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If M is a non-compact complete non-negatively curved n-dimensional Riemannian manifold, then M contains a compact, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S ( S is called the soul of M .) In particular, if M has strictly positive curvature everywhere, then it is diffeomorphic to R < sup > n </ sup >.
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If and M
* If S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )
* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =
* If a set S is in M and S is congruent to T then T is also in M and a ( S )
* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =
* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:
If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.
Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.
If ( m, n ) is regular and M and N have i and j prime factors respectively, then ( m, n ) is said to be of type ( i, j ).
If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space.
If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.
If used, the word „ gothic “ was used to describe ( mostly early ) works of F. M. Dostoevsky.
If ψ is satisfiable in a structure M, then certainly so is φ and if ψ is refutable, then is provable, and then so is ¬ φ, thus φ is refutable.
If is satisfiable in a structure M, then, considering, we see that is satisfiable as well.
If the password is correct, then M releases the transferred sum to B ( 3b ), usually minus a small commission.
If M is an R module and is its ring of endomorphisms, then if and only if there is a unique idempotent e in E such that and.
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 non-compact
If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples ( such as the complement of a trefoil knot ) where a manifold may have a finite volume geometric structure of either type.
* If π < sub > 1 </ sub >( M ) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or non-compact.
If and complete
If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.
If the problem is enlarged to require a complete coverage of feed states, Af operations are needed by the dynamic program and Af by the direct search.
If the student drops out of high school, the break with the school is even more complete.
If laborers are merely commodities competing against each other in a market place like so many bags of wheat and corn ( unsupported, by the way, by any agricultural subsidy ), then they may be pardoned for reacting with complete antagonism to a system that imposes such status upon them.
If an opponent clearly cannot dodge an attack, there is no reason to complete it.
If dynamic re-routing is established, even if one or more nodes depart and abandon a downloading file, for example, the remaining nodes should still have the data needed to complete the download.
If insufficient yarn of a single dye lot is bought to complete a project, additional skeins of the same dye lot can sometimes be obtained from other yarn stores or online.
If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.
He is still seeking greater autonomy from China, although Dolma Gyari, deputy speaker of the parliament-in-exile has stated " If the middle path fails in the short term, we will be forced to opt for complete independence or selfdetermination as per the UN charter ".
If, on the other hand, the project is only 10 % complete at week 8, the project is significantly over budget and behind schedule.
If we allow to incorporate the algebraic sign, this becomes a sum and implies that there is a function of state which is conserved over a complete cycle.
If an eligible receiver on the passing team legally catches the ball it is a complete pass and the receiver may attempt to advance the ball.
If a forward pass is caught near a sideline or endline it is a complete pass ( or an interception ) only if a receiver catches the ball in bounds.
If some specific deductive system of first-order logic is sound and complete, then is it " perfect " ( a formula is provable iff it is a semantic consequence of the axioms ), thus equivalent to any other deductive system with the same quality ( any proof in one system can be converted into the other ).
If and then the limit is where is the complete elliptic integral of the first kind
If brass is used after this ring appears, it risks a crack, or worse, a complete head separation, which will leave the forward portion of the brass lodged in the chamber of the gun.
If they turned to illegality it was due to the fact that honest toil only benefited the employers and often entailed a complete loss of dignity, while any complaints resulted in the sack ; to avoid starvation through lack of work it was necessary to beg or steal, and to avoid conscription into the army many of them had to go on the run.
If the British failed to meet the deadline, the Congress would call upon all Indians to fight for complete independence.
If a knitter buys insufficient yarn of a single dye lot to complete a project, additional skeins of the same dye lot can sometimes be obtained from other yarn stores or online.
If it is rational, the proof is complete, and
If there is any way to avoid the emergence of the city-state, not to speak of self-serving bourgeois " cooperatives ," it is through a municipalization of political life that is so complete that politics embraces not only what we call the public sphere but material means of life as well.
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