is said to be well-defined or a class invariant under the relation ~.

The well-known historian of physics, C. P. Snow, says about him, " If Fermi had been born a few years earlier, one could well imagine him discovering Rutherford's atomic nucleus, and then developing Bohr's theory of the hydrogen atom.

If the cause of the crash is uncertain, this number is rendered as 48454C50, which stands for " HELP " in hexadecimal ASCII characters ( 48 = H, 45 = E, 4C = L, 50 = P ).

If the quantity of powder stored for personal use exceeds 75 kg, then a Propellant Magazine Licence ( Type P ) is required.

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 >)

If the base manifold is four-dimensional, the Kaluza – Klein manifold P is five-dimensional.

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, 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 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.