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 ).

f. Thus, every pair of ( mutually ) inverse elements gives rise to two idempotents, and ex

Thus, to have equal kinetic energies, the velocities of two different molecules would have to be in inverse proportion to the square roots of their masses.

Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered an operator that takes zero arguments.

Thus a class has an inverse if and only if there is an ideal J such that IJ is a principal ideal.

Thus, the two distinct notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case ( see below ).

Thus the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of algebraic fractions, that produces a single rational function with a numerator and denominator usually of high degree.

Thus one may describe the functors as being " inverse up to isomorphism ".

The category Top of topological spaces has as morphisms the continuous functions, where a function f is continuous if the inverse image f < sup > − 1 </ sup >( O ) of any open set in the codomain of f is open in the domain of f. Thus any continuous function f from a space X to a space Y defines an inverse mapping f < sup > − 1 </ sup > from Ω ( Y ) to Ω ( X ).

Thus, an article by the 20-year-old Hambardzumyan opened up a whole area of research on inverse problems and became the foundation of an entire discipline.

Thus a limaçon can be defined as the inverse of a conic where the center of inversion is one of the foci.

Thus, our sampling of the DTFT causes the inverse transform to become periodic.

Thus, inverse quadratic interpolation is seldom used as a stand-alone algorithm.

Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup.

Thus a cartesian section consists of a choice of one object x < sub > S </ sub > in F < sub > S </ sub > for each object S in E, and for each morphism f: T → S a choice of an inverse image m < sub > f </ sub >: x < sub > T </ sub > → x < sub > S </ sub >.

Thus, by the properties of inversive geometry, since the figure traced by point D is the inverse of the figure traced by point B, if B traces a circle passing through the center of inversion O, then D is constrained to trace a straight line.

Thus inverse dimensions identify subconcepts for a superconcept.

Thus the government simultaneously undertook the vast burden of social security which had traditionally been privately discharged, and created a national scarcity which has engendered calamitous problems of social security.

Thus the distribution of Symbiodinium on coral reefs and its role in coral bleaching presents one of the most complex and interesting current problems in reef ecology.

Thus, the theory ‘ is continually evolving as it is brought to bear on solving problems of a research or practical nature ’.

Thus the problems theoretical statistics considers include:

Thus in 1918, Belgium was already confronted with the problems associated with occupation that most European countries only discovered at the end of World War II.

Thus, economic transformation created severe problems in maintaining social welfare since local governments were unable to assume financial or operational responsibility for these functions.

Thus, the growing body of genome information can also be tapped in a more general way to address global problems by applying a comparative approach.

Thus, reinforcement learning is particularly well suited to problems which include a long-term versus short-term reward trade-off.

Thus the law sought to solve the twin problems of increasing the number of men eligible for military service ( thereby boosting Rome's military strength ) and also providing for homeless war veterans.

Thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems.

Thus, transvestic fetishism is not diagnosed unless it causes significant problems for the person concerned.

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, in scenarios where the TCP port used by XMPP is blocked, a server can listen on the normal HTTP port and the traffic should pass without problems.

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, Strauss agrees with the Socrates of the Phaedrus, where the Greek indicates that, insofar as writing does not respond when questioned, good writing provokes questions in the reader — questions that orient the reader towards an understanding of problems the author thought about with utmost seriousness.

Thus, it seems that a diet can be functional ( it keeps us going ) and dysfunctional ( it causes health problems ) at the same time.

Thus large computational problems can be solved more cost effectively by using the aggregate power and memory of many computers.

Thus, DLB is similar in some ways to both the dementia resulting from Alzheimer's disease and the movement problems of Parkinson's disease.

Thus compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as angle trisection and doubling of the cube.

Thus there is reason to believe that ophiolites are indeed oceanic mantle and crust ; however, certain problems arise when looking closer.

Thus, the grid had become the main driver of remote customers ’ power costs and power quality problems, which became more acute as digital equipment required extremely reliable electricity., Efficiency gains no longer come from increasing generating capacity, but from smaller units located closer to sites of demand.,

Thus swept-forward wings are unstable in a fashion similar to the low-speed problems of a conventional swept wing.

Thus, despite the problems, Langston has braved the storms and produced outstanding graduates.

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.