* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

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

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If a Banach algebra has unit 1, then 1 cannot be a commutator ; i. e., for any x, y ∈ A.

If x is held fixed, then the Bessel functions are entire functions of α.

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

If x is a member of A, then it is also said that x belongs to A, or that x is in A.

If ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >) are the Cartesian coordinates and ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, u < sub > 3 </ sub >) are the orthogonal coordinates, then

If the Council accepts these amendments then the legislation is approved.

If the Supreme Court grants certiorari and accepts the case, it will receive written briefs from each side ( and any amici curiae or friends of the court — usually interested third parties with some expertise to bear on the subject ) and schedule oral arguments.

If one accepts that Luke was in fact the author of the Gospel bearing his name and also the Acts of the Apostles, certain details of his personal life can be reasonably assumed.

If any branch of the tree halts with an " accept " condition, we say that the NTM accepts the input.

If the defendant accepts these suggestions and changes his penalty proposition, the court approves it and passes the verdict according to the plea agreement.

If the state accepts them ( or fails to act at all ), both the reserving state and the accepting state are relieved of the reserved legal obligation as concerns their legal obligations to each other ( accepting the reservation does not change the accepting state's legal obligations as concerns other parties to the treaty ).

If the accused accepts NJP, he or she, plus a representative if desired, will attend the hearing conducted by the commander.

If the first person accepts the hypothesis that " All ravens are black " then, according to the argument, about 50 objects whose colors were previously in doubt ( the ravens ) are now thought to be black, while nothing different is thought about the remaining objects ( the non-ravens ).

" If one accepts this theory, then the Induction becomes vital to interpretation, as it serves to undermine any questions of the seriousness of Katherina's closing sentiments.

If one accepts the theory that the speech is sincere, then the final scene must be interpreted literally.

If she accepts, she is tonsured in a formal service during which she is given the outer robe ( Exorassa ) and veil ( Epanokamelavkion ) to wear, and ( because she is now dead to the world ) receives a new name.

If the publisher accepts unsolicited manuscripts, then the manuscript is placed in the slush pile, which publisher's readers sift through to identify manuscripts of sufficient quality or revenue potential to be referred to acquisitions editors for review.

If the chair accepts the point of order, it is said to be sustained or ruled well-taken.

If the dealer accepts then the elder hand must propose a discard and the dealer should deal the same number of fresh cards from the pack ; following which the dealer must then also make an exchange of at least one card.

If the proposal was accepted, then the elder hand can make another proposal, if desired, and can go on making proposals as long as the dealer accepts them.

* If, after the trump card is turned up, and before the dealer accepts or refuses, or, if there is no proposal, before he plays, it be discovered that he has dealt himself more than five cards, the non-dealer may claim a fresh deal.

* If, after the trump card is turned up, and before the dealer accepts or refuses, or, if there is no proposal, before he plays, it be discovered that the dealer has fewer than five cards, the non-dealer may permit the dealer to complete his hand from the stock, or may claim a fresh deal.

If the final state is an accepting state, then the automaton accepts the word.

When Achilles demands that " If you accept A and B and C, you must accept Z ," the Tortoise remarks that that's another hypothetical proposition, and suggests even if it accepts C, it could still fail to conclude Z if it did not see the truth of:

If the next unelected candidate has died or declines the offer, it is offered to each subsequent candidate on the list until one accepts the seat, such as was the case upon the resignation of Darren Hughes in March 2011, where Louisa Wall was elected after the five list candidates above her on the New Zealand Labour Party's list declined the seat.

If it is valid, it accepts ; otherwise, it rejects.

If one accepts the validity of the idea of social reality, scientifically, it must be amenable to measurement, something which has been explored particularly in relation to trust.

If one accepts this connection, then the Tukharas (= Rishikas

* If N accepts ( which we know it does in at most f ( n ) operations ), this means that K rejects (, ), so (, ) is not in H < sub > f </ sub >, and thus N does not accept in f ( n ) steps.

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 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.