Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space, which in turn is equal to the Euler characteristic χ ( X ).

Thus, besides the traditional numeric and logical constants, Euler introduces the following additional types:

Thus far the advances made have been almost entirely along functional lines.

Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.

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 at the same time that William Henry Harrison was preparing to pacify the aborigines of Indiana Territory and winning fame at the battle of Tippecanoe, Anglo-Saxon settlement made a great leap into the center of the North American continent to the west of the American agricultural frontier.

Thus we hear of abbots going out to hunt, with their men carrying bows and arrows ; keeping horses, dogs and huntsmen ; and special mention is made of an abbot of Leicester, c. 1360, who was the most skilled of all the nobility in hare hunting.

Thus we have the story of his riding across a stick ( horse made of stick ) with his children and upon being discovered by a friend desiring that he not mention till he himself were the father of children ; and because of the affection of his son Archidamus ' for Cleonymus, he saved Sphodrias, Cleonymus ' father, from execution for his incursion into the Piraeus, and dishonorable retreat, in 378 BC.

Thus, in the Hindu schools, if a claim was made that could not be substantiated by appeal to the textual canon, it would be considered as ridiculous as a claim that the sky was green and, conversely, a claim which could not be substantiated via conventional means might still be justified through textual reference, differentiating this from the epistemology of modern science.

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 a small voltage on the grid can be made to control a much larger voltage on the anode.

Thus, for example, the same equation can be made to produce both a lyrical and melodic piece of music in the style of the mid-nineteenth century, and a fantastically dissonant cacophony more reminiscent of the avant-garde music of the 1950s and 1960s.

Thus made invisible, he appears to have entered the building as Clark Kent and exited seconds later as Superman.

Thus, it might be used to challenge the sharp distinction made in U. S. v. E. C. Knight ( 1895 ) between commerce, which was subject to federal regulation, and manufacturing, which was not.

Thus if we reason from the statement " Pegasus flies " to the statement " Pegasus exists ", we are not asserting that Pegasus is made up of atoms, but rather that Pegasus exists in a particular worldview, the worldview of classical myth.

Thus, Baresi made his World Cup debut in 1990.

Thus, although Scots were only 10 per cent of the British population, they made up 15 per cent of the national armed forces and eventually accounted for 20 per cent of the dead.

Thus he became distinguished as the man " who made Israel to sin.

Thus, Muhammad, by saying this, clearly made it a point to the Arabs to not make any distinction between the great apostles of God.

Thus, law has an internal morality that goes beyond the social rules by which valid laws are made.

Thus the sky might be called naturalistically él-ker “ squall-vat ” ( Markús Skeggjason: Eiríksdrápa 3 ) or described in mythical terms as Ymis haus “ Ymir ’ s skull ” ( Arnórr jarlaskáld: Magnúsdrápa 19 ), referring to the idea that the sky was made out of the skull of the primeval giant Ymir.

Thus familiarized with the phenomena associated with the movements of recent glaciers, Agassiz was prepared for a discovery which he made in 1840, in conjunction with William Buckland.

Thus, a dough made from juice, etc., is of doubtful validity as mitzva matzo and may be used for the mitzva only in cases of illness or age.

Thus it made sense to not map the memory between programs for IPC, but instead migrate the program code being used into the local space of the program.

Thus, on the basis of remarks made by President Kim Il Sung in 1977 concerning school attendance, the population that year was calculated at 17. 2 million persons.

Thus, energy is no longer released when such nuclei are made by fusion ; instead, energy is absorbed in such processes.

Thus they add two equivalents of bromine whereas an alkene adds only one equivalent.

Thus the overall ratios of the three coinages remained roughly equivalent.

Thus, asymmetric algorithm keys must be longer for equivalent resistance to attack than symmetric algorithm keys.

Thus, in nutrition, the terms calorie and kilocalorie refer to equivalent units.

Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.

Thus, despite widely held popular belief outside the Orthodox cultures, there is not one bishop at the head of the Orthodox Church ; references to the Patriarch of Constantinople as a leader equivalent or comparable to a pope in the Roman Catholic Church are mistaken.

) Thus far, no published FFT algorithm has achieved fewer than complex-number additions ( or their equivalent ) for power-of-two.

Thus, the statement that f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element with respect to the function composition operation ∘ on functions from S to S.

Thus the following two statements are equivalent:

) 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 a vacuum of 26 inHg gauge is equivalent to an absolute pressure of 30 inHg ( typical atmospheric pressure ) − 26 inHg = 4 inHg.

Thus, it may be an Eastern equivalent of the European toponym " Piedmont.

Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others.

Two computers P and Q are called Turing equivalent if P can simulate Q and Q can simulate P. Thus, a Turing-complete system is one that can simulate a Turing machine ; and, per the Church-Turing thesis, that any real-world computer can be simulated by a Turing machine, it is Turing equivalent to a Turing machine.

Thus two arrows and in space represent the same free vector if they have the same magnitude and direction: that is, they are equivalent if the quadrilateral ABB ′ A ′ is a parallelogram.

Thus, " earl " and " earldom " were taken as equivalent to the continental use of " count " and " county ".

Thus there is no simple way to find an equivalent in the proleptic Julian calendar of a date quoted using the Roman pre-Julian calendar ( AUC or by reference to consuls ).

Thus, the use of the term " diocese " referring to geography is the most equivalent in the United Methodist Church, whereas each annual conference is part of one episcopal area ( though that area may contain more than one conference ).

Thus, the standard form of corporal punishment in US schools ( licks with a paddle ) is often referred to as a spanking, whereas its pre-1997 English equivalent ( strokes of the cane ) would never have been so described.

Thus public key systems require longer key lengths than symmetric systems for an equivalent level of security.

Thus, the definition of T < sub > 3 </ sub > may cite T < sub > 0 </ sub >, T < sub > 1 </ sub >, or T < sub > 2½ </ sub > instead of T < sub > 2 </ sub > ( Hausdorffness ); all are equivalent in the context of regular spaces.

Thus Tychonoff's theorem joins several other basic theorems ( e. g. that every nonzero vector space has a basis ) in being equivalent to AC.

Thus Virpiranta sees Efnisien as a Sun god and therefore the Welsh equivalent to Esus.