Then A is an algebra over K if the following identities hold for any three elements x, y, and z of A, and all elements (" scalars ") a and b of K:

Then, since span ker T, there exists a set of scalars such that:

Then by the Feynman rules, the correction ( from both scalars ) is:

Then the joint distribution of X and Y is completely determined by our channel and by our choice of, the marginal distribution of messages we choose to send over the channel.

Then it stalled, changed direction completely, became hurricane-strength and progressing west-to-east through the Caribbean Sea.

Then ρ will be the finest completely regular topology on X which is coarser than τ.

Then, completely confident in his own safety, the narrator comments on the sturdiness of the building and raps upon the wall he had built around his wife's body.

Then, carvers cut the parts with no character off the board with knives, so that the characters were cut in relief, completely different from those cut in intaglio.

Then, in 1960, Pope John XXIII issued the Code of Rubrics, completely ending the ranking of feast days by doubles etc., and replacing it by a ranking, applied not only to feast days but to all liturgical days, as I, II, III, and IV class days.

Then as a teenager, that completely changed.

Then Y is completely determined by X, so that X and Y are perfectly dependent, but their correlation is zero ; they are uncorrelated.

Then there was the completely unexpected, late night visit by Eartha Kitt.

Then, " in 1988, the Public Protection Committee unanimously recommended that the California legislature completely abolish the state's UPL laws.

Then late in the first half Catchings went down with another injury ; she had completely torn her Achilles tendon.

Then in 1959 he embarked on a completely new career and by 1963, now married with two children, he was a fully qualified glacier pilot, flying to remote and inaccessible icy areas-a vocation which brought him considerable fame until he finally stopped flying in 1987 at the age of 72.

Then, the sports society proceeded to completely omit all Yugoslav Army symbols from the crest.

Then they are left outside during hot, sunny days and cold, freezing nights until they become completely dehydrated.

Then the signal beam first passes through the center for maximal amplification, but in later passes the diameter is increased to stay below the damage-threshold, to avoid amplification the outer parts of the beam, thus increasing beam quality and cutting off some amplified spontaneous emission and to completely deplete the inversion in the gain medium.

Then some troops he had out raiding returned, fell upon the forces attacking Bugha and completely routed them.

Then in 1948 the Soviet edition of the book was scrapped completely.

Then ( 1 ) is completely integrable at each point of A × B if and only if

# Captain William Bainbridge, USN, reported the surrender of HM frigate Java to USS Constitution on 29 December 1812 by the following minutes taken during the action: " At 4. 5 Having silenced the fire of the enemy completely and his colours in main Rigging being Supposed he had Struck, Then hawl'd about the Courses to shoot ahead to repair our rigging, which was extremely cut, leaving the enemy a complete wreck, soon after discovered that The enemies flag was still flying hove too to repair Some of our damages.

Then it was completely destroyed when the asteroid exploded, right after Hulk, Angel, and Iceman managed to escape.

Then in April 1657, Blake completely destroyed another Spanish merchant convoy — the West Indian Fleet — that had stopped at Santa Cruz harbour on its return voyage.

Then she gave me a completely unrealistic eating plan which involved very little meat or fish and lots of food that disagreed with my system like avocado — which makes me sick — and cucumber.

Then each sentence of the draft SPM, displayed on a giant screen, was discussed at length by the delegates and often ended up completely rewritten.

Then they began to move, and once in retreat they fled for dear life … discipline was completely forgotten ".

Then it added: `` It is not possible to determine how extensive these ill effects will be -- nor how many people will be affected ''.

Then one must extract the principles, analogies and statements by various courts of what they consider important to determine how the next court is likely to rule on the facts of the present case.

Then they requested data from researchers, conducting analyses to determine which criteria required change, with instructions to be conservative.

Then follows a site investigation of soil, rock, fault distribution and bedrock properties on and below an area of interest to determine their engineering properties including how they will interact with, on or in a proposed construction.

Then, it is undecidable to determine whether the language decided by an arbitrary Turing machine lies in S.

Then each production department met to determine what each episode needed in terms of costumes, lights, and sets.

Then, detailed study was undertaken to determine the painting's original form, using scientific tests ( especially infrared reflectoscopy and microscopic core-samples ), and original cartoons preserved in the Royal Library at Windsor Castle.

Then in 2004, the Pacific League played five fewer games than the Central League teams during the regular season and used a new playoff format to determine its champion.

Then, he sought to determine in what contexts winks were used, and whether, as one moved about a region, winks remained meaningful in the same way.

Then the receiver samples at predefined intervals to determine the values of the bits in the byte ; this is plesiochronous since it depends on the transmitter to transmit at roughly the same rate the receiver expects, without coordination of the rate while the bits are being transmitted.

* Then, the BPI process owners should determine what outcomes would add value to the organization's objectives and how best to align its processes to achieve those outcomes ( TO-BE ).

Then we could equate F < sub > E </ sub > with w. Also, determining F < sub > E </ sub > proves difficult because the mass of the oil drop is difficult to determine without reverting back to the use of Stokes ' Law.

Then quantum mechanics allows us to determine the ψ function of the partial system B from the measurements made, and from the ψ function of the total system.

Then investigation to the foreign producer is conducted to determine if the allegation is valid.

Then a second and final phase of the tour opposed the first ( Rosewall ) and the second ( Laver ) of the first phase to determine the final winner ( the third ( Buchholz ) met the fourth ( Gimeno )).

This rule base would be searched and the third and fourth rules would be selected, because their consequents ( Then Fritz is green, Then Fritz is yellow ) match the goal ( to determine Fritz's color ).

Then the object's velocity is multiplied by this approximate light-time to determine its approximate displacement through space during that time.

Then having found the weight of the shortest Hamilton path, determine which edges are in the path by asking for each edge whether there is a Hamiltonian path with weight for the graph modified so that edge has weight ( if there is no such path in the modified graph, then edge must be in the shortest path for the original graph ).

Then the user can determine how much he is willing to pay in order to protect his / her business from those conditions in case they occurred based on his / her cost-benefit analysis and appetite for risk.

Then the performance of black, Hispanic, and low-income students was analyzed to determine whether these groups were performing better than average for similar students in the state.

Then, with the supervision of the umpires, a coin is tossed to determine who bats first.

Then, he / she will go through unspecified " tests " of their ability to determine what " evil " is, as well as election sessions in which a small handful of Group members will interview the candidate and assess his or her loyalty to the Group.

Then the craftsmen must determine the thickness, temper and alloy of the sheet he will use to make the tank.