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If S has the additional property that is a coherent sheaf and locally generates S over ( that is, when we pass to the stalk of the sheaf S at a point x of X, which is a graded algebra whose degree-zero elements form the ring then the degree-one elements form a finitely-generated module over and also generate the stalk as an algebra over it ) then we may make a further construction.
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If and S
** 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 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 )
* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )
* 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:
* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.
More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient
If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.
If and has
`` If Blue Throat has his way he'll keep us all cooped up in here for days '', he said.
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 the man on the sidewalk is surprised at this question, it has served as an exclamation.
If any one of them has any power to veto the Secretary General's decisions the nature of the organization will have changed.
If this new Bible does not increase in significance by repeated readings throughout the years, it will not survive the ages as has the King James Version.
If the case is thus determined by us to be domestic, the court has no jurisdiction.
If this capacity had not failed them, they would see that their enemy has made a disastrous miscalculation.
If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.
If the superintendents do not receive more cooperation from Handlers, it has been suggested that licensed Judges also be qualified to judge this Class.
If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.
If design head has a deep cavity, clay lid will be quite thick at this point ; ;
If you run into excess plug fouling on one truck, check to be sure that the rig has a thermostat.
If we manage to keep track of a Bombus queen after she has left her feeding place, we may discover the snug little hideout which she has fixed up for herself when she woke up from her winter sleep.
If cell Af has not previously been assigned as the information cell of a form in the text-form list, it is now assigned as the information cell of Af.
If of the founders of glottochronology Swadesh has escaped our steady plodding, and Lees has repudiated his own share in the founding, that is no reason why we should swerve.
If the transferor has substantial assets other than the claim, it seems reasonable to assume no corporation would be willing to acquire all of its properties in the dim hope of collecting a claim for refund of taxes.
If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.
If the distant patron of the suburban branch has been frightened away from downtown by traffic problems, however, the city store can only pressure the politicians to do something about the highways or await the completion of the federal highway program.
If a statement has been assigned an address in the index word area
If she has not had such experiences, the female's normal adolescent degree of indecision will be compounded.
If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.
If Barnett doesn't call a special session in 1961, it will be the first year in the last decade that the Legislature has not met in regular or special session.
If the church has followed the plan of cultivation of prospects and carried through a program of membership preparation as outlined earlier in this book, the process of assimilation and growth will be well under way.
If and additional
If we examine the three types of change from the point of view of their internal structure we find an additional profound difference between the third and the first two, one that accounts for the notable difference between the responses they evoke.
If your computation on Form 1040 or Form 1040A shows you owe additional tax, it should be remitted with your return unless you owe less than $1, in which case it is forgiven.
If you file Form 1040A and the District Director computes your tax, you will be sent a bill if additional tax is due.
If you would feel happier with full collision insurance, there is a small additional charge, again varying from country to country and depending on the term of such insurance.
If the trend of general business activity follows the pattern suggested here, we are likely to see additional steps by the Federal Reserve authorities to ease the availability of credit.
If an alpha channel is used in an image, it is common to also multiply the color by the alpha value, to save on additional multiplications during compositing.
If the universe is flat, there must be an additional component making up 73 % ( in addition to the 23 % dark matter and 4 % baryons ) of the energy density of the universe.
If a customer uses more than 300 GB a month, Bell will also be able to implement an additional charge of 75 cents per gigabyte.
If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.
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 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 additional solute is added to a saturated solution, it will not dissolve, except in certain circumstances, when supersaturation may occur.
If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation.
If available, inhabitants may take potassium iodide at the rate of 130 mg / day per adult ( 65 mg / day per child ) as an additional measure to protect the thyroid gland from the uptake of dangerous radioactive iodine, a component of most fallout and reactor waste.
If Language A borrowed a word from Language B, or both borrowed the word from a third language or inherited it from a common ancestor, and later the word shifted in meaning or acquired additional meanings in at least one of these languages, a native speaker of one language will face a false friend when learning the other.
If we compress data in a manner that assumes q ( X ) is the distribution underlying some data, when, in reality, p ( X ) is the correct distribution, the Kullback – Leibler divergence is the number of average additional bits per datum necessary for compression.
If a merchant received a loan to fund his shipment, he would pay the lender an additional sum in exchange for the lender's guarantee to cancel the loan should the shipment be stolen or lost at sea.
If the items are stored in a linked list, then the list can be sorted with O ( 1 ) additional space.
If convicted of these charges, Lindh could have received up to three life sentences and 90 additional years in prison.
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 a particular KR is inadequate in some way, that set of problematic KR elements may be transformed by importing them into a KBS, modified and operated on to eliminate the problematic elements or augmented with additional knowledge imported from other sources, and then exported into a different, more appropriate KR.
If there is an additional magnitude of value or a loss of value after transformation compared with before then the relation between values ( proportional to labor ) and prices ( proportional to total capital advanced ) is incomplete.
If no additional energy is supplied by any other mechanism, the nucleus will not fission, but will merely absorb the neutron, as happens when U-238 absorbs slow and even some fraction of fast neutrons, to become U-239.
If each additional layer contains less oil, the final painting will crack and peel.
If the bidder makes more or fewer tricks than were bid, he / she loses 10 points for the first over or under trick + 15 points for each additional over or under trick.
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