Thus isometry is simply bending and twisting of a surface without internal crumpling or tearing, in other words without extra tension, compression or shear.

Thus each pin line has two shear points, one where the driver and spacer pins meet, and one where the spacer and key pins meet.

Thus, the maximum shear stress will occur either in the web of maximum shear flow or minimum thickness.

Thus, the viscosity of a shear-thickening fluid is dependent on the shear rate.

Thus they are the lines which join the points with equal maximum shear stress magnitude.

Thus, a simple liquid will not typically support a transverse acoustic phonon, or shear wave.

Thus, they joined two powerful political families with strains of madness running through them, both renowned for cruel behavior.

Thus there is scope for future research into cultivation of safer strains.

Thus outbred strains of most laboratory animals are also available.

Thus, all strains of canine parvovirus will affect dogs, wolves, and foxes, but only some of them will infect cats.

Thus variants could arise during incubation, giving rise to new strains.

Thus, for example in Escherichia coli, identifying strains carrying toxin genes is more important than having a population genetics-based evaluation of prevalent strains.

Thus many of the most adventurous strains in South African music were pursued outside its borders for several decades.

Thus we are compelled to face the urbanization of the South -- an urbanization which, despite its dramatic and overwhelming effects upon the Southern culture, has been utterly ignored by the bulk of Southern writers.

Thus, paradoxically, the beat writers resort to `` religious '' metaphors: they are in search of mana, the spiritual, the numinous, but not anything connected with formal religion.

Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus there are polynomials Af such that Af.

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

Thus pencils of tangents to Q are transformed into pencils of tangents.

Thus it follows that the secants of **zg are all invariant.

Thus, although the agenda of external assistance in the economic sphere are cumulative, and many of the policies suggested for nations in the earlier stages remain relevant, the basic purpose of American economic policy during the later stages of development should be to assure that movement into a stage of self-sustaining growth is not prevented by lack of foreign exchange.

Thus the first lines of one of Charles Wesley's hymns are as follows.

Thus in his friends are stranger than his sisters' strong stresses are normal for his and sisters', but in his friends are stranger than his sisters strong stresses are normal for friends and sisters.

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

Thus, the white middle and lower classes are arrayed against the Negro middle and lower classes.

Thus, during any given year ( A ) if the revenues from the residential service are $7,000,000, ( B ) if the operating expenses imputed to this class of service come to $5,000,000, and ( C ) if the net investment in ( or value of ) the plant and equipment deemed devoted to this service amounts to $30,000,000, the cost analyst will report that residential service, in the aggregate, has yielded a return of $2,000,000 or 6-2/3 per cent.

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, the three main categories of antisubmarine warfare operations are defense of shipping, defense of naval forces, and area defense.

Thus, the most productive areas are not necessarily the most stressed.

Thus a number of operations of the order of Af are required.

Thus, if we are to observe American folklore in the twentieth century, we will do well to establish the relationships between folklore, nationalism and imperialism at the outset.

Thus, in `` The Story Of Ruth '' we have Ruth, Naomi, and Boaz and sets that are meticulously authentic.

" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.

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 for example two extant plays, The Phoenicean Women and Iphigenia at Aulis, are significantly corrupted by interpolations ( the latter possibly being completed post mortem by the poet's son ) and the very authorship of Rhesus is a matter of dispute.

Thus, both Carson and Armstrong ultimately contributed significantly to the science and technology of radio.

Thus, while the outcome of any particular hand significantly involves chance, the long-run expectations of the players are determined by their actions chosen on the basis of probability, psychology and game theory.

Thus, when FEV is genetically deleted from the mouse genome, male mice will instantly attack other males, whereas their wild-type counterparts take significantly longer to initiate violent behaviour.

Thus, on a weight-for-weight basis they are significantly more energetic than normal condensed explosives.

Thus, by bringing large numbers of entrepreneurial adventurers to the region, the Gold Rush significantly contributed to the economic development of Western Canada, Alaska and the Pacific Northwest.

Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.

Thus it is seen that a polyp is an animal of very simple structure, a living fossil that has not changed significantly for about half a billion years ( per generally accepted dating of Cambrian sedimentary rock ).

Thus, he notes that hiring decisions at Hitachi, during the time of his study, were " regulated by very strict qualification standards ", and not very significantly influenced by ascribed characteristics.

Thus, the Navajo Nation's census figures for Kayenta Chapter are significantly different from those of Kayenta proper.

In one direction, the Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower ( in the worst case, as a function of the number of vertices ) than finding a Hamiltonian cycle.

Thus the Matthew Bible, though largely unrecognized, significantly shaped and influenced English Bible versions in the centuries that followed its first appearance.

Thus women suffered significantly from the transition away from subsistence food production towards specialization and trade.

Thus a significantly important alternative route was established ( see also Roman roads in Britain ).

Thus, the implementation of new technologies along with a change in consumption habits could significantly affect the likelihood of the consequences of Olduvai Theory.

Thus ended one of Hess's closest professional associations and significantly compromised one of his deepest friendships.

Thus, applying an edge detection algorithm to an image may significantly reduce the amount of data to be processed and may therefore filter out information that may be regarded as less relevant, while preserving the important structural properties of an image.

Thus their rate of valley cutting was not significantly affected.

Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly.

Thus the number of H-1B visas issued each year is significantly more than the 65, 000 cap, with 117, 409 having been issued in 2010.

Thus IRT provides significantly greater flexibility in situations where different samples or test forms are used.

Thus, the major difference between the two is the cocoa butter, whose absence makes hot cocoa significantly lower in fat than hot chocolate while still preserving all the antioxidants found in chocolate.

Thus the destruction capability is greatly multiplied by MIRV, when the number of enemy silos does not significantly increase.

Thus, the western territories contributed to making Wartislaw significantly independent from the Polish dukes.