Thus in 565 or 566 Justinian's successor Justin II sent his son-in-law Baduarius as magister militum ( field commander ) to lead a Byzantine army against Alboin in support of Cunimund, ending in the Lombards ' complete defeat.

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, 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 field protection may be the primary mechanism that saved the population, but it is likely that field protection would not have been increased without CITES protection.

Thus the CGS system never gained wide general use outside the field of science.

Thus, the divergence of the vector field F can be expressed as:

Thus a field is a particular type of skew field, and not all skew fields are fields.

Thus, a field due to any particular particle or time-varying electric or magnetic field contributes to the fields present in the same space due to other causes.

Thus, a changing electric field generates a magnetic field ; conversely a changing magnetic field generates an electric field.

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

In the field of high-energy particle physics, the fundamental velocity unit is the speed of light c. Thus, dividing energy in eV by the speed of light in vacuum, one can describe the momentum of an electron in units of eV / c.

Thus, the electromagnetic field may be viewed as a dynamic entity that causes other charges and currents to move, and which is also affected by them.

Thus, this law can be applied to generate a magnetic field and run an electric motor.

Thus, a piece of iron in its lowest energy state (" unmagnetized ") generally has little or no net magnetic field.

Thus, the whole war could be decided by a single field battle ; victory was enforced by ransoming the fallen back to the defeated, called the " Custom of the Greeks ".

Thus, the Jones vector represents ( relative ) amplitude and ( relative ) phase of electric field in x and y directions.

Thus, even when the applied field is removed, the electrons in the material maintain a parallel orientation.

In particular, one should think of the condition number as being ( very roughly ) the rate at which the solution, x, will change with respect to a change in b. Thus, if the condition number is large, even a small error in b may cause a large error in x.

Thus the Sidewinder team replaced a potentially complex control system with a simple mechanical solution.

Thus, using Euler's theorem we can say that the solution must be of the form:

Thus, in such systems, the Demiurge acts as a solution to the problem of evil.

Thus, a solution requires that no two queens share the same row, column, or diagonal.

Thus, in such systems, the Demiurge acts as a solution to the problem of evil.

Thus, while he recognised the benefits of citrus fruit ( although he weakened the effect by switching to a boiled concentrated or " rob ", the production of which unfortunately destroyed the vitamin C ), he never advocated citrus juice as a single solution.

Thus, Olbers ' paradox can not decide between a finite ( e. g. some variants of the Big Bang model ) and an infinite ( Steady State theory or static universe ) solution.

Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell's equation is found.

Thus the solution is 2, not ½ as would be the result of an incorrect analysis.

Thus the interior of the sphere is a lacuna for the solution.

Thus the total pressure above solution of A and B would be

Thus the decision-maker is a satisficer, one seeking a satisfactory solution rather than the optimal one.

Thus he lost two crucial days before finally adopting the solution, however obvious, of his predecessor.

Thus, for a fixed path length, UV / Vis spectroscopy can be used to determine the concentration of the absorber in a solution.

Thus if state vectors, and each solve the linear equation on ψ, then would also be a solution, in which each is a coefficient.

Thus, a salt water solution can be used when a longer period of retention is desired, such as to soften an impaction.

Thus, if the solution is to be said to work, then it is to be said to work solely within the given formal system, and not otherwise.

Thus in most " natural " settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic formula with real coefficients.

Thus, a solution does not generally exist for an arbitrary choice of strain components.

in a ring R. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

Thus, in this case, the equation bx = a has no solution if a is not equal to 0, and has any x as a solution if a equals 0.

Thus, marketing research may also be described as the systematic and objective identification, collection, analysis, and dissemination of information for the purpose of assisting management in decision making related to the identification and solution of problems and opportunities in marketing.

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, 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, the Kolmogorov complexity of the raw file encoding this bitmap is much less than 1. 62 million.

Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

“ Acts, then is a continuation of the Lucan Gospel, not in the sense that it relates what Jesus continued to do, but how his followers carried out his commission under the guidance of his Spirit .” Thus, part of the answer to the purpose of Acts is that Luke is writing to Theophilus, who is also mentioned in Luke 1: 3, in order to explain to him the occurrences that take place in the church that fulfill Jesus ’ promise to his disciples that “ you will be baptized with, the Holy Spirit not many days from now ” ( Acts 1: 5 ).

Thus, for example, V ← V + 1 is commonly understood to be a shorthand for ( V ,( V ) + 1 ).

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, there are 35 possible basic contracts ( five at each of the seven levels ); 1 being the lowest, followed by 1 etc., up to 7NT, the highest possible bid.

Thus a circular geosynchronous orbit has a radius of 42164 km ; 42164 / 6378. 1

Thus, the red plane shows the points with x = 1, the blue plane shows the points with z = 1, and the yellow plane shows the points with y =− 1.

Thus for instance the determinant of a matrix with integer coefficients will be an integer, and the matrix has an inverse with integer coefficients if and only if this determinant is 1 or − 1 ( these being the only invertible elements of the integers ).

Thus, in Zoroaster's revelation, Ahura Mazda was perceived to be the creator of only the good ( Yasna 31. 4 ), the " supreme benevolent providence " ( Yasna 43. 11 ), that will ultimately triumph ( Yasna 48. 1 ).

Thus, if n < sub > r </ sub > is the average number of times a new civilization reappears on the same planet where a previous civilization once has appeared and ended, then the total number of civilizations on such a planet would be ( 1 + n < sub > r </ sub >), which is the actual reappearance factor added to the equation.

Thus it is 1 volt ( 1 joule per coulomb, 1 J / C ) multiplied by the electron charge ( 1 e, or ).