Thus, it's really only necessary to encode three of the four possibilities as dummy variables.

Thus, the " dummy " is always first or last to play in the 2-player version.

Thus, an interaction dummy ( product of two dummies ) can alter the dependent variable from the value that it gets when the two dummies are considered individually.

Thus, it is important to recognize the situations when a trump opening lead is called for – usually, they arise when both declarer and dummy have bid other suits but found the trump fit in the third one.

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

Thus, variable only depends directly on variable, rather than depending on the entire set of variables.

Thus, the variable named ' Car ' can have subscripts " Door ", " Steering Wheel " and " Engine ", each of which can contain a value and have subscripts of their own.

Thus a variable which can store up to nine digits would be " packed " into 5 bytes.

Thus in determining whether to shut down a firm should compare total revenue to total variable costs ( VC ) rather than total costs ( FC + VC ).

Thus a is considered to be a parameter: it is less variable than the variable x or y, but it is not an explicit constant like the exponent 2.

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 it was suggested that Vega showed occasional low-amplitude pulsations associated with a Delta Scuti variable.

Thus the concept of a random function is one example of a random element and hence is a generalization of the simpler idea of a random variable.

Thus a polynomial in one variable x can be written as

Thus according to the central limit theorem, we can model jitter as a gaussian random variable.

Thus an algorithm to find isolated real roots of a low-degree polynomial in one variable may bear little resemblance to an algorithm for complex roots of a " black-box " function which is not even known to be differentiable.

Thus, the German Historical School focused on specific dynamic institutions as the largest variable in changes in political economy.

Thus, if more than 1 qualitative variable is included in the regression, it is important to note that the omitted category should be chosen as the benchmark category and all comparisons will be made in relation to that category.

Thus there are many sorts of quantifiers, two for each sort of variable.

Thus I ( X ; X ) ≥ I ( X ; Y ), and one can formulate the basic principle that a variable contains at least as much information about itself as any other variable can provide.

Thus the two-dimensional Fourier transform of the initial function is the one variable Fourier transform of the Radon transform of that function.

Thus in an image processing context the original image ƒ can be recovered from the ' sinogram ' data Rƒ by applying a ramp filter ( in the variable ) and then back-projecting.

Thus, research into alien Overtechnology culminated in the development of the variable fighters for the U. N. Spacy.

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: