Thus the number of preorders is the sum of the number of partial orders on every partition.

Thus, all partial derivatives of temperature with respect to space may either be zero or have nonzero values, but all derivatives of temperature at any point with respect to time are uniformly zero.

Thus, the total number of Germans worldwide lies between 66 and 160 million, depending on the criteria applied ( native speakers, single-ancestry ethnic Germans, partial German ancestry, etc.

Thus, the higher the diffusing capacity, the more gas will be transferred into the lung per unit time for a given gradient in partial pressure ( or concentration ) of the gas.

Thus, the " non-Self " doctrine is presented in the Mañjuśrī-nāma-saṅgīti ( and in certain tantric texts ) as a merely partial, incomplete truth rather than as an absolute verity.

Thus man mistakes his partial ability to transcend himself for the ability to prove his absolute authority over his own life and world.

Thus, the partial pressure of CO < sub > 2 </ sub > in the gas has increased until Henry's law is obeyed.

Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species ' concentrations in the mixture remaining constant, and at constant temperature.

Since any vector v is a linear combination ∑ v < sup > j </ sup > e < sub > j </ sub > of its components, df is uniquely determined by df < sub > p </ sub >( e < sub > j </ sub >) for each j and each p ∈ U, which are just the partial derivatives of f on U. Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x < sup > 1 </ sup >, x < sup > 2 </ sup >,... x < sup > n </ sup > are themselves functions on U, and so define differential 1-forms dx < sup > 1 </ sup >, dx < sup > 2 </ sup >, ..., dx < sup > n </ sup >.

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, the outcome of a full partial fraction expansion expresses that function as a sum of fractions, where:

Thus in general a schedule is a partial order of operations, containing ( embedding ) the partial orders of all its transactions.

Thus, if the resulting enlarged image is recorded on contrasty photographic paper, the partial cancellation emphasizes the high frequency ( fine detail ) information in the original, without loss of highlight or shadow detail.

Thus the complete partial orders with Scott continuous maps form a cartesian closed category.

Thus, if v is a weak α-th partial derivative of u, we may denote it by D < sup > α </ sup > u := v.

Thus, the Ethiopian's conversion can also be interpreted as a partial fulfillment of the mission presented in to bring the Gospel to the " ends of the earth ".

Thus, when O < sub > 2 </ sub > partial pressure is low, hemoglobin-bound O < sub > 2 </ sub > is more readily transferred to myoglobin.

Thus Dunkl operators represent a meaningful generalization of partial derivatives.

Thus temporarily there is partial duplication of the content, so it is useful to put an explanation in the edit summary.

Thus, partial contraposition can be obtained conditionally in an " E " type proposition with a change in quantity.

Thus in regions where malaria exerts or has exerted a strong selective pressure, sickle cell anemia has been selected for its conferred partial resistance to the disease.

Thus a partial isometry is also sometimes defined as a closed partially defined isometric map.

Thus, if a road is built across the fault as in the figure panel Time 1, it is perpendicular to the fault trace at the point E, where the fault is locked.

Thus a scholar might attempt to trace how a particular literary idea or motif traveled between nations over time.

Thus for the pragmatist, verifiability as an operational definition ( or test ) of the empirical meaning of a statement requires that the speaker know how to apply the statement, and when not to apply it, and be able to trace the consequences of the statement in situations both real and hypothetical.

Thus a brother / sister from the same Idip means that they can trace their origin to the same mother or father.

Thus, rare organisms like Chloroflexus aurantiacus that can survive using either respiration or photosynthesis are of interest in on-going attempts to trace the evolution of photosynthesis.

Thus we can trace the development of the legend from a letter, but no image in Eusebius, to an image painted by a court painter in Addai ; then to a miracle caused by the letter in Procopius, which becomes a miracle caused by a miraculously-created image supernaturally made when Jesus pressed a cloth to his wet face in Evagrius.

' Thus the sage will be able to work in harmony with Tao to accomplish what is needed, and, working in perfect harmony with the Tao, leave no trace of having done it.

Thus the founder of Rome, Romulus, was traditionally nursed only by a she-wolf, whereas Telephus, to whom the Attalids trace their ancestry, is shown in the frieze being suckled by a she-lion.

" Thus the romance would trace back the model of ideal civilization that it proposes, a model also for the orders of chivalry created from the 14th century onwards, to a legendary origin where the glory of Alexander is united with the fame of Arthur.

Thus the first trace map is given by

Thus, the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of g ( x, p ) with the Wigner function.

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, the trace of the hat matrix is n / k.

If we call P the plane that is perpendicular to Δ, then the normals to the planes all belong to P. Thus, the poles of the planes belonging to the same zone are on the trace of the plane perpendicular to the axis.

Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.

Thus jazz is transmuted into something holy, the sacred road to integration of being.

Thus the fictional detective is much more than a simple businessman.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus in both types attention is focused on the community itself, and its phenomenological life.

Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

Thus human perception and human volition is the immanent cause of all social change and this most truly when the change reaches the civilizational level.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

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 science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.

Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.

Thus there is a clearer division of authority, administrative and legislative.

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, the need for the B-70 as a strategic weapon system is doubtful.

Thus technical efficiency is achieved at the expense of actual experience.

Thus, there is an added incentive to stay on the job.

Thus, the Span of its ossification was shortened and the center's ability to `` catch up '' in ossification is demonstrated.

Thus T is not diagonalizable.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Thus, the study of the solutions to the equation Af is reduced to the study of the space of solutions of a differential equation of the form Af.

Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.

Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.