* 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 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 it does, however, it is unique in a strong sense: given any other inverse limit X ′ there exists a unique isomorphism X ′ → X commuting with the projection maps.

If n = 2k is even, then let W ′ be an isotropic subspace complementary to W. If n = 2k + 1 is odd let W ′ be a maximal isotropic subspace with W ∩ W ′

If the functor F: C ← D has two right adjoints G and G ′, then G and G ′ are naturally isomorphic.

If L ′ is of finite height, or at least verifies the ascending chain condition ( all ascending sequences are ultimately stationary ), then such an x ′ may be obtained as the stationary limit of the ascending sequence x ′< sub > n </ sub > defined by induction as follows: x ′< sub > 0 </ sub >=⊥ ( the least element of L ′) and x ′< sub > n + 1 </ sub >= f ′( x ′< sub > n </ sub >).

If it is a pseudovector, it will be transformed to v ′ =-Rv.

If it does, however, it is unique in a strong sense: given another direct limit X ′ there exists a unique isomorphism X ′ → X commuting with the canonical morphisms.

If M ′ is another almost complex manifold of the same dimension, then it is cobordant to M if and only if the Chern numbers of M ′ coincide with those of M.

If more than one intein is encoded in the corresponding gene, the inteins are given a numerical suffix starting from 5 ′ to 3 ′ or in order of their identification ( for example, " Msm dnaB-1 ").

As before, suppose that E is a fibre bundle with structure group G. In the special case when G has a free and transitive left action on F ′, so that F ′ is a principal homogeneous space for the left action of G on itself, then the associated bundle E ′ is called the principal G-bundle associated to the fibre bundle E. If, moreover, the new fibre F ′ is identified with G ( so that F ′ inherits a right action of G as well as a left action ), then the right action of G on F ′ induces a right action of G on E ′.

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.

This state is separable if yielding and It is inseparable if If a state is inseparable, it is called an entangled state.

* From the above property, one can deduce the following: If X is a separable space having an uncountable closed discrete subspace, then X cannot be normal.

If a normed space X is separable, then the weak -* topology is metrizable.

If the Hamiltonian is not an explicit function of time, the equation is separable into its spatial and temporal parts.

If the training data are linearly separable, we can select two hyperplanes in a way that they separate the data and there are no points between them, and then try to maximize their distance.

If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting so-called Kummer extension is an abelian extension ( if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension ).

If the requirement for the countable chain condition is replaced with the requirement that R contains a countable dense subset ( i. e., R is a separable space ) then the answer is indeed yes: any such set R is necessarily isomorphic to R.

" If they are separable, function will differ from original reality and exist independently, and in that way function will have its own original reality.

If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS.

If P ( x ) is separable, and its roots form a group ( a subgroup of the field K ), then P ( x ) is an additive polynomial.

* If is an algebraic field extension, and if are separable over F, then and are separable over F. In particular, the set of all elements in E separable over F forms a field.

* If is such that and are separable extensions, then is separable.

* If is a finite degree separable extension, then it has a primitive element ; i. e., there exists with.

* If is a finite degree normal extension, and if, then K is purely inseparable over F and E is separable over K.

If is a finite degree extension, the degree: F is referred to as the separable part of the degree of the extension ( or the separable degree of E / F ), and is often denoted by: F < sub > sep </ sub > or: F < sub > s </ sub >.

If is not separable ( i. e., inseparable ), then: F < sub > sep </ sub > is necessarily a non-trivial divisor of: F, and the quotient is necessarily a power of the characteristic of F.

If such an intermediate extension does exist, and if: F is finite, then if S is defined as in the previous paragraph,: F < sub > sep </ sub >=: F =: K. One known proof of this result depends on the primitive element theorem, but there does exist a proof of this result independent of the primitive element theorem ( both proofs use the fact that if is a purely inseparable extension, and if f in F is a separable irreducible polynomial, then f remains irreducible in K ).

If F is any field, the separable closure F < sup > sep </ sup > of F is the field of all elements in an algebraic closure of F that are separable over F. This is the maximal Galois extension of F. By definition, F is perfect if and only if its separable and algebraic closures coincide ( in particular, the notion of a separable closure is only interesting for imperfect fields ).