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If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A ; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.
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If and K
If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.
If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).
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 a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.
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.
If their existence past the K – T boundary can be confirmed, these hadrosaurids would be considered a Dead Clade Walking.
Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.
If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.
If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.
If the X ion is K or Na, the mica is a " common " mica, whereas if the X ion is Ca, the mica is classed as a " brittle " mica.
If Carol has a hand like < font color = red > K ♥</ font > J ♠, she may choose to fold.
: If the temperature T of a body is quantified as 300 kelvin ( in which T is the quantity symbol, 300 the value, and K is the unit ), this is written
If and is
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 and only
If there were only darkness, all would be clear.
If there were only the mess, all would be clear ; ;
Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.
If it is not one of his best books, it can only be considered unsatisfactory when compared with his own Garibaldi.
If only for this modest masterpiece of military history, Blenheim is likely to be read and reread long after newer interpretations have perhaps altered our picture of the Marlborough wars.
If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.
If he had any worries, it was only the small ones, about Mother in New York, and his daughter Edwina and what she might be doing at this hour, with her Aunt Asia, in Philadelphia.
If only the fucken weather wasn't so lousy!!
Somebody, got to be somebody If I don't put my two cents in soon, somebody else will I know they're waitin only for one thing: for the bastards what done it to be nailed.
If only Simms Purdew could do that, whatever the thing he remembered and told.
`` If he would only leave the East '', said Mary.
If only there was a clock for him to crawl against.
If the artist would study his work more thoroughly and move certain units in his design, often only slightly, finer pictures would result.
If you have a full-time doctor now, can he be replaced with a part-time doctor or one who serves on a fee-per-case basis only??
If you wish to budget closely on transportation, saving your extra dollars to indulge in luxuries, one agency lists the small Fiat 500 at only $1.26 a day plus $.03 a kilometer and the Fiat 2100 Station Wagon, seating six, at just $1.10 a day and $.105 a kilometer.
`` If I could only think of something at the studio, near me, to absorb his boundless energy '', I said.
If this is the case, one would expect that not only the various procedures just mentioned which alter the hypothalamic balance would influence emotional state and behavior but that emotion itself would act likewise.
If adjectival meanings show relatively low retentiveness of stems, as I am confident will prove to be the case in most languages of the world, why should our basic lists include 15 per cent of these unstable forms, but only 8 per cent of animals and plants which replace much more slowly??
If only state funds were used to pay for the vocational education, it could be argued that the state should not have to bear the cost of vocational training which would benefit employers in other states.
If so, then it is clear that in saying the suffering was bad we are not expressing our feelings only.
If goodness and badness lie in attitudes only and hence are brought into being by them, those men who greeted death and misery with childishly merry laughter are taking the only sensible line.
If there is nothing evil in these things, if they get their moral complexion only from our feeling about them, why shouldn't they be greeted with a cheer??
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