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 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.

Thus mail armour proved to be sufficient protection in most situations.

Thus, for example, the funerary god Anubis was portrayed as a jackal, a creature whose scavenging habits threatened the preservation of the body, in an effort to counter this threat and employ it for protection.

Thus Finland had to face its big neighbour on its own, without any greater power's protection.

Thus a treaty of protection was signed in 1602 with Pope Clement VIII, which came into force in 1631.

Thus the courts had no power to compel the sovereign to be bound by the courts, as they were created by the sovereign for the protection of his or her subjects.

Thus, the major threat in these areas is fire management ( protection of private assets is a key objective and so past fire regimes may not reflect environmental needs ), weed invasion and lack of connectivity between patches.

Thus the evolution of atmospheric phenomena can be followed, ensuring real-time coverage of short-lived dynamic events, especially severe local storms and tropical cyclones — two meteorological events that directly affect public safety, protection of property, and ultimately, economic health and development.

Thus, both signs could have been symbols of Auseklis, originally used for protection against evil spirits.

Thus, stricter rules were created for the Study tour overall not only for legal purposes but also for the protection of the students.

Thus the dead also needed protection, so they were sometimes buried with spells that could destroy Apep.

Thus, in North America, protected areas were about safeguarding dramatic and sublime scenery ; in Africa, the concern was with game parks ; in Europe, landscape protection was more common.

Thus the Supreme Court's actual decision never hinged on the equal protection claims.

Thus, world conscience is a concept that overlaps with the Gaia hypothesis in advocating a balance of moral, legal, scientific and economic solutions to modern transnational problems such as global poverty and climate change, through strategies such as environmental ethics, climate ethics, natural conservation, ecology, cosmopolitanism, sustainability and sustainable development, biosequestration and legal protection of the biosphere and biodiversity.

Thus, the Ayyubids required Nubia to guarantee the protection of Aswan and Upper Egypt, but like their Fatimid predecessors, were discouraged from further expansion by the poverty of the region.

Thus the collective heritage of the clan, the dùthchas, gave the right to settle the land to which the chiefs and leading gentry provided protection and authority as trustees for the people.

Thus, pyrolysis reaction mechanisms and the pyrolysis properties of materials are important in fire protection engineering for passive fire protection.

Thus hares are adapted to the lack of physical protection, relative to that afforded by a burrow.

Thus it managed a mixed portfolio of issues: housing and planning, local government, public buildings, environmental protection and, initially, transport-James Callaghan gave transport its own department again in 1976.

Thus, Aero Continente lost its insurance protection, as Global Aerospace, a partly US-based company which was responsible for this matter, was forced to cease its involvement with the airline.

Thus, plinkers do not generally use typical range commands and it is not uncommon for plinkers who do not use prescription lenses to go without eye protection.

Thus, the effect of UV-A leads to skin darkening, but this is only cosmetic since it does not lead to greatly increased production of melanin and therefore also to little increase in protection against UV-B, or protection against sunburn.

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 member of an industrial union comes to regard his officers as business agents who may proceed without interference or recall ; ;

Thus the student of literature may sometimes find it helpful to classify a poem or an essay as being in idea or in ideal content or subject matter typical or atypical of its period.

Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.

Thus, a child's Skeletal Age `` dots '' may be classified as `` advanced '' when they appear above the middle curve, `` moderate '' when they appear immediately above or below the middle curve, and `` delayed '' when they appear below the lower curve.

Thus he may be referring to some concrete thing, or incident, in his immediate environment by some symbolic-sounding, hyperbolic reference to transcendental events on the global scene.

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 while his theory or technique may not be oversubscribed, it is commonplace for bullish and bearish positions to become temporarily over-subscribed.

Thus, if public pressure sets the effective limit to the price that the industry may charge, this pressure is itself a function of the wage rate.

Thus, when specifically permitted, the operand of a given line on the Autocoder coding sheet may be continued in the operand of from one to four additional lines which immediately follow.

Thus creativity may run all the way from making a cake, building a chicken coop, or producing a book, to founding a business, creating a League of Nations or, developing a mature character.

Thus a man who is butting a stone wall at the office may become unusually aggressive in bed -- the one place he can still be champion.

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, giving grades and promotions based only on an individual's performance relative to a small local group, as is common, may reduce cooperative behaviors in the group.

Thus, while altruistic persons may under some circumstances be outcompeted by less altruistic persons at the individual level, according to group selection theory the opposite may occur at the group level where groups consisting of the more altruistic persons may outcompete groups consisting of the less altruistic persons.

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, a court officer taking possession of goods under a court order may use force if reasonably necessary.

Thus, a gene for antibiotic resistance that evolves via natural selection may be shared.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

Thus it may be roughly translated as " protector of man ".

Thus, three eclipse calculations may be evidence for the suggestion that Romulus reigned from 746 BC to 709 BC, and Rome was founded during 745 BC.

Thus if the instrument depends on the pressure or suction effect alone, and this pressure or suction is measured against the air pressure in an ordinary room, in which the doors and windows are carefully closed and a newspaper is then burnt up the chimney, an effect may be produced equal to a wind of 10 mi / h ( 16 km / h ); and the opening of a window in rough weather, or the opening of a door, may entirely alter the registration.