Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.

Thus the initial assumption that p and q − 1 are relatively prime is untenable.

Thus there is no way of discerning which, if any, ethical properties exist ; by Occam's Razor, the simplest assumption is that none do.

Thus we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.

Thus, " assumption " connotes other associations than the contemporary standard sense of “ that which is assumed or taken for granted ; a supposition, postulate ” ( only the 11th of 12 senses of “ assumption ,” and the 10th of 11 senses of “ assume ”).

Thus it seems that the previous assumption of a close relationship between Tylopoda and ruminants is simply due to the fact that all other close relatives ( whales, pigs etc ) are so divergent in their adaptations as to have obsured most indications of relationship, or at least those visible to phenetic analyses.

where is the point biserial correlation of item i. Thus, if the assumption holds, where there is a higher discrimination there will generally be a higher point-biserial correlation.

It makes a bold and positive assertion and therefore requires warrant in excess of that which attends the Molinist assumption that there are true counterfactuals about creaturely free actions .” Latter, Craig points out “ Anti – Molinists have not even begun the task of showing that counterfactuals of creaturely freedom are members of the set of propositions or statements which require truth – makers if they are to be true .” Thus the grounding objector must prove a universal negative regarding the falsity of counterfactuals of freedom or he must explain his theory of the basis for truth and prove that theory true.

Thus, requiring that the DDH assumption holds in a group is a more restricting requirement.

Thus, if one is going to make that assumption, then one would want to use the simplest definition.

Thus, the user's assumption of a one-to-one correspondence between the visual appearance of a name and the named entity breaks down.

Thus the common assumption that Mozart's grave was unmarked because he was too poor is false.

Thus all differences were leveled, and all contrasts erased, in a realm of no distinction, and the harmonious balance of the Lo Shu square could effectively symbolize the world in balanced harmony around a powerful central axis.

Thus, the emperor could draw on sources not available to those with less puissant ancestors.

Thus he maintained that there must be blackness as well as whiteness in snow ; how otherwise could it be turned into dark water?

Thus, the above conditions could be re-written as follows:

Thus, services could be moved to a different machine and, so long as they kept the same service name, there was no need for users to do anything different to continue accessing the service.

Thus they could be used in place of the more expensive Athlon MP in dual socket A motherboards.

Thus, the new charter changed the manner in which the military could exercise its moderating power.

Thus, small minority parties cannot easily enter the Bundestag and prevent the formation of stable majority governments as they could under the Weimar constitution.

Thus, in the Hindu schools, if a claim was made that could not be substantiated by appeal to the textual canon, it would be considered as ridiculous as a claim that the sky was green and, conversely, a claim which could not be substantiated via conventional means might still be justified through textual reference, differentiating this from the epistemology of modern science.

Thus the view from Whitehall early in 1916: If defeat was not imminent, neither was victory ; and the outcome of the war of attrition on the Western Front could not be predicted.

Thus, only the immediate purchaser could recover for a product defect, and if a part was built up out of parts from parts manufacturers, the ultimate buyer could not recover for injury caused by a defect in the part.

Thus, Confucius could not achieve the idealistic reform that he wanted and restore the legitimate rule of the duke, returning to the period of the Duke of Zhou.

Thus, a catalyst that could change the equilibrium would be a perpetual motion machine, a contradiction to the laws of thermodynamics.

Thus a construction of the following type could conceivably arise: " It nearly knocked me off me plates — he was wearing a syrup!

Thus, in Wilson v. U. S. ( 1911 ), he asserted that corporate officers could not resist a subpoena for company records by invoking the Fifth Amendment's privilege against self-incrimination.

Thus, in the event of their death, their estate could pass to a competent heir.

Thus, doublespeak might have theories that have parallels with these theories and could possibly help us to understand the theories underlying doublespeak better.

Thus, an emissary to the Ottoman Empire could expect to be arrested and imprisoned upon the outbreak of hostilities between their State and the empire.

Thus, SETE could not claim copyright on photographs or panoramas of Paris incorporating the lit tower.

Thus, churches actually had money-changing tables at which each coin would be examined separately and a token of actual worth given to the layperson so that he or she could be seen by the other parishioners as putting money in the basket during that part of the service.

Thus, the Audion could not only detect and amplify radio signals, it could transmit them as well.

Thus, since the 19th century many historical Jesus scholars have argued that only one of the two traditions could be authentic.

Thus, to cite but one example, the Pax Britannica of the nineteenth century, whether with the British navy ruling the seas or with the City of London ruling world finance, was strictly national in motivation, however much other nations ( e.g., the United States ) may have incidentally benefited.

Thus, the avant-garde choreographers have extended the scope of materials available for dance composition.

Thus he complains, with considerable justice, that the Tory writers have resorted to libel instead of answering his arguments.

Thus we will have a society consisting of the planners or conditioners, and the controlled.

Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.

Thus far the advances made have been almost entirely along functional lines.

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 have no part, and want none, in current discussions of the relative importance of science, the social studies, the humanities, the creative arts.

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, 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 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 in Mary wrote an account of the trip first strong stress on Mary marks Mary as the first in a series of people who wrote accounts of the trip, strong stress on wrote marks the writing as the first of a series of actions of Mary's concerned with an account of her trip ( about which she may later have made speeches, for example ), and strong stress on trip makes the trip the first of a series of subjects about which Mary wrote accounts.

Thus when yes, I have is the response to have you finished reading the paper??

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 far, the cases which have come before the courts have involved only the issue of referral where the job is vacant due to a strike -- condition ( 1 ) in the Regulation of the Secretary.

Thus he will be in a position to disabuse the Soviet leader of any notions he may have about grave Allied disunity.

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, in `` The Story Of Ruth '' we have Ruth, Naomi, and Boaz and sets that are meticulously authentic.

Thus, although some things may be certain, they have little to do with Dasein's sense of care and existential anxiety, e. g., in the face of death.

Thus, over the past 1, 500 years, some Christians have used the term Arian to refer to those groups that see themselves as worshiping Jesus Christ or respecting his teachings, but do not hold to the Nicene creed.

Thus we have the De officiis ministrorum, De viduis, De virginitate and De paenitentia.

Thus we have the story of his riding across a stick ( horse made of stick ) with his children and upon being discovered by a friend desiring that he not mention till he himself were the father of children ; and because of the affection of his son Archidamus ' for Cleonymus, he saved Sphodrias, Cleonymus ' father, from execution for his incursion into the Piraeus, and dishonorable retreat, in 378 BC.