* 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.

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 these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.

If an origin is also chosen, this can be decomposed as an affine transformation that sends, namely

If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation.

If the affine transformation can be decomposed into isometries and a transformation given by a diagonal matrix, we have directionally differential scaling and the diagonal elements ( the eigenvalues ) are the scale factors in two or three perpendicular directions.

If is an affine transformation of where c is an vector of constants and B is a constant matrix, then y has a multivariate normal distribution with expected value and variance BΣB < sup > T </ sup > i. e.,.

If the world file describes an image that is rotated from the axis of the target projection, however, then A, D, B and E must be derived from the required affine transformation ( see below ).

If V is a vector space, one distinguishes " vector hyperplanes " ( which are subspaces, and therefore must pass through the origin ) and " affine hyperplanes " ( which need not pass through the origin ; they can be obtained by translation of a vector hyperplane ).

If these texture coordinates are linearly interpolated across the screen, the result is affine texture mapping.

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

If it is a smooth affine variety, then all extensions of locally free sheaves split, so group has an alternative definition.

If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m ( m − 1 )/ 2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of where is the local ring at P and is its integral closure.

If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2.

If denotes an affine connection, then the curvature tensor is the tensor defined by

If the manifold is equipped with an affine connection ( a covariant derivative or connection on the tangent bundle ), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.

If ∇ is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.

* If is a matrix and lies in its column space, the set of solutions of the equation is an affine space over the subspace of solutions of.

If one chooses a base point ( as zero ), then an affine space becomes a vector space, which one may then projectivize, but this requires a choice.

2 ) If X is an affine L-variety, defined by

If, on the other hand, the kernel assumes negative values, such as the sinc function, then the value of the filtered signal will instead be an affine combination of the input values, and may fall outside of the minimum and maximum of the input signal, resulting in undershoot and overshoot, as in the Gibbs phenomenon.

If X is an affine algebraic variety, and if U is an open subset of X, then K < sub > X </ sub >( U ) will be the field of fractions of the ring of regular functions on U. Because X is affine, the ring of regular functions on U will be a localization of the global sections of X, and consequently K < sub > X </ sub > will be the constant sheaf whose value is the fraction field of the global sections of X.

If X is integral but not affine, then any non-empty affine open set will be dense in X.

If X has dimension at most n and F is a torsion sheaf then these cohomology groups with compact support vanish if q > 2n, and if in addition X is affine of finite type over a separably closed field the cohomology groups vanish for q > n ( for the last statement, see SGA 4, XIV, Cor. 3. 2 ).