Thus, Margenau remarks: `` A large number of unrelated epicycles was needed to explain the observations, but otherwise the ( Ptolemaic ) system served well and with quantitative precision.

Thus, in the last few years, a number of programs which looked very promising at the time their development was commenced have since been completely eliminated.

Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.

Thus the multiplicity of Af for a given T must be an even number.

Thus, casework involving a limited number of interviews is still to be regarded in terms of the quality of service rendered rather than of the quantity of time expended.

Thus the middle number, 5, represented Sung-Shan in Honan, Central China ; ;

Thus a simple count of the number of distinct symbols is an important clue to the nature of an unknown script.

Thus although most species in the order are herbaceous, some no more than 15 cm high, there are a number of climbers ( e. g. some species of Asparagus ), as well as several genera forming trees ( e. g. Agave, Cordyline, Yucca, Dracaena ), some of which can exceed 10 m in height.

Thus, almost all positive integers are composite ( not prime ), however there are still an infinite number of primes.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

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 ).

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Thus, when the fluid finally does strike the object, it is forced to change its properties -- temperature, density, pressure, and Mach number -- in an extremely violent and irreversible fashion called a shock wave.

Thus, in the mid-1990s the army sought to reduce the number of active-duty generals.

Thus the discovery report by the Berkeley group reads: " It is suggested that element 97 be given the name berkelium ( symbol Bk ) after the city of Berkeley in a manner similar to that used in naming its chemical homologue terbium ( atomic number 65 ) whose name was derived from the town of Ytterby, Sweden, where the rare earth minerals were first found.

Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

Thus, a PICMG backplane can provide any number and any mix of ISA, PCI, PCI-X, and PCI-e slots, limited only by the ability of the SBC to interface to and drive those slots.

Thus, a number of Allied soldiers received the Bronze Star Medal in World War II, as well as U. N. soldiers in the Korean War, Vietnamese and allied forces in the Vietnam War, and coalition forces in recent military operations such as the Gulf War, Operation Enduring Freedom and the Iraq War.

Thus, all carbon isotopes have nearly identical chemical properties because they all have six protons and six electrons, even though carbon atoms may differ in number of neutrons.

Thus, a poison that covers surface randomly will tend to reduce the number of uncontaminated large planes but leave proportionally more smaller sites free, thus changing the hydrogenation vs. hydrogenolysis selectivity.

Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space.

Thus, the three main categories of antisubmarine warfare operations are defense of shipping, defense of naval forces, and area defense.

Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system.

Thus, for the general, non-associative case, binary operations can be represented with binary trees.

Thus the record and array data structures are based on computing the addresses of data items with arithmetic operations ; while the linked data structures are based on storing addresses of data items within the structure itself.

Thus, many typical household applications already in existence like word processing and email were designed specifically to avoid using floating point operations.

Thus, cloud ceiling and flight visibility are the most important variables for safe operations during all phases of flight.

Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

Thus the retina performs operations on the image to enhance the edges of objects within its visual field.

Thus, a distinction must be drawn between peacekeeping and other operations aimed at peace.

Thus, for example,, which is just the statement that and are the only two ways of pairing three elements of a magma with two operations.

Thus the Solid State depended, at the heart of its operations, on a vacuum tube.

Thus the Orthodox emphasis would be that Christ died, not to appease an angry and vindictive Father, or to avert the wrath of God, but to change people so that they may become divine, that is to say, become God in his energies or operations but not in his essence or identity as God ( see theosis ).

Thus, MNCs are likely to adapt production processes in many of their operations to conform to the standards of the most rigorous jurisdiction in which they operate ( this tends to be either the USA, Japan, or the EU ).

Thus, many other party leaders sought sanctuary in exile in India to set up operations against the new system.

Thus, the deaths of Ögedei and Möngke in 1241 and 1259, respectively, necessitated the withdrawal of Mongol leaders ( and troops ) from the outskirts of Vienna and Venice ( in 1241 ) and from Syria ( in 1259 ), hamstringing military operations against the Austrians and Mamluks that might otherwise have continued.

Thus any algorithm that can be expressed primarily as operations on arrays and matrices can run almost as quickly as the equivalent C code.

Thus it is a chapter in a larger narrative that includes many ideas and fields, from the folk wisdom of thrift to a profusion of applied-science successors, including time and motion study, the Efficiency Movement ( which was the broader cultural echo of scientific management's impact on business managers specifically ), Fordism, operations management, operations research, industrial engineering, manufacturing engineering, logistics, business process management, business process reengineering, lean manufacturing, and Six Sigma.

Thus in general a schedule is a partial order of operations, containing ( embedding ) the partial orders of all its transactions.

Thus the amount of time taken and the number of elementary operations performed by the algorithm differ by at most a constant factor.

Thus we can write when describing the intrinsic operations: