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If and number
If in any one calculation Ptolemy had had to invoke 83 epicycles all at once, while Copernicus never required more than one third this number, then ( in the sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.
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 a nation wished to get a head start in physical fitness over all other nations, it would start its kindergarten students on a program of gymnastics the day they entered and thus eliminate a large number of the problems that plague American schools.
If no specific organization plan exists limiting the number of scientists at each salary level, the result is a department top-heavy with high-level, high-salaried personnel ''.
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 a wedge-shaped coating of increasing thickness is removed from a substrate by an instrument like the Hesiometer with a knife of constant rake angle, a number of removal mechanisms are often observed which depend upon the thickness of the coating.
If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.
If Rock shows up during the number -- or you hear anything -- give me the signal ''.
If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.
If so, it might be worth while to assign a future jazz show to a different department -- one with enough confidence in the musical material to cut down on the number of performers and give them a little room to display their talents.
* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.
If its minimal polynomial has degree, then the algebraic number is said to be of degree.
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 >.
* 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 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 ).
If a ) testimonies conflict one another, b ) there are a small number of witnesses, c ) the speaker has no integrity, d ) the speaker is overly hesitant or bold, or e ) the speaker is known to have motives for lying, then the epistemologist has reason to be skeptical of the speaker's claims.
If we take in our hand any volume ; of divinity or school metaphysics, for instance ; let us ask, Does it contain any abstract reasoning concerning quantity or number?
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 the two carbon atoms at the double bond are linked to a different number of hydrogen atoms, the halogen is found preferentially at the carbon with fewer hydrogen substituents ( Markovnikov's rule ).
If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.
If there is no number to its left, simply look at the column headed " 1 " in the previous row.
If the game has been scored correctly, the total number of plate appearances for a team should equal the total of that team's runs, men left on base, and men put out.
If the other player also knows to offer sacrifices, the expert also has to manipulate the number of available sacrifices through earlier play.

If and byes
If adopted, the two higher-placed clubs in the play-offs would have enjoyed first-round byes and home advantage in the semi-finals.
If runs are to be scored as byes, the umpire will hold up one open palm above the head.
If the wicket-keeper misses the ball and it travels all the way to the boundary, the batting team scores four byes, just as if the batsman had hit the ball to the boundary for four runs.
If the ball was a no ball, any extras scored that would otherwise be byes are instead recorded as no balls, not as byes.
If the ball was a wide, again any extras are scored as wides and not as byes.
If the ball reaches the boundary, whether or not the batsmen ran, then four byes are awarded.
If ( with the same proviso's ) the ball reaches the boundary, whether or not the batsmen ran, then four leg byes are awarded.
If the ball strikes a hand which for whatever reason is not holding the bat, then leg byes may be scored.
If the ball deflects off the batsman's body and travels all the way to the boundary, the batting team immediately scores four leg byes, similar to if the ball had been hit to the boundary for a four.
If the batsman leaves the ball by not offering a shot, no runs are recorded, unless the ball reaches the boundary and four byes are recorded.

If and causes
If placing a tile causes three or four chains to merge, then the merger steps are handled between the largest and second-largest chain, then with the third-largest chain, and finally with the smallest chain.
If the wind blows into the mouth of a tube it causes an increase of pressure on one side of the manometer.
* If a player's injury causes the play-clock to stop, the player must sit out at least one play ; and
** If a team is penalized in the final minute of a half and the penalty causes the clock to stop, the opposing team now has the right to have 10 seconds run off the clock in addition to the yardage penalty.
If the company's stock is publicly traded, a Chapter 11 filing generally causes it to be delisted from its primary stock exchange if listed on the New York Stock Exchange, the American Stock Exchange, or the NASDAQ.
If these molecules are biological molecules in DNA, this causes lasting damage.
If there have been declines in national IQs this may have causes other than those proposed above.
If the pressure is steady, there is no more stimulus ; thus, typically these neurons respond with a transient depolarization during the initial deformation and again when the pressure is removed, which causes the corpuscle to change shape again.
If a supercontinent covers a portion of the earth ’ s surface where heat from the mantle accumulates under the crust, it causes the land mass to dome upward.
If a previous sound change causes X, Y > Y ( features X and Y merge as Y ), a new one cannot affect only original X's.
If food or liquid does enter the trachea and contacts the vocal folds because of a failure of this safeguard (" going down the wrong pipe "), it causes a cough reflex to expel the matter in order to prevent choking.
If the null hypothesis that is rejected, then the alternative hypothesis that and equivalently that causes y cannot be rejected.
If the impact causes the head to move, the injury may be worsened, because the brain may ricochet inside the skull causing additional impacts, or the brain may stay relatively still ( due to inertia ) but be hit by the moving skull ( both are contrecoup injuries ).
If competitors ' products are not interoperable ( due to causes such as patents, trade secrets or coordination failures ), the result may well be monopoly or market failure.
If the aquifer is confined by low-permeability layers, the reduced water pressure in the sand and gravel causes slow drainage of water from the adjoining confining layers.
If Parkinson's disease has been excluded, the differential diagnosis or list of potential causes for Parkinsonism syndrome includes:
If Imatinib doesn ’ t work, patients with advanced chronic myelogenous leukemia can use nilotinib, another inhibitor to the malfunction enzyme that causes the leukemia.
If a robbery is foiled before it can be completed, an alternative offence under section 8 ( 2 ) of the 1968 Act is assault ; any act which intentionally or recklessly causes another to fear the immediate and unlawful use of force, with an intent to rob, will suffice.
If cerebral edema is suspected because of confusion, recurrent vomiting or other symptoms, computed tomography may be performed to assess its severity and to exclude other causes such as stroke.
If these impedances are not equal, it causes a condition called standing waves, in which the power is reflected back from the antenna toward the transmitter, wasting power and sometimes overheating the transmitter.
If alternative causes have been eliminated, steroids or immunosuppressants ( such as azathioprine ) may be considered in this scenario.
If these fail to join up it causes a cleft palate.
If the motion goes far enough, this motion causes the lever arms to pull down on a thrust bearing, which moves a beam linkage, which reduces the aperture of a throttle valve.
The Council of Trent taught that only the priest who celebrated Mass was bound by divine law to receive communion under both species, and that Christ, whole and entire, and a true sacrament are received under either form alone, and therefore, as regards its fruits, those who receive one species only are not deprived of any grace necessary to salvation "; and it decreed: " If anyone says that the holy Catholic Church was not moved by just causes and reasons that laymen and clerics when not consecrating should communicate under the form of bread only, or has erred in this, let him be anathema.
If the input signal causes the control grid to become positive ( where current flow begins ), nonlinearity is to be expected.

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