Thus Af denotes the state of the feed to the R-stage process, and Af the state of the product from the last stage.

where D denotes the continuum derivative operator, mapping f to its derivative f. The expansion is valid when both sides act on analytic functions, for sufficiently small h. Thus, T < sub > h </ sub >= e < sup > D </ sup >, and formally inverting the exponential yields

Thus, the derivative of f ( x ) becomes for an infinitesimal, where st (·) denotes the standard part function, which associates to every finite hyperreal the unique real infinitely close to it.

Thus, velocity is the time rate of change of position, and the dot denotes the derivative with respect to time.

Thus, if its inputs are an array x of length N, we can imagine extending this array to ( x, – x < sub > R </ sub >, – x, x < sub > R </ sub >, ...) and so on, where x < sub > R </ sub > denotes x in reverse order.

: Thus, the MDCT of 2N inputs ( a, b, c, d ) is exactly equivalent to a DCT-IV of the N inputs: (– c < sub > R </ sub >– d, a – b < sub > R </ sub >), where R denotes reversal as above.

According to Monier-Williams ( 1964 ), the etymology of sukha is " said to be su + kha and to mean originally ' having a good axle-hole '...." Thus, for instance, in the Rig Veda sukha denotes " running swiftly or easily " ( applied, e. g., to chariots ).

Thus begins an epic adventure of survival and a quest for freedom that takes him across the Sahara desert to Mogadore, Morocco.

Thus the stage was set for the great war, for which the epic of Mahabharata is known most of all.

Thus the epic tale is told from father to son, down through the ages.

Thus, the three completed books of Cowley's great ( albeit unfinished ) English epic, The Civill Warre ( otherwise spelled " The Civil War "), was finally published in full for the first time in 1973.

Thus it often favors epic plots and heroic scenes over historical accuracy.

Thus, antonyms are often found next to each other throughout the poem, reinforcing the idea that everything that was lost in the first epic is going to be regained by the end of the mini-epic.

Thus a satisfactory conclusion about the epic ’ s origins cannot be drawn based on the lifespans of historical heroic figures .... Jiangbian pointed out that the foundation for the origin of epic is ethnic folk culture.

Thus, Beowulf can be the national epic of England, yet England is part of Western Civilization and thus also subscribes to the Iliad and Odyssey of Homer and the Aeneid of Virgil as pan-national epics.

Thus the epic tale is told from father to son, down through the ages.

This epic has the story in how the Mayon Volcano was made. Thus ends Ibalon.

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, the Commission acted with a sense of social responsibility within the area of its own convictions about the problem of government support to private education.

Thus, the combined efficiency of the elements replaced by the two fiber plates ( with a combined efficiency of 0.25 ) is 0.043 or about six times less than that of the two fiber plates.

Thus if the gyro and platform-controller combination maintains the platform with zero angular deviation about the **yf axis, the system can be rotated with an angular velocity Af if a torque is supplied to the gyro output axis Aj.

Thus he will be in a position to disabuse the Soviet leader of any notions he may have about grave Allied disunity.

Thus for example Heraclitus ' The Allegorist ' quoted fr. 326 and part of fr. 6, about ships in a storm, in his study on Homer's use of allegory.

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, a 3072-bit Diffie-Hellman key has about the same strength as a 3072-bit RSA key.

Thus, the total applied torque about the origin is given by

Thus Claudius ' private words about his own policies and motives were lost to history.

Thus conclusions about actual agriculture in the Letters are as likely to evoke Latium as Attica.

Thus the CSF turns over about 3. 7 times a day.

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, the Sheikh of al-Azhar in Cairo, Muhammad Sayid Tantawy, who is the leading religious authority in the Sunni Muslim establishment in Egypt, alongside the Grand Mufti of Egypt, said the following about fatwās issued by himself or the entire Dar al-Ifta:

Thus we have the important result that when a discrete data sequence, s, is proportional to samples of an underlying continuous function, s ( t ), one can deduce something about the continuous Fourier transform, S ( ƒ ).

Thus 2. 5 cm < sup > 3 </ sup > contain about a googol Planck spaces.

" Thus irreal works such as Italo Calvino's Cosmicomics and Jorge Luis Borges ' Ficciones can be seen as an attempt to find a new allegorical language to explain our changed perceptions of the world that have been brought about by our scientific and technical culture, especially concepts such as quantum physics or the theory of relativity.

Thus, there may sometimes be confusion about the blade lengths, depending on which shaku value is being assumed when converting to metric or US measurements.

Thus typical dimensions were about long by wide by deep.

Thus, the locomotive is loaded or unloaded in about a third of the time.

) Thus lighting energy accounts for about four or five million barrels of oil ( equivalent ) per day.

Thus commercial lighting alone consumes about four to five million barrels per day ( equivalent ) of petroleum, in line with the alternate rationale above to estimate U. S. lighting energy consumption.

" Thus we " make up our minds ," " change our minds " or are " of two minds " about something.

Thus, an additional 6 % of the total energy of fission is also released eventually as non-prompt ionizing radiation, and this is about evenly divided between gamma and beta ray energy.