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 the hand value exceeds 21 points, it busts, and all bets on it are immediately forfeit.

If the ambient temperature exceeds T < sub > m </ sub >, the homologous pairing of the sticky ends would not be stable because the high temperature disrupts hydrogen bonding.

If the minibus exceeds 100 km / h, the beeping will turn into a sustained tone.

If the quantity of powder stored for personal use exceeds 75 kg, then a Propellant Magazine Licence ( Type P ) is required.

If the number of tricks exceeds the number of players, players who bid nil ( zero ) score 20 if they take no tricks, otherwise-10.

If the number of dies — the integrated circuits that will eventually become chips — etched on a wafer exceeds a failure threshold ( i. e. too many failed dies on one wafer ), the wafer is scrapped rather than investing in further processing.

If the mass of the core exceeds the Chandrasekhar limit, electron degeneracy pressure will be unable to support its weight against the force of gravity, and the core will undergo sudden, catastrophic collapse to form a neutron star or ( in the case of cores that exceed the Tolman-Oppenheimer-Volkoff limit ), a black hole.

If made very pure, tungsten retains its hardness ( which exceeds that of many steels ), and becomes malleable enough that it can be worked easily.

If the demand for power exceeds the supply, generation plants and transmission equipment can shut down which, in the worst cases, can lead to a major regional blackout, such as occurred in the US Northeast blackouts of 1965, 1977, 2003, and in 1996 and 2011.

If the sender does not receive an acknowledgment before the timeout, it usually re-transmits the frame / packet until the sender receives an acknowledgment or exceeds a predefined number of re-transmissions.

If the clearance from the nearest anticipated path position, over the expected range of Earth radius k-factor, exceeds 0. 6 of the first Fresnel zone radius, the feature is not normally considered an obstruction.

If the pressure exceeds design limits, the plug pops out to depressurize the pot.

If actual spending for a given year exceeds the spending target for that year, reimbursement rates are adjusted downward by decreasing the Conversion Factor ( CF ) for RBRVS RVUs.

If the energy absorbed exceeds this potential, then the electron is emitted as a free electron.

Recently, it has been suggested that patients be screened for iron overload using serum ferritin as a marker: If serum ferritin exceeds 1000 ng / mL, iron overload is very likely the cause.

If liability exceeds assets, negative equity exists.

If a population exceeds the size outlined by its cognitive limitations, the group undergoes a schism.

If their combined mass exceeds the Oppenheimer-Volkoff limit then the two will collapse into a black hole, resulting in a supernova that disperses the outer layers of the star.

If the thermal resistance of the fluid / sphere interface exceeds that thermal resistance offered by the interior of the metal sphere, the Biot number will be less than one.

If the number of errors within a code word exceeds the error-correcting code's capability, it fails to recover the original code word.

If the pressure exceeds the limit of the safety valve, water in the form of gas is lost.

If the POSITIVE balance exceeds the agreed terms, then additional fees may be charged and higher interest rates may apply.

If a train crew's route, or tour of duty, exceeds a single shift, or is in conflict with any rules pertaining to a legal or contractual limit to the number of hours that can be worked, more than one crew may be assigned, each with its own conductor, while onboard service crew members aboard passenger trains normally remain on duty for the entire run, including their assigned meal and sleep breaks.

If the torque applied to one of the drive wheels exceeds the threshold of traction, then that wheel will spin, and thus only provide torque at each other driven wheel limited by the sliding friction at the slipping wheel.

If the luminosity exceeds the Eddington limit, then the radiation pressure drives an outflow.