* 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 < 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 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 there was collusion between an outside murderer and a member of the household it would be an elementary precaution to check on the door later.

If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their perpendicular height of incidence, i. e. their distance from the axis.

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

If a " positive list " approach were taken, only species evaluated and approved for the positive list would be permitted in trade, thus lightening the review burden for member states and the Secretariat, and also preventing inadvertent legal trade threats to poorly known species.

* Overrunning clutch or freewheel: If some external force makes the driven member rotate faster than the driver, the clutch effectively disengages.

If the driving member rotates in the direction that would unwind the spring the spring expands minutely and slips although with some drag.

If one member ran away or married into another household or died, the rest of family continued to be responsible for the tax obligation of the family unit.

If no election is to take place within 120 days of the vacancy then the district chairman is to call together the members of the district and they are to elect a member who will serve until the next town election.

To avoid debates about the possibility of in-vitro fertilization, artificial insemination, or cloning, the original Darwin Awards book applied the following " deserted island " test to potential winners: If the person were unable to reproduce when stranded on a deserted island with a fertile member of the opposite sex, he or she would be considered sterile.

If a planet orbits one member of a binary star system, then an uppercase letter for the star will be followed by a lowercase letter for the planet.

If any member of the offensive team is downed while in possession of the ball behind his own team's goal line, this is called a safety and scores two points for the defensive team.

If the board of directors has judged that a member bank is performing or behaving poorly, it will report this to the Board of Governors.

A bureau chief stated, " If any member had been excluded it would have been the same thing, it has nothing to do with Fox or the White House or the substance of the issues ".

If Philip II Augustus supported Philip of Swabia, member of the House of Hohenstaufen, then Richard Lionheart supported Otto IV, member of the House of Welf.

If a member of the executive committee quits or dies, the other four members appoint his or her replacement.

Pei's background in architecture was seen as a considerable asset ; one member of the committee told him: " If you know how to build you should also know how to destroy.

If Justus was a member of the second group of missionaries, then he arrived with a gift of books and " all things which were needed for worship and the ministry of the Church ".

Unification Church member Kristopher Esplin told Reuters what is normally done if the word is seen in media sources: " If it's printed in newspapers, we will respond, write to the editor, that sort of thing.

* If baptism is a sign that a person is a member of God's covenant community, and if the children of believers are members of that community, it follows that the children of believers should receive the sign that they are members of God's covenant community by being baptized, as an infant is entitled to a passport that indicates the child as a member of a particular country.

If a person elected prime minister is neither a member of the Lok Sabha nor Rajya Sabha, then he must become a member of the Lok Sabha or Rajya Sabha within six months.

If the members of the task group agree that the name is appropriate, it can be retained for use when there is a request from a member of the scientific community that a specific feature be named.