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 X is a topological space, there is a natural way of transforming X /~ into a topological space ; see quotient space for the details.

If N is a closed normal subgroup of a profinite group G, then the factor group G / N is profinite ; the topology arising from the profiniteness agrees with the quotient topology.

If I is a right ideal of R, then R / I is simple if and only if I is a maximal right ideal: If M is a non-zero proper submodule of R / I, then the preimage of M under the quotient map is a right ideal which is not equal to R and which properly contains I.

If a finite difference is divided by b − a, one gets a difference quotient.

If H is a subgroup of G, the set of left or right cosets G / H is a topological space when given the quotient topology ( the finest topology on G / H which makes the natural projection q: G → G / H continuous ).

Since a one sided maximal ideal A is not necessarily two-sided, the quotient R / A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J ( R ).

If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module.

If N is the nilradical of commutative ring R, then the quotient ring R / N has no nilpotent elements.

If we add the relation x < sup > 2 </ sup > = 1 to the presentation of Dic < sub > n </ sub > one obtains a presentation of the dihedral group Dih < sub > 2n </ sub >, so the quotient group Dic < sub > n </ sub >/< x < sup > 2 </ sup >> is isomorphic to Dih < sub > n </ sub >.

If X is a diffeological space and ~ is some equivalence relation on X, then the quotient set X /~ has the diffeology generated by all compositions of plots of X with the projection from X to X /~.

If e < sub > 1 </ sub >, ... e < sub > d </ sub > is a basis of V, the unital zero algebra is the quotient of the polynomial ring k ..., E < sub > n </ sub > by the ideal generated by the E < sub > i </ sub > E < sub > j </ sub > for every pair ( i, j ).

If we try to use the quotient to compute f '( 0 ), however, an undefined value will result, since | x | is nondifferentiable at x = 0.

If N is a normal subgroup of G, then the index of N in G is also equal to the order of the quotient group G / N, since this is defined in terms of a group structure on the set of cosets of N in G.

If G and H are finite groups, then the index of H in G is equal to the quotient of the orders of the two groups:

If ΔP is infinitesimal, then the difference quotient is a derivative, otherwise it is a divided difference:

If p is a regular cover, then Aut ( p ) is naturally isomorphic to a quotient of.

If G is not simply connected, then the lattice P ( G ) is smaller than P ( g ) and their quotient is isomorphic to the fundamental group of G.

If it is not, there are three possible problems: the multiplication is wrong, the subtraction is wrong, or a greater quotient is needed.

If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

If R is a quotient of by a homogeneous ideal, then the canonical surjection induces the closed immersion

If X and Y are algebraic structures of some fixed type ( such as groups, rings, or vector spaces ), and if the function f from X to Y is a homomorphism, then ker f will be a subalgebra of the direct product X × X. Subalgebras of X × X that are also equivalence relations ( called congruence relations ) are important in abstract algebra, because they define the most general notion of quotient algebra.

If G is an extension of Q by N, then G is a group, N is a normal subgroup of G and the quotient group G / N is isomorphic to group Q.

If X is the plane with the origin missing, and G is the infinite cyclic group generated by ( x, y )→( 2x, y / 2 ) then this action is wandering but not properly discontinuous, and the quotient space is non-Hausdorff.

# If adiabats and isotherms are graphed severally at regular changes of entropy and temperature, respectively ( like altitude on a contour map ), then as the eye moves towards the axes ( towards the south-west ), it sees the density of isotherms stay constant, but it sees the density of adiabats grow.

If A is a fixed element of a ring ℜ, the first additional relation can also be interpreted as a Leibniz rule for the map given by B ↦.

If G is a group, and g is a fixed element of G, then the conjugation map

If the user is unable to identify what is being demonstrated in a reasonable fashion, the map may be regarded as useless.

If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors.

If a map contains a reducible configuration, then the map can be reduced to a smaller map.

If f: X → Y is a continuous map, x < sub > 0 </ sub > ∈ X and y < sub > 0 </ sub > ∈ Y with f ( x < sub > 0 </ sub >) = y < sub > 0 </ sub >, then every loop in X with base point x < sub > 0 </ sub > can be composed with f to yield a loop in Y with base point y < sub > 0 </ sub >.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to " solid ground " where the problem is easier to understand and work with.

If G is any subgroup of GL < sub > n </ sub >( R ), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

If a player dies all their weapons are lost and they receive the spawn weapons for the current map, usually the gauntlet and machine gun.

If a pushwall exits the boundaries of the level, the game quits with the error message " PushWall Attempting to escape off the edge of the map ".

If the tangent space is defined via curves, the map is defined as

More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G. If B is another finite generating set for G, then the word problem over the generating set B is equivalent to the word problem over the generating set A.

If Φ is a unital positive map, then for every normal element a in its domain, we have Φ ( a * a ) ≥ Φ ( a *) Φ ( a ) and Φ ( a * a ) ≥ Φ ( a ) Φ ( a *).

If a Borel function happens to be a section of some map, it is called a Borel section.

If K is a field, then for every vector space V over K we have a " natural " injective linear map from the vector space into its double dual.