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 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, 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, 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 at the same time that William Henry Harrison was preparing to pacify the aborigines of Indiana Territory and winning fame at the battle of Tippecanoe, Anglo-Saxon settlement made a great leap into the center of the North American continent to the west of the American agricultural frontier.

According to the Christian doctrine of Universal Reconciliation, the Greek New Testament scriptures use the word " eon " to mean a long period ( perhaps 1000 years ) and the word " eonian " to mean " during a long period "; Thus there was a time before the eons, and the eonian period is finite.

Thus, the time of the ion arrival can be used to infer the ion type itself, if the evaporation time is known.

Thus for an ion which traverses a 1 m flight path, across a time of 2000 ns, given an initial accelerating voltage of 5000 V and noting that one amu is 1 × 10 < sup >− 27 </ sup > kg, the mass-to-charge ratio ( more correctly the mass-to-ionisation value ratio ) becomes Z amu / charge.

Thus, in time, the bishop changed from being the leader of a single church confined to an urban area to being the leader of the churches of a given geographical area.

Thus, Walsh modified the vertical passing scheme he had learned during his time with the Raiders, designing a horizontal passing system that relied on quick, short throws-often spreading the ball across the entire width of the field.

Thus, by monitoring eye movements, we can study what information is being processed at a given time.

Thus, the most stable configuration of nuclei and electrons is one in which the electrons spend more time between nuclei, than anywhere else in space.

Thus, according to Aquinas, there must have been a time when nothing existed.

Thus, the first Olympic medals in curling, which at the time was played outside, were awarded for the 1924 Winter Games, with the gold medal won by Great Britain and Ireland, two silver medals by Sweden, and the bronze by France.

Thus, for example, the CGS unit of pressure, barye, is related to the CGS base units of length, mass, and time in the same way as the SI unit of pressure, pascal, is related to the SI base units of length, mass, and time:

Thus, Cyril followed his uncle in a position that had become powerful and influential, rivalling that of the prefect in a time of turmoil and frequently violent conflict between the cosmopolitan city's Pagan, Jewish, and Christian inhabitants.

Thus at any point in time most of their content resides in nonvolatile storage.

Thus the probability of two failures in a same RAID group in time proximity is much smaller ( approximately the probability squared, i. e., multiplied by itself ).

Thus query optimization typically tries to approximate the optimum by comparing several common-sense alternatives to provide in a reasonable time a " good enough " plan which typically does not deviate much from the best possible result.

Thus at any point in time most of their content resides in nonvolatile storage.

Thus the probability of two failures in a same RAID group in time proximity is much smaller ( approximately the probability squared, i. e., multiplied by itself ).

Thus query optimization typically tries to approximate the optimum by comparing several common-sense alternatives to provide in a reasonable time a " good enough " plan which typically does not deviate much from the best possible result.

Thus, when either field is changing in time, then a field of the other is necessarily induced.