Thus for map projections we must introduce the concept of point scale, which is a function of position, and strive to keep its variation within narrow bounds.

Thus the variance is a measure of the amount of variation of the values of that variable, taking account of all possible values and their probabilities or weightings ( not just the extremes which give the range ).

Thus, variation 3 is a canon at the unison, variation 6 is a canon at the second ( the second entry begins the interval of a second above the first ), variation 9 is a canon at the third, and so on until variation 27, which is a canon at the ninth.

Thus, knowledge of psychology is incomplete without knowledge of variation.

Thus, the force on a particle due to a scalar potential,, can be interpreted as a Lagrange multiplier determining the change in action ( transfer of potential to kinetic energy ) following a variation in the particle's constrained trajectory.

Thus, wavelength ratios observed for different ionic species in the absorption spectra of quasi stellar objects ( QSO or quasars ) place a limit on any variation in the fine-structure constant to less than 1 part in 1 million out to a distance in space ( and time ) of z = 3 ( about 6500 megaparsecs or 11. 5 billion light years ); as the fine-structure constant is determined by the relation between the speed of light ( c ), Planck's constant ( h ) and the electron charge ( e ), these physical constants are constrained as well.

Thus even a small variation in the eye spacing is unusual and should be exaggerated, whereas a correspondingly small change in the nose size relative to the mean would not be unusual enough to be worthy of exaggeration.

Thus the motion of no confidence is required to be at the same time as a motion of confidence for a new candidate ( this variation is called a constructive vote of no confidence ).

Thus, despite the occasional revival by GMs Vladimir Akopian, Vladimir Malaniuk and Jeroen Piket, this variation is also unusual.

Thus the structure displays a marked variation in architectural styles from Aibak to that of Tughlaq dynasty.

Thus scene two is a variation on a single note ( B natural, heard continuously in the scene, and the only note heard in the powerful orchestral crescendos at the end of act two, scene two ); scene three is a variation on a rhythmic pattern, with every major thematic element constructed around this pattern ; scene four is a variation on a chord, used exclusively for the whole scene ; the orchestral interlude is a freely composed passage that is firmly grounded in the key of d minor ; the final scene is a moto perpetuum, a ' variation on a single rhythm ' ( the quaver ).

Thus, there is an immense potential of variation of the aforementioned subunits.

Thus, robustness reduces the amount of heritable genetic variation on which selection can act.

Thus varying diameters are used for the new tones for variation.

Thus, in each family, the potential for variation would tend to narrow, quite dramatically, with each generation, and, so it would go for the entire population with every trait.

Thus, even though variations that arise in response to altered conditions are environmental and non-heritable ( contrary to Darwin's assumptions ), some of the continuous variation in natural species could have a genetic basis, and could serve as the Mendelian basis for Darwinian gradualism.

Thus, this hypervariable region of VP2 is the obvious target for the molecular techniques applied for IBDV detection and strain variation studies.

Thus, he effectively uses a variation of the post-Crisis continuity whereby the Golden and Silver Age heroes both exist in a unified timeline rather than separate realities.

Thus a company may vary pricing by location, but then offer bulk discounts as well.

Thus, there are bulk discounts, special pricing for long-term commitments, non-peak discounts, discounts on high-demand goods to incentivize buying lower-demand goods, rebates, and many others.

Thus pricing is very important in marketing.

Thus, it can be said safely that usage of average costs for pricing may be considered a reasonable approximation of marginal cost decision making.

Thus knowing how to price caps is also sufficient for pricing swaptions.

Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

Thus, when you have prepared your foundation and laid the floor, these can be trucked to the site and erected with a small crew of friends in a weekend.

Thus the film can be applied to back-lighted translucent plastics faces ; ;

Thus, direct comparisons can be drawn with free burning arcs which have been studied in detail during the past years and decades by numerous investigators ( Ref. 3 ).

Thus, for aqueous media, we can think of the idealized organic active as an oleophilic or hydrophobic surface-active agent, and of an idealized builder as a oleophobic or hydrophilic surface-active agent.

Thus, the larvae of the intruder can develop at the expense of the rightful inhabitants and the store of beebread.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus, one can think of a dictionary entry as a word rather than a form.

Thus the unstressed it of it rarely snows here gets its significance from its use with snows: nothing can snow snow but `` it ''.

Thus in it's incredible what that boy can eat dominant stress is likely to be on incredible, and eat will have strong stress also.

Thus, besides the training provided to youth in school, the existence of the school program can have supplementary benefits to industry which make it an asset to industrial development efforts.

Thus, for non-negative changes in the basic wage rate, the industry becomes the active wage-setter, since any increase in the basic wage rate can occur only by reason of industry acquiescence.

Thus if the gyro and platform-controller combination maintains the platform with zero angular deviation about the **yf axis, the system can be rotated with an angular velocity Af if a torque is supplied to the gyro output axis Aj.

Thus a man who is butting a stone wall at the office may become unusually aggressive in bed -- the one place he can still be champion.

Thus, man can readily deduce that the primary objective end of the conjugal act is procreation, the propagation of the race.

Thus, while it remains possible that the Babylonians and/or the Pythagoreans may perhaps have had the magic square of three before the Chinese did, more definite evidence will have to turn up from the Middle East or the Classical World before China can lose her claim to the earliest known magic square by more than a thousand years.

" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.

Thus an algorithm can be an algebraic equation such as y

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

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.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

Thus areas can be measured in square metres ( m < sup > 2 </ sup >), square centimetres ( cm < sup > 2 </ sup >), square millimetres ( mm < sup > 2 </ sup >), square kilometres ( km < sup > 2 </ sup >), square feet ( ft < sup > 2 </ sup >), square yards ( yd < sup > 2 </ sup >), square miles ( mi < sup > 2 </ sup >), and so forth.

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: