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 u is an upper bound for S. To see that it is a least upper bound, notice that the limit of ( u < sub > n </ sub > − l < sub > n </ sub >) is 0, and so l =

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 areas can be measured in square metres ( m < sup > 2 </ sup >), square centimetres ( cm < sup > 2 </ sup >), square millimetres ( mm < sup > 2 </ sup >), square kilometres ( km < sup > 2 </ sup >), square feet ( ft < sup > 2 </ sup >), square yards ( yd < sup > 2 </ sup >), square miles ( mi < sup > 2 </ sup >), and so forth.

Thus, if P is a program in L < sub > 2 </ sub > which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

Thus this class of molecules belong to the D < sub > 2d </ sub > point group.

Thus, few drops of diamminesilver ( I ) hydroxide ( Ag ( NH < sub > 3 </ sub >)< sub > 2 </ sub > OH ) reacts with terminal alkynes signaled by formation of a white precipitate of the silver acetylide.

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

Thus for an ion which traverses a 1 m flight path, across a time of 2000 ns, given an initial accelerating voltage of 5000 V and noting that one amu is 1 × 10 < sup >− 27 </ sup > kg, the mass-to-charge ratio ( more correctly the mass-to-ionisation value ratio ) becomes Z amu / charge.

Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

Thus, oxygen is reduced to water and the original electron acceptors NAD < sup >+</ sup > and quinone are regenerated.

Thus, for example, the English form is is an apparent cognate of German ist, Latin est and Russian < nowiki > jest '</ nowiki >, even though the Germanic, Italic, and Slavic language groups split at least three thousand years ago.

Thus, the molecular formula of ethanol may be written in a paper in conformational, three-dimensional, full two-dimensional ( indicating every bond with no three-dimensional directions ), compressed two-dimensional ( CH < sub > 3 </ sub >– CH < sub > 2 </ sub >– OH ), separating the functional group from another part of the molecule ( C < sub > 2 </ sub > H < sub > 5 </ sub > OH ), or by its atomic constituents ( C < sub > 2 </ sub > H < sub > 6 </ sub > O ), according to what is discussed.

Thus ℵ < sub > 0 </ sub > is a regular cardinal.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

Thus the Green's function describes the influence at of the data f and g. For the case of the interior of a sphere of radius a, the Green's function may be obtained by means of a reflection ( Sommerfeld, 1949 ): the source point P at distance ρ from the center of the sphere is reflected along its radial line to a point P that is at a distance

Thus, if the two densities are equal, ρ < sub > o </ sub >

Thus we define the representations of G on an Hilbert space H to be those group homomorphisms, ρ, which arise from continuous actions of G on H. We say that a representation ρ is unitary if ρ ( g ) is a unitary operator for all g ∈ G ; i. e., for all v, w ∈ H. ( I. e.

Thus the density matrix describing ρ is diagonal in the energy representation.

# There is a positive real number r, called the Perron root or the Perron – Frobenius eigenvalue, such that r is an eigenvalue of A and any other eigenvalue λ ( possibly, complex ) is strictly smaller than r in absolute value, | λ | < r. Thus, the spectral radius ρ ( A ) is equal to r.

Thus the only positive eigenvectors are those associated with ρ ( A ).

Thus, if the system is finite ( finite dimensional matrix representation ), the entropy S ( ρ ) quantifies the departure of the system from a pure state.

Thus, middle C would be heard as the note B < sub > 3 </ sub > just below it, and so on.

Thus, when light is normally incident on the grating, the diffracted light will have maxima at angles θ < sub > m </ sub > given by:

where θ is the angle between the vectors Ω < sub > P </ sub > and L. Thus, if the gyroscope's spin slows down ( for example, due to friction ), its angular momentum decreases and so the rate of precession increases.

Thus, psi ( ψ ) may be written as ps, 4 or y ; xi ( ξ ) as ks, x or 3 ; and theta ( θ ) as th or 8.

Thus, if θ is one value of arg ( z ), the other values are given by arg ( z ) = θ + 2nπ, where n is any integer ≠ 0.

The function f is computable, so there is a formula δ representing f in T. Thus for each formula θ, T proves

Thus, the angle θ does not change as its apex point is moved around on the circle ( green, blue and gold angles ).

Thus, a vacuum with ⟨ θ ⟩= 0 will shift to a different vacuum with ⟨ θ ⟩=− ε.

Thus vector ( x, y, z ) is the unit axis of rotation scaled by tan < sup > θ </ sup >⁄< sub > 2 </ sub >.

Thus, it is a special case of a phase shift gate ( next ) with θ = π.

Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ.

Thus we have a formula for the rotation matrix R corresponding to an axis angle representation θ:

Thus the maximum distance is obtained if < var > θ </ var > is 45 degrees.

Thus, when given a Poisson regression model θ and an input vector, the predicted mean of the associated Poisson distribution is given by