* 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 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 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 she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.

If a branch extended out too far, each man held it back for the next, and if they met a low overhang, each warned the other.

If you are an income beneficiary of property held in trust or an heir, legatee, or devisee, you may deduct allowable depreciation and depletion, if not deductible by the estate or trust.

If these things be so, then the evidence of the senses must be held in slight esteem.

If a flame is held before the lips while these words are spoken, it flickers more during aspirated nitrate than during unaspirated night rate.

If the celebration takes place outside of Mass, it can be held in the home, in a hospital or institution, or in church.

If the daughter is an only child or her sisters are deceased and have no living issue, she ( or her heir ) is vested with the title ; otherwise, since a peerage cannot be shared nor divided, the dignity goes into abeyance between the sisters or their heirs, and is held by no one.

If a king held sway over a large kingdom, such as when the Mercians dominated the East Anglians, the relationship would have been more equal than in the case the Mercian dominance of the Hwicce, which was a comparatively small kingdom.

If a coalition collapses, a confidence vote is held or a motion of no confidence is taken.

If the pawl were held away from the sleeve the clutch would continue to drive the load without slipping.

If a rotating frame is chosen so that just the angular position of an object is held fixed, more complicated motion, such as elliptical and open orbits, appears because the centripetal and centrifugal forces will not balance.

He felt differently about the Muslim leader Muhammed Ali Jinnah, but was aware of his power, stating " If it could be said that any single man held the future of India in the palm of his hand in 1947, that man was Mohammad Ali Jinnah.

If liberty is taken to mean " the ability to exercise one's rights as provided for by the law and nature " then this is true, but if it means " the state when one is not held to nor required to perform anything against their will " then this is clearly false.

If the frequency deviation is held constant and the modulation frequency increased, the spacing between spectra increases.

If the detention is held to be unlawful, the prisoner can usually then be released or bailed by order of the court without having to be produced before it.

If the defendant waives a jury trial, a bench trial is held.

If they voted " yes ", then an ostracism would be held two months later.

If the inverting input is held at ground ( 0 V ) directly or by a resistor, and the input voltage V < sub > in </ sub > applied to the non-inverting input is positive, the output will be maximum positive ; if V < sub > in </ sub > is negative, the output will be maximum negative.

If a judge determines that there is sufficient evidence to believe that the defendant committed the crime, it is said that the defendant is " held to answer " or " bound over " ( in U. S. jurisdictions ).

If the former option were to win, the referendum would have been held again every 8 years.

If the latter option were to win, a separate referendum would be held where Puerto Ricans would have been given the option of being admitted as a US State " on equal footing with the other states ", or becoming a " sovereign nation, either fully independent from or in free association with the United States.

If these tournament rounds were held in one competition, up to 512 contestants would participate in the tournament, which is difficult to believe for a single contest.

If requested to do so by a petition signed by a majority of the membership of the Seanad, and one-third of the membership of the Dáil, the President may, after consultation with the Council of State, decline to sign into law a bill ( other than a bill to amend the constitution ) he / she considers to be of great " national importance " until it has been approved by either the people in an ordinary referendum or the Dáil reassembling after a general election, held within eight months.

If he swore to cease hostilities against the nation who held him captive, he could be repatriated or exchanged but could not serve against his former captors in a military capacity.