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, different combinations of properties and relations produce different objects.

Thus, the task is to find universal properties that uniquely determine the objects of interest.

Thus, computational chemistry can assist the experimental chemist or it can challenge the experimental chemist to find entirely new chemical objects.

Thus macroscopic objects tend toward being neutral overall, but macroscopic objects are rarely perfectly net neutral.

Thus the 32-bit integer and 32-bit floating-point objects can simply use the value directly ; whereas the 64-bit integer and 64-bit floating-point cannot use this method.

Thus, for example, two objects may be group isomorphic without being ring isomorphic, since the latter isomorphism selects the additional structure of the multiplicative operator.

Thus when we realize the nature of knowledge, Berkeley's argument is seen to be wrong in substance as well as in form, and his grounds for supposing that ' idea '- i. e. the objects apprehended-must be mental, are found to have no validity whatever.

Thus the often-used term body needs to stand for a wide assortment of objects, including particles, projectiles, spacecraft, stars, parts of machinery, parts of solids, parts of fluids ( gases and liquids ), etc.

Thus, a set of input objects is gathered and corresponding outputs are also gathered, either from human experts or from measurements.

Thus, six objects may be divided into piles of 5 + 1 or 4 + 2, but not 3 + 3.

Thus, Paley's use of the watch ( and other mechanical objects like it ) continued a long and fruitful tradition of analogical reasoning that was well received by those who read Natural Theology when it was published in 1802.

Thus, small < sup > 169 </ sup > Yb samples ( which emit gamma rays ) act like tiny X-ray machines useful for radiography of small objects.

: and this exceeds unity by only about r /( 2N-2b ) if N-b is large compared to r. Thus the weight of evidence provided by the sight of a white shoe is positive, but is small if the number of ravens is known to be small compared to the number of non-black objects.

Thus fingers are commonly used as sensory probes to ascertain properties of objects encountered in the world, and so they are prone to injury.

Thus, one way to show two objects of C are distinct ( up to isomorphism ) is to show that their images under f are distinct ( i. e. not isomorphic ).

Thus, " I entitle transcendental all knowledge which is occupied not so much with objects as with the mode of our knowledge of objects insofar as this mode of knowledge is to be possible a priori.

Thus the retina performs operations on the image to enhance the edges of objects within its visual field.

Thus the physical content of the theory has to do not with objects themselves, but the relations between them.

" Thus ... unquestionably incident to sovereign power to erect corporations to that of the United States, in relation to the objects entrusted to the management of the government.

Thus, they wrote in a very metaphorical and suggestive manner, endowing particular images or objects with symbolic meaning.

Thus the axioms of ZFC refer only to sets, not to urelements ( elements of sets which are not themselves sets ) or classes ( collections of mathematical objects defined by a property shared by their members ).

Thus congruent objects are always geometrically similar, but geometrical similarity additionally allows uniform scaling.