Thus Af where the maximization is over all admissible policies Af, and Af is related to Af by ( 5 ).

Thus, when Dartmouth's Winter Carnival -- widely recognized as the greatest, wildest, roaringest college weekend anywhere, any time -- was broadcast over a national television hookup, Prexy John Sloan Dickey appeared on the screen in rugged winter garb, topped off by a tam-o'-shanter which he confessed had been acquired from a Smith girl.

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 Ambrose refused to be drawn into a false conflict over which particular local church had the " right " liturgical form where there was no substantial problem.

Thus in 1285 he declared war on his uncle, James II of Majorca, and conquered both Majorca ( 1285 ) and Ibiza ( 1286 ), effectively reassuming suzerainty over the Kingdom of Majorca.

Thus, the Hungarian supremacy over Halych disappeared.

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, over time, the expression " a cappella " ( Italian for " in the manner of the chapel ") came to mean exclusively vocal music in contradistinction to the spreading use of the organ in cathedrals.

Thus, if the score was 3 – 2 with the bases loaded in the bottom of the ninth, and the batter smacked an " over the fence home run ", the game would end at 4 – 3, with the batter only allowed a double, and the runners officially stopped on 2nd and 3rd ( since they were not needed to win the game ).

Thus, if a ball went over the fence fair, and curved behind the foul pole, it was ruled foul.

Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, i. e. acting on every point in it.

Thus the CSF turns over about 3. 7 times a day.

Thus a spacecraft can determine its relative position by tracking such asteroids across the star background, which appears fixed over such timescales.

Thus, we should prioritize social reform over attempts to account for consequences, individual virtue or duty ( although these may be worthwhile attempts, provided social reform is provided for ).

Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.

Thus, before the Aswan High Dam was completed in 1971, the White Nile watered the Egyptian stretch of the river throughout the year, whereas the Blue Nile, carrying seasonal rain from Ethiopia, caused the Nile to overflow its banks and deposit a layer of fertile mud over adjacent fields.

Thus, over time we would expect each traditional song to become aesthetically ever more appealing — it would be collectively composed to perfection, as it were, by the community.

Thus John asserts Jesus ' innate superiority over all divine messengers, whether angels or prophets.

Thus methane has a potential of 25 over 100 years but 72 over 20 years ; conversely sulfur hexafluoride has a GWP of 22, 800 over 100 years but 16, 300 over 20 years ( IPCC TAR ).

Thus, if Adar was over and Spring had not yet arrived, an additional month was observed.

Thus they were involved in the struggle for power within France as princes but they also had a religious authority over Roman Catholicism in France as King.

Thus, the writer decided to hold one experimental section of the functional preparation for marriage course in the spring semester of 1960 exclusively for persons already married -- that is, prerequisite: `` marriage ''.

Thus, a robust surveillance system involving human clinicians and veterinarians may identify a bioweapons attack early in the course of an epidemic, permitting the prophylaxis of disease in the vast majority of people ( and / or animals ) exposed but not yet ill.

Thus there were two classes of baccalarii: the baccalarii cursores, theological candidates passed for admission to the divinity course ; and the baccalarii dispositi, who, having completed this course, were entitled to proceed to the higher degrees.

Thus a description of the cat during the course of the experiment — having been entangled with the state of a subatomic particle — becomes a " blur " of " living and dead cat.

Thus, the wise course on the eleventh spin would be to bet on red, not on black: exactly the opposite of the layman's analysis.

Thus, while dialectical methods are necessary to find truth in theoretical matters, rhetorical methods are required in practical matters such as adjudicating somebody's guilt or innocence when charged in a court of law, or adjudicating a prudent course of action to be taken in a deliberative assembly.

Thus began one of the most famous apprenticeships in the musical theatre, as Hammerstein designed a kind of course for Sondheim on the construction of a musical.

Thus, a robust surveillance system involving human clinicians and veterinarians may identify a bioweapons attack early in the course of an epidemic, permitting the prophylaxis of disease in the vast majority of people ( and / or animals ) exposed but not yet ill.

Thus the majority of the Commonwealth's prime ministers agreed that there was " no alternative to course ( 3 )".

" Thus more embassies were dispatched to Anxi, Yancai later joined the Alans, Lijian under the Seleucids, Tiaozhi, and Tianzhu India … As a rule, rather more than ten such missions went forward in the course of a year, and at the least five or six.

Thus the value of each team fluctuates during the course of the betting.

Thus in theory, ribavirin might remain a reproductive hazard for as long as 6 months after a course of the drug has ended.

Thus, the counterfactual brings into sharp relief the question of how important Hitler was as an individual and how his personal fate shaped the course of the war and, ultimately, of world history.

Thus, while supporting the extinction periodicity, it appears to be inconsistent with the Nemesis hypothesis, though of course not inconsistent with other kinds of dark stellar companions.

Thus, the most obedient were led ( usually by Spider Woman ) to the next higher world, with physical changes occurring both in the people in the course of their journey, and in the environment of the next world.

Thus, for example, a government budget that is balanced over the course of the business cycle is considered to represent a neutral fiscal policy stance.

" Thus it was held that notwithstanding the right to an appeal, if the situation disclosed be such that to take the ordinary course by appeal would of itself subject the complainant to irreparable loss, the writ should issue notwithstanding no objection was made below ; that the matter of judicial courtesy should yield to substantial personal rights of litigants, such as a sacrifice of their liberty.

Thus during the session 1778 – 1779, in addition to his mathematical work, he delivered an original course of lectures on morals.

Thus in the golf course example, both the recreation and the negative impacts from deforestation, associated habitat and biodiversity loss, and pesticide toxicity would be estimated across those generations and some amortization applied to determine whether the golf course was a public benefit or a public bad from the point of view of that seventh generation.

Thus, Hébert was one of the proponents of " parcours ", an obstacle course, which is now standard in the military training and led to the development of civilian fitness trails and confidence courses.

Thus it was that the Portuguese ship, having been blown off course from China to Okinawa made their way to Tanegashima, and not directly to Japan proper.

Thus, the formal instruction of law as a course of study did not commence until the middle of the 19th century.