Thus, u < sub > ρ </ sub > and u < sub > θ </ sub > form a local Cartesian coordinate system attached to the particle, and tied to the path traveled by the particle.

Thus, u + v is an element of W, too.

Carl Friedrich Gauss in his 1827 Disquisitiones generales circa superficies curvas ( General investigations of curved surfaces ) considered a surface parametrically, with the Cartesian coordinates x, y, and z of points on the surface depending on two auxiliary variables u and v. Thus a parametric surface is ( in today's terms ) a vector valued function

Thus, a and b are the eigenvalues corresponding to u and v, respectively.

Thus for a given characteristic strength of gravity, the amount of drag for a given speed u can be made arbitrarily small by increasing the speed v of the ultramundane corpuscles.

Thus, we can obtain the function u ( x ) through knowledge of the Green's function in equation ( 1 ), and the source term on the right hand side in equation ( 2 ).

Thus φ ↔ ψ (< u >#( φ )</ u >) is provable in T, as desired.

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

Thus, through the twentieth century, Roman Catholic theology and spirituality yet teaches us that Jesus Christ in the Eucharist has meaning for us as individuals, a theme elaborated especially well by Dietrich von Hildebrand in < u > Transformation in Christ </ u >.

Thus it appears that u should be countable.

Thus it is possible to recognize that a particular set u is countable, but not countable in a particular model of set theory, because there is no set in the model that gives a one-to-one correspondence between u and the natural numbers in that model.

Thus Bahá ' u ' lláh's eldest daughter, Bahíyyih ( given name, Fatimih ), was given the title of the Greatest Holy Leaf.

Thus, " complete " or " perfect teeth " refers to the complete bunodont dentition of ( 3-1-4-3 / 3-1-4-3 ) Prothero, Donald and Foss, Scott < u > The Evolution of Artiodactyls </ u > The Johns Hopkins University Press.

Thus, the Bahá ' í teachings recognize the divine origin of several world religions as different stages in the history of one religion, while believing that the revelation of Bahá ' u ' lláh is the most recent ( though not the last ), and therefore the most relevant to modern society.

Thus, the Bahá ' í teachings recognize the divine origin of several world religions as different stages in the history of one religion, while believing that the revelation of Bahá ' u ' lláh is the most recent ( though not the last — that there will never be a last ), and therefore the most relevant to modern society.

Thus, we can contrast Inuviniq takujara-I see < u > the </ u > dead person-with the table for non-specific verbs above.

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.