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, 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 Euclidean geometry corresponds to the choice of the group E ( 3 ) of distance-preserving transformations of the Euclidean space R < sup > 3 </ sup >, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.

Thus, the most common letter in English, the letter " E ," has the shortest code, a single dot.

Thus optical amplification, and laser operation, can take place at a frequency of ν < sub > 32 </ sub > ( E < sub > 3 </ sub >- E < sub > 2 </ sub > = hν < sub > 32 </ sub >).

Thus Dumézil conceives of Víðarr as a spatial god ( per the hypothesized Víð -/ wide root-O. E.

While this equation suggests that the activation energy is dependent on temperature, in regimes in which the Arrhenius equation is valid this is cancelled by the temperature dependence of k. Thus, E < sub > a </ sub > can be evaluated from the reaction rate coefficient at any temperature ( within the validity of the Arrhenius equation ).

Thus the total time is O ( E log E ) = O ( E log V ).

Thus, according to Paul L. Maier, the majority of Herod biographers and theologians hold that the Massacre of the Innocents is " legend and not historical ": Geza Vermes and E. P. Sanders, for example, regard the story as part of a creative hagiography.

Thus, by charging a higher per unit price and a lower lump-sum fee, the firm has generated area E more profit than if it had charged a lower per-unit price and a higher lump-sum fee.

Thus, what we call the year 3053 A. D. ( the year the novel begins ) is in the novel the year 1108 A. E.

Thus the three notes ( E, C and A ) are played in quick succession at relative ease to the player.

Light's wavenumber is proportional to frequency, and therefore also proportional to light's quantum energy E. Thus,.

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.

Given two tensor bundles E → M and F → M, a map A: Γ ( E ) → Γ ( F ) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A ( fs ,...) = fA ( s ,...) in each argument, where f is a smooth function on M. Thus a tensor is not only a linear map on the vector space of sections, but a C < sup >∞</ sup >( M )- linear map on the module of sections.

Thus, a peak in E ’’ is observed at the frequency.

Thus, on July 19, 1990, the Konkan Railway Corporation Limited ( KRCL ) was incorporated as a public limited company under the Companies Act, 1956, with its headquarters at CBD Belapur in Navi Mumbai and E. Sreedharan, a senior railway official, as its first Chairman and Managing Director.

Thus, replacing E by K we may assume that f < sub > n </ sub > converge to f pointwise on E. Next, by the definition of the Lebesgue Integral, it is enough to show that if φ is any non-negative simple function less than or equal to f, then

Thus the set of strings of length k in the domain is exactly the set of such strings already enumerated.

Thus, if ( M, d ) and ( N, d ) are two metric spaces, and, then there is a constant k such that

Thus, the initial step corresponds to k =

Thus, r < sub > k </ sub > is smaller than its predecessor r < sub > k − 1 </ sub > for all k ≥ 0.

Thus there is a number x ≡ ( q − 1 )< sup >( p − 2 )</ sup > for which ( q − 1 )· x ≡ 1 ( mod p ), and therefore a number k for which ( q − 1 )· x − 1 = kp.

Thus, the " kilo -" symbol, k, can be used to produce km, kg, and kW, ( kilometre, kilogram, and kilowatt ).

Thus for English the " e " box is the largest while the " j ", " k ", " q ", " x ", and " z " boxes are the smallest.

Thus, the DCT-I corresponds to the boundary conditions: x < sub > n </ sub > is even around n = 0 and even around n = N-1 ; similarly for X < sub > k </ sub >.

Thus, if one of the coordinates, say q < sub > k </ sub >, does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that

Thus, the absence of the ignorable coordinate q < sub > k </ sub > from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of q < sub > k </ sub >; the Lagrangian is invariant, and is said to exhibit a symmetry under such transformations.

Thus, every element of Dic < sub > n </ sub > can be uniquely written as a < sup > k </ sup > x < sup > j </ sup >, where 0 ≤ k < 2n and j = 0 or 1.

Thus, if k and n are coprime, z < sup > k </ sup > is also a primitive nth root of unity, and therefore there are φ ( n ) ( where φ is Euler's totient function ) distinct primitive nth roots of unity.

Thus, given the field K, any algebra can be specified up to isomorphism by giving its dimension ( say n ), and specifying n < sup > 3 </ sup > structure coefficients c < sub > i, j, k </ sub >, which are scalars.

Thus, in mathematical physics, the structure coefficients are often written c < sub > i, j </ sub >< sup > k </ sup >, and their defining rule is written using the Einstein notation as

Thus, the DST-I corresponds to the boundary conditions: x < sub > n </ sub > is odd around n =- 1 and odd around n = N ; similarly for X < sub > k </ sub >.

Thus, mD represents a case with hypomania and major depression.

Thus, at a given angular rate of rotation, a concave surface represents the stable situation, and the more rapid the rotation, the more concave this surface.

Thus in Modern Greek, Eta is pronounced and represents the sound / i / ( a close front unrounded vowel ).

Thus, in the word ship ( pronounced ), the digraph ⟨ sh ⟩ ( two letters ) represents the sound.

Thus if a represents Socrates then Phil ( a ) asserts the first proposition, p ; if a instead represents Plato then Phil ( a ) asserts the second proposition, q.

Thus, the Jones vector represents ( relative ) amplitude and ( relative ) phase of electric field in x and y directions.

) Thus the main sequence represents the primary hydrogen-burning stage of a star's lifetime.

Thus, in several languages of the Banks Islands, including Mwotlap, the simple m stands for, but an m with a macron ( m ̄) is a labial-velar nasal ; while the simple n stands for the common alveolar nasal, an n with macron ( n ̄) represents the velar nasal ; the vowel ē stands for a ( short ) higher by contrast with plain e ; likewise ō contrasts with plain o.

Thus Kim Jong-il was, until 2011, in substance head of the state, but theoretically the chairman of the SPA Presidium represents the state and is responsible for foreign affairs, such as reception.

Thus represents a sequence of three phonemes,, ( the word push in standard English ), while represents the phonetic sequence of sounds ( aspirated " p "),, ( the usual pronunciation of push ).

Thus AB represents shadows and reflections of physical things, and BC the physical things themselves.

Thus, a cap of 13 million tons could be credited to Japan ( which represents about 4 % of its base-year emissions ).

Thus, in the language of some Jewish poets, the beloved one's curls indicate the mysteries of the Deity ; sensuous pleasures, and especially intoxication, typify the highest degree of divine love as ecstatic contemplation ; while the wine-room represents merely the state through which the human qualities merge or are exalted into those of God.

Thus for a transition, K represents the core level hole, the relaxing electron's initial state, and the emitted electron's initial energy state.

Thus, the reading is given by the number of whole divisions that are visible on the scale of the frame, multiplied by 25 ( the number of thousandths of an inch that each division represents ), plus the number of that division on the thimble which coincides with the axial zero line on the frame.

Thus, on these processors, the bit represents an ' is_quiet ' flag.

Thus, on these processors, the bit represents an ' is_signaling ' flag.

Thus the set of functions, where g is any one-argument function, represents the entire set of functions in variables x, y that could have produced the x-partial derivative 2x + y.

Thus if S. tchadensis is an ancestral relative of the chimpanzees ( or gorillas ) then it represents the first known member of their lineage.

Thus the premium represents the ability to instantly participate in the fruits of the fund manager's decisions.

Thus, maximin in the original position represents a formulation of social equality.

Thus a cask standing on end represents the mountain on which Christ is tempted by the Devil ; thunder is imitated by the report of a gun ; in order to signify that the Devil had entered into him, Judas holds a bird of black plumage before his mouth and makes it flutter.