Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

* If the balance factor of P is 2, then the left subtree outweighs the right subtree of the given node, and the balance factor of the left child ( L ) must be checked.

* If the balance factor of L is + 1, a single right rotation ( with P as the root ) is needed ( Left-Left case ).

* If the balance factor of L is-1, two different rotations are needed.

* 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 at least 50 cars had been built, sportscars like the GT40 and the Lola T70 were allowed, with a maximum of 5. 0 L. John Wyer's revised 4. 7 litre ( Bored to 4. 9 litres, and o-rings cut and installed between the deck and head to prevent head gasket failure, a common problem found with the 4. 7 engine.

If the cause of the crash is uncertain, this number is rendered as 48454C50, which stands for " HELP " in hexadecimal ASCII characters ( 48 = H, 45 = E, 4C = L, 50 = P ).

Clark L. Hull, probably the first major empirical researcher in the field, wrote If a subject after submitting to the hypnotic procedure shows no genuine increase in susceptibility to any suggestions whatever, there seems no point in calling him hypnotised ...

If the lower case L is used as the symbol, it is sometimes rendered as a cursive ℓ to help distinguish it from the capital " I ", although this usage has no official approval by any international bureau.

* Case 1: If L is a whole number, then the value will be found halfway between positions L and L + 1.

* Case 2: If L is a fraction, round to the nearest whole number.

If one considers a version of QCD with N < sub > f </ sub > flavors of massless quarks, then there is a global ( chiral ) flavor symmetry group SU < sub > L </ sub >( N < sub > f </ sub >) × SU < sub > R </ sub >( N < sub > f </ sub >) × U < sub > B </ sub >( 1 ) × U < sub > A </ sub >( 1 ).

Another more recent story involving travel to the future is Louis-Sébastien Mercier's L ' An 2440, rêve s ' il en fût jamais (" The Year 2440: A Dream If Ever There Were One "), a utopian novel in which the main character is transported to the year 2440.

If X and Y are topological vector spaces, the space L ( X, Y ) of continuous linear operators f: X → Y may carry a variety of different possible topologies.

If the array of weights consists of N weights spaced evenly over the length L

If J is a category with two objects and two parallel morphisms from object 1 to object 2 then a diagram of type J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms.

If one is given an object L of C together with a natural isomorphism Φ: Hom (–, L ) → Cone (–, F ), the object L will be a limit of F with the limiting cone given by Φ < sub > L </ sub >( id < sub > L </ sub >).

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 S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.

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 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 r is the degree of the primitive generator polynomial, then the maximal total block length is, and the associated code is able to detect any single-bit or double-bit errors.

If we use the generator polynomial, where is a primitive polynomial of degree, then the maximal total block length is, and the code is able to detect single, double, and triple errors.

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 unital commutative ring with an ideal m, then k = R / m is a field if and only if m is a maximal ideal.

) If formulated in von Neumann – Bernays – Gödel set theory, the surreal numbers are the largest possible ordered field ; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals ; it has also been shown that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.

If the patient were completely non-responsive to the maximal recommended dose of sertraline ( 200 mg ), further increasing the dose did not significantly improve the response rates.

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical.

If a given manifold admits a geometric structure, then it admits one whose model is maximal.

If the rank of this matrix at a point P on the curve has the maximal value of n − 1, then the point is a smooth point.

If a concave ( i. e., concave downward ) function has an " apex ", any point at the apex is a maximal extremum.