Thus there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X.

Thus it is meaningful to speak of a " presentation of an equivalence relation ," i. e., a presentation of the corresponding groupoid ;

Thus, S < sub > i </ sub > is a subset of an equivalence class of R < sub > L </ sub >.

Thus, the existence of a finite automaton recognizing L implies that the Myhill – Nerode relation has a finite number of equivalence classes, at most equal to the number of states of the automaton, and the existence of a finite number of equivalence classes implies the existence of an automaton with that many states.

Thus, it is a highly idiomatic translation, and as such falls on the extreme dynamic end of the dynamic and formal equivalence spectrum.

Thus the classical solutions to G Chern – Simons theory are the flat connections of principal G-bundles on M. Flat connections are determined entirely by holonomies around noncontractible cycles on the base M. More precisely, they are in one to one correspondence with equivalence classes of homomorphisms from the fundamental group of M to the gauge group G up to conjugation.

Thus, the mass-energy equivalence, combined with the Weak Equivalence Principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an object.

Thus, evaluators need to take into account two important concepts when administering data collection tools: lexical equivalence and conceptual equivalence ( ibid ).

Thus the pH will equal the pK < sub > a </ sub > which occurs half-way to the equivalence point.

Thus the relation of logical equivalence partitions all primary arithmetic expressions into two equivalence classes: those that simplify to the Cross, and those that simplify to the void.

Thus Q defines an inner product on dom T. Let H < sub > 1 </ sub > be the completion of dom T with respect to Q. H < sub > 1 </ sub > is an abstractly defined space ; for instance its elements can be represented as equivalence classes of Cauchy sequences of elements of dom T. It is not obvious that all elements in H < sub > 1 </ sub > can identified with elements of H. However, the following can be proved:

Thus, after flashing furniture, the equivalence of chair-chair is detected more rapidly than stove-stove.

Thus they left open the possibility that the Ape Index may be more significant when there is a greater degree of equivalence between the other traits under consideration.

Thus a novel might be translated with greater use of dynamic equivalence so that it may read well, while in diplomacy or in some business settings people may insist on formal equivalence because they believe that fidelity to the grammatical structure of the language equals greater accuracy.

Thus we can determine the equivalence ratio of the given mixture as,

Thus the relevant notion of topological equivalence is a considerable weakening of the naïve C < sup > 1 </ sup > conjugacy of vector fields.

Thus, to talk of a method in relation to deconstruction, especially regarding its ethico-political implications, would appear to go directly against the current of Derrida's philosophical adventure.

Thus, Melnyk's rebuttal, stated very crudely, is that one should define physicalism in relation to current physics, and hold a similar attitude towards its truth as most scientists hold towards the truth of currently accepted scientific theories.

Thus, much of human development has been in relation to zoonotic, not epidemic, diseases.

( Thus Aristotle first suggested a reciprocal or circular causality-as a relation of mutual dependence, action, or influence of cause and effect.

Thus, the effect of the accidental has to be understood in relation to the " natural " meaning of the note's staff position.

Thus, for the de Broglie – Bohm theory, the particle's spin is not an intrinsic property of the particle — instead spin is, so to speak, in the wave function of the particle in relation to the particular device being used to measure the spin.

" 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 there is not the same direct relation between magnetic optical activity and molecular stereochemistry which would be expected, because it is found in natural optical activity.

Thus, wavelength ratios observed for different ionic species in the absorption spectra of quasi stellar objects ( QSO or quasars ) place a limit on any variation in the fine-structure constant to less than 1 part in 1 million out to a distance in space ( and time ) of z = 3 ( about 6500 megaparsecs or 11. 5 billion light years ); as the fine-structure constant is determined by the relation between the speed of light ( c ), Planck's constant ( h ) and the electron charge ( e ), these physical constants are constrained as well.

Thus, the issue becomes an existentialist query about whether nature can exist in a humanist critique, and whether we can discern the " other's " views in relation to our actions on their behalf.

Thus, a candidate key is a minimal set of attributes that functionally determine all of the attributes in a relation.

Thus the study of the relation between the fundamental groups of X and is not so straightforward.

Thus, if more than 1 qualitative variable is included in the regression, it is important to note that the omitted category should be chosen as the benchmark category and all comparisons will be made in relation to that category.

Thus in practical terms Venizelos ' authority over his commander of the army was diminished due to the obvious relation between the Crown Prince and the King.

Thus, according to the theory, even the vacuum has a vastly complex structure and all calculations of quantum field theory must be made in relation to this model of the vacuum.

Thus, Henry's veneration as a saint and his relation to King Eric seem to have emerged in the historical record at the same time in the mid-1290s with strong support from the church.

Thus, book i. opens with an account of the Trinity and his relation to creation ; then follows a similar series of chapters about angels, their attributes, powers, orders, etc., down to such minute points such as their methods of communicating thought, on which matter the author decides, in his own person, that they have a kind of intelligible speech, and that with angels, to think and to speak are not the same process.

Thus, at the same time that Přemyslid rulers used the German alliance to consolidate their rule against a perpetually rebellious regional nobility, they struggled to retain their autonomy in relation to the empire.

Thus, " artisanal " is sometimes used in marketing and advertising as a buzz word to describe or imply some relation with the crafting of handmade food products, such as bread, beverages, or cheese.

Thus Lewis saw the logical positivists as failing to distinguish between " linguistic " meaning, namely the logical relations among terms, and " empirical " meaning, namely the relation expressions have to experience.

" Thus, a government attorney only needs to certify that information will ' likely ' be obtained in relation to an ' ongoing criminal investigation '.

Thus the given recurrence relation T ( n ) was in Θ ( n < sup > 3 </ sup >).

Thus Af where the maximization is over all admissible policies Af, and Af is related to Af by ( 5 ).

Thus, when Dartmouth's Winter Carnival -- widely recognized as the greatest, wildest, roaringest college weekend anywhere, any time -- was broadcast over a national television hookup, Prexy John Sloan Dickey appeared on the screen in rugged winter garb, topped off by a tam-o'-shanter which he confessed had been acquired from a Smith girl.

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 Ambrose refused to be drawn into a false conflict over which particular local church had the " right " liturgical form where there was no substantial problem.

Thus in 1285 he declared war on his uncle, James II of Majorca, and conquered both Majorca ( 1285 ) and Ibiza ( 1286 ), effectively reassuming suzerainty over the Kingdom of Majorca.

Thus, the Hungarian supremacy over Halych disappeared.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Thus, over time, the expression " a cappella " ( Italian for " in the manner of the chapel ") came to mean exclusively vocal music in contradistinction to the spreading use of the organ in cathedrals.

Thus, if the score was 3 – 2 with the bases loaded in the bottom of the ninth, and the batter smacked an " over the fence home run ", the game would end at 4 – 3, with the batter only allowed a double, and the runners officially stopped on 2nd and 3rd ( since they were not needed to win the game ).

Thus, if a ball went over the fence fair, and curved behind the foul pole, it was ruled foul.

Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, i. e. acting on every point in it.

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

Thus a spacecraft can determine its relative position by tracking such asteroids across the star background, which appears fixed over such timescales.

Thus, we should prioritize social reform over attempts to account for consequences, individual virtue or duty ( although these may be worthwhile attempts, provided social reform is provided for ).

Thus, before the Aswan High Dam was completed in 1971, the White Nile watered the Egyptian stretch of the river throughout the year, whereas the Blue Nile, carrying seasonal rain from Ethiopia, caused the Nile to overflow its banks and deposit a layer of fertile mud over adjacent fields.

Thus, over time we would expect each traditional song to become aesthetically ever more appealing — it would be collectively composed to perfection, as it were, by the community.

Thus John asserts Jesus ' innate superiority over all divine messengers, whether angels or prophets.

Thus methane has a potential of 25 over 100 years but 72 over 20 years ; conversely sulfur hexafluoride has a GWP of 22, 800 over 100 years but 16, 300 over 20 years ( IPCC TAR ).

Thus, if Adar was over and Spring had not yet arrived, an additional month was observed.

Thus they were involved in the struggle for power within France as princes but they also had a religious authority over Roman Catholicism in France as King.