Thus the Af are projections which correspond to some direct-sum decomposition of the space V.

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 his persistence values for some stem frequencies per meaning are: stem identical in 8 languages, 100% ; ;

Thus some provinces contained tribes which detested each other, and to them independence meant an opportunity for war.

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 although most species in the order are herbaceous, some no more than 15 cm high, there are a number of climbers ( e. g. some species of Asparagus ), as well as several genera forming trees ( e. g. Agave, Cordyline, Yucca, Dracaena ), some of which can exceed 10 m in height.

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 while Persian-speakers were concentrated on the lower rungs of the work force, managers tended to be brought in from some distance.

Thus, during the abdication crisis of 1936, caused by Edward VIII's desire to marry Wallis Simpson, the consent of all realms concerned, along with, in some cases, new acts of parliament, was required in Britain and throughout the British Dominions to allow for Edward's stepping aside and to ensure that if he had any children they would have no claim to the thrones.

Thus aesthetic judgments might be seen to be based on the senses, emotions, intellectual opinions, will, desires, culture, preferences, values, subconscious behavior, conscious decision, training, instinct, sociological institutions, or some complex combination of these, depending on exactly which theory one employs.

For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is

Thus one finds brass instruments made of wood, like the alphorn, the cornett, the serpent and the didgeridoo, while some woodwind instruments are made of brass, like the saxophone.

Thus fines and noncustodial sentences may address the crimes seen as least serious, with lengthy imprisonment or ( in some jurisdictions ) capital punishment reserved for the most serious.

Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space.

Thus literary criticism became highly theoretical and some of those practicing it began referring to the theoretical dimension of their work as " critical theory "-a philosophically inspired theory of literary criticism.

Thus miniatures were no longer required for game play, although some players continued to use them as a visual reference.

Thus, though the former understanding certainly enjoys the greatest popularity, the identification of Plotinus ’ opponents as Gnostic is not without some contention.

Thus, while D-Cinema is a defined standard, though one that is still partly being framed by SMPTE as of 2007, E-Cinema may be anything, ranging from a DVD player connected to a consumer projector to something that approaches the quality of D-Cinema without conforming to some of the standards.

Thus in some cases a single entity can essentially maintain controlling interest with only 33. 4 % of the outstanding shares.

Thus, some individuals and institutions will enter into a derivative contract to speculate on the value of the underlying asset, betting that the party seeking insurance will be wrong about the future value of the underlying asset.

Thus, in the example cited above Fromm rests his whole case on the premise that the workers are being deprived unconsciously, unknowingly, of fulfillment, and then supports this with survey data reporting conscious, experienced frustrations.

Thus, if what is at issue is whether `` All S is P '', it is indifferent whether `` Some S is not P '' or `` No S is P '', since in either case the judgment in question is false.

Thus, a Rabbi, a Catholic priest, and an agnostic might agree that, in this particular case, the best approach is to withhold extraordinary medical care, while disagreeing on the reasons that support their individual positions.

Thus, an analog signal is one represented by a continuous stream of data, in this case along an electrical circuit in the form of voltage, current or charge changes ( compare with digital signals below ).

Thus, the designers of ATM utilized small data cells to reduce jitter ( delay variance, in this case ) in the multiplexing of data streams.

Thus, for the general, non-associative case, binary operations can be represented with binary trees.

Thus, mD represents a case with hypomania and major depression.

Thus, a man is properly charged with premeditated murder if the circumstances surrounding his case closely resemble the exemplar premeditated murder case.

Thus, though often de facto the case, Conservative Judaism's halakhic system does not inherently see Orthodox halakhic practice as acceptable and legitimate halakhic practice for a Conservative Jew.

Thus ion implantation is a special case of

Thus, these facts are presupposed by the question, and in this case an entrapment, because it narrows the respondent to a single answer, and the fallacy of many questions has been committed.

Thus it is true in every case above except case 2, because this is the only case when is true but is not.

#* Proof: Then so Thus However, is prime, so or In the former case, hence ( which is a contradiction, as neither 1 nor 0 is prime ) or In the latter case, or If however, which is not prime.

Thus, not only the existence of the crime depends on there being a previous legal provision declaring it to be a penal offense ( nullum crimen sine praevia lege ), but also, for a specific penalty to be imposed in a certain case, it is also necessary that the penal legislation in force at the time when the crime was committed ranked the penalty to be imposed as one of the possible sanctions to that crime ( nulla poena sine praevia lege ).

Thus the praetor peregrini essentially created new substantive law for each case.

Thus, issues such as the ability to grant pre-trial relief, procedure and form, as well as statutes of limitations are classified as “ procedure ” and are always subject to domestic law where the divorce case is pending.

Thus, the statement " there are no contradictions in the Principia system " cannot be proven in the Principia system unless there are contradictions in the system ( in which case it can be proven both true and false ).

Thus, it is not enough to demonstrate that you have suffered the wrong in order to win a tort case, you must also have legally recognized damages that were directly or indirectly caused by the tortfeasor as a result of the tort, and be able to prove the extent of those damages.

Thus, it is correctly written either way, and is only incorrect when specification is first made that the word is being used as a unit, or else a symbol of the unit, and then the incorrect letter case for the specified use is employed.

Thus science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.

Thus, Cobb's totals may be even greater than is recorded.

Thus while bumblebees have been found to be about ten times more efficient pollinators on cucurbits, the total efficiency of a colony of honey bees is much greater due to greater numbers.

Thus, it has been known for many years that, due to repulsive Coulombic interactions, electrically charged macromolecules in an aqueous environment can exhibit long-range crystal-like correlations with interparticle separation distances, often being considerably greater than the individual particle diameter.

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

Thus in the physical world it is difficult to give examples of numbers that compare to the vastly greater googolplex.

Hence, those with less pigmented skin survived and had children at higher rates because their skin allowed more UV light for the production of vitamin D. Thus, the skin of those in the group that left the African continent and went far north gradually developed adaptations for relatively greater translucence compared to equatorial hues.

Thus, an ionic bond is considered a bond where the ionic character is greater than the covalent character.

Thus, an ionic bond is considered a bond where the ionic character is greater than the covalent character.

Thus is greater than, where, etc.

Thus, inherited behavioral mechanisms that allowed an organism a greater chance of surviving and / or reproducing in the past are more likely to survive in present organisms.

A person ’ s satisfaction is not part of any greater satisfaction .’ Thus the aggregation of utility becomes futile as both pain and happiness are intrinsic to and inseparable from the consciousness in which they are felt, rendering impossible the task of adding up the various pleasures of multiple individuals.

Thus the greater part of the valley of the Esk and the whole of Liddesdale can be considered to be northern English dialects rather than Scots ones.

Thus, a high Q tuned circuit in a radio receiver would be more difficult to tune, but would have greater selectivity, it would do a better job of filtering out signals from other stations that lie nearby on the spectrum.

Thus, despite the population of France being greater than the population of all of the German states that participated in the war, the Germans mobilized more soldiers for battle.

Thus range encoding can achieve greater compression ratios than the one-bit-per-symbol lower bound on Huffman encoding and it does not suffer the inefficiencies that Huffman does when dealing with probabilities that are not exact powers of two.

) Thus, the group of conformal transformations in spaces of dimension greater than 2 are much more restricted than the planar case, where the Riemann mapping theorem provides a large group of conformal transformations.

Thus, the longer the biological half-life of the substance the greater the risk of chronic poisoning, even if environmental levels of the toxin are not very high.

Thus, a program will achieve greater performance if it uses memory while it is cached in the upper levels of the memory hierarchy and avoids bringing other data into the upper levels of the hierarchy that will displace data that will be used shortly in the future.

Thus appeared a definite distinction, which eventually had the effect of restricting to the greater barons alone the privileges and duties of peerage.

Thus, while East Tennessee and greater Knox County voted decisively against secession in 1861, the city of Knoxville favored secession by a 2-1 margin.

Thus, the attempt to maintain a given surface temperature in an environment of faster heat loss results in both the perception of lower temperatures and an actual greater heat loss increasing the risk of adverse effects such as frostbite, hypothermia, and death.

Thus, lighting a larger area to the same level of lux requires a greater number of lumens.