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 a number of operations of the order of Af are required.

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 this class of molecules belong to the D < sub > 2d </ sub > point group.

Thus, molecules exist as electrically neutral units, unlike ions.

Thus, any real object cannot remain the same, also because of movement of its physical parts ( even much bigger than molecules ).

Thus when there are many molecules in the gas, there will be an equal number moving in one direction as any other hence demonstrating isotropy.

Thus, the molecules are free to flow and their center of mass positions are randomly distributed as in a liquid, but still maintain their long-range directional order.

Thus the molecules and atoms which make up the human body are vibrating, colliding, and moving.

Thus, IUPAC recommendations are more closely followed for simple compounds, but not complex molecules.

Thus, concentrated solutions of polymers are far rarer than those of small molecules.

Thus, the specific processes that can be probed with PET are virtually limitless, and radiotracers for new target molecules and processes are continuing to be synthesized ; as of this writing there are already dozens in clinical use and hundreds applied in research.

Thus to calculate the stoichiometry by mass, the number of molecules required for each reactant is expressed in moles and multiplied by the molar mass of each to give the mass of each reactant per mole of reaction.

Thus he deduced that the G-protein was a transducer that accepted glucagon molecules and affected the cell.

Thus the mechanical energy of the blades is transferred to the gas molecules.

Thus, heavy molecules are pumped much more efficiently than light molecules.

Thus, by definition, one mole of pure < sup > 12 </ sup > C has a mass of exactly 12 g. It also follows from the definition that X moles of any substance will contain the same number of molecules as X moles of any other substance.

Thus, the heat used to transform the working body of molecules from one state to the next cannot be used to do external work, e. g., to push the piston.

Thus, the molecules are " held in " the liquid more strongly when a second molecule is present.

Thus, in the limit of ideal gas behavior ( which many gases approximate except at low temperatures and / or extremes of pressure ) this property reduces differences in gas volumetric heat capacity to simple differences in the heat capacities of individual molecules.

Thus the heat capacity can be equivalently calculated as the product of the mass m of the body and the specific heat capacity c for the material, or the product of the number of moles of molecules present n and the molar specific heat capacity.

Thus, to have equal kinetic energies, the velocities of two different molecules would have to be in inverse proportion to the square roots of their masses.

Thus, the elastic modulus of the bilayer is primarily determined by how much extra area is exposed to water when the lipid molecules are stretched apart.

Thus, in order for gene transcription to be activated, the cell must encounter signaling molecules secreted by other cells in its environment.

Thus the field can be seen as a precision form of chemical engineering that includes protein engineering, the creation of protein molecules, a process that occurs naturally in biochemistry, e. g., prion reproduction.

Thus, lighter molecules have a higher speed.