* If numbers have mean X, then.

* 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 the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

If a detector was placed at a distance of 1 m, the ion flight times would be X and Y ns.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

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.

* Theorem If X is a normed space, then X ′ is a Banach space.

If X ′ is separable, then X is separable.

If F is also surjective, then the Banach space X is called reflexive.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

If there is a bounded linear operator from X onto Y, then Y is reflexive.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

In the biography, Paul Allen wrote, regarding a suggestion in Cosmopolitan that his plays were becoming autobiographical: " If we take that to mean that his plays tell his own life story, he still hasn't started.

If the rearcourt attacker plays a dropshot, his partner will move into the forecourt to threaten the net reply.

* If the offence fails to make ten yards in three plays, the defence takes over on downs.

The song If You Wanna Be Happy by Jimmy Soul plays in the background.

If the modem plays a melody consisting of 1000 tones per second, the symbol rate is 1000 symbols / second, or baud.

If new Shakespearean plays were being written especially for presentation at the Blackfriars ' theatre after 1608, they could not have been written by Edward de Vere.

If no range is specified, the stream is played from the beginning and plays to the end, or, if the stream is paused, it is resumed at the point it was paused.

# Opposite corner: If the opponent is in the corner, the player plays the opposite corner.

If O plays a corner, a perfect X player has already played the corner opposite his first and proceeds to play a 3rd corner, stopping O's 3-in-a-row and making his own fork.

** If O plays a corner or side-middle first, X is guaranteed to win:

*** If O plays a side-middle, X takes the only corner that O's blocking won't make 2 in a row.

* If a player can do an approach and a withdrawal at the same time, he has to choose which one he plays.

If White plays 1.

" If the next player also plays a seven, he announces " have a very nice day " and the player after that draws two penalty cards.

If many of these plays have been judged less compelling than his earlier work, his later work, in which satiric fury is tempered and broadened, also includes three of his acknowledged masterpieces.

If the closest boules from each team are an equal distance from the jack, then the team that played last plays again.

* 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 king be the card first played by the dealer, he may announce it at any time before he plays again.

If a contestant plays a false hidden immunity idol, the host would toss it into the fire pit.

* If a contestant plays the hidden immunity idol, any votes cast for that contestant will not count, and the person with the next largest number of votes will be eliminated.

If White plays a regular opening and Black responds in an unconventional way, the opening is not categorized A00.

If South ( declarer ) is on lead he can lead to the queen ; that is, he leads a small spade and, if West plays low, plays the queen from dummy.

If working in a zinc mine, which he once did for 87-1/2 cents an hour, paid more than playing center field for the Yankees, Mantle would work in a zinc mine.

If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism of commutative rings.

If, in an unsharp image, a patch of light corresponds to an object point, the center of gravity of the patch may be regarded as the image point, this being the point where the plane receiving the image, e. g., a focusing screen, intersects the ray passing through the middle of the stop.

If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have

If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant.

If one cube has an A atom at its center, the atom is on a site usually occupied by a B atom, and is thus an antisite defect.

If the panegyric detailing the ceremony implied that the true center of the Empire was not Rome, but where the Emperor sat ("... the capital of the Empire appeared to be there, where the two emperors met "), it simply echoed what had already been stated by the historian Herodian in the early third century: " Rome is where the emperor is ".

If multiple symmetry axes are present, they will necessarily intersect at a single point, and this is usually taken to be the center.

If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity.

If we draw an ellipse twice as long as it is wide, and draw the circle centered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel to the shorter axis the length within the circle is twice the length within the ellipse.

Formulae connecting a tangential angle, the angle anchored at the ellipse's center ( called also the polar angle from the ellipse center ), and the parametric angle t < ref > If the ellipse is illustrated as a meridional one for the earth, the tangential angle is equal to geodetic latitude, the angle is the geocentric latitude, and parametric angle t is a parametric ( or reduced ) latitude of auxiliary circle </ ref > are:

If the applied voltage is large enough, the liquid crystal molecules in the center of the layer are almost completely untwisted and the polarization of the incident light is not rotated as it passes through the liquid crystal layer.

If inversion through the center of symmetry in a molecule results in the same phases for the molecular orbital, then the MO is said to have gerade, g, symmetry.

If inversion through the center of symmetry in a molecule results in a phase change for the molecular orbital, then the MO is said to have ungerade, u, symmetry.

If all the particles are moving, the center of mass will generally be moving as well.

If the center of mass is moving at velocity, the momentum is:

If the top has mass and its center of mass is at a distance from the pivot point, its gravitational potential is

If G is a p-group, then so is G / Z, and so it too has a nontrivial center.

If the cross stands at the center of Paul's theology, so does the Resurrection: unless the one died the death of all, the all would have little to celebrate in the resurrection of the one.

If two intersecting planes pass through its center, then they will subdivide the sphere into four lunes or biangles, the vertices of which all coincide with the antipodal points lying on the line of intersection of the planes.

If the planes don't pass through the sphere's center, then the intersection is called spheric section.