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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.
from Brown Corpus
Some Related Sentences
Thus and if
Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.
Thus, if public pressure sets the effective limit to the price that the industry may charge, this pressure is itself a function of the wage rate.
Thus, during any given year ( A ) if the revenues from the residential service are $7,000,000, ( B ) if the operating expenses imputed to this class of service come to $5,000,000, and ( C ) if the net investment in ( or value of ) the plant and equipment deemed devoted to this service amounts to $30,000,000, the cost analyst will report that residential service, in the aggregate, has yielded a return of $2,000,000 or 6-2/3 per cent.
Thus if the gyro and platform-controller combination maintains the platform with zero angular deviation about the **yf axis, the system can be rotated with an angular velocity Af if a torque is supplied to the gyro output axis Aj.
Thus, if we are to observe American folklore in the twentieth century, we will do well to establish the relationships between folklore, nationalism and imperialism at the outset.
Thus, if what is at issue is whether `` All S is P '', it is indifferent whether `` Some S is not P '' or `` No S is P '', since in either case the judgment in question is false.
For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.
Thus throwing a rock at someone for the purpose of hitting him is a battery if the rock in fact strikes the person and is an assault if the rock misses.
Thus, a court officer taking possession of goods under a court order may use force if reasonably necessary.
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 every historian were to claim that there was a solar eclipse in the year 1600, then though we might at first naively regard that as in violation of natural laws, we'd come to accept it as a fact.
Thus and E
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 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 >).
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, 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, E ( k ) dk represents the contribution to the kinetic energy from all the Fourier modes with k < | k | < k + dk, and therefore,
Thus, what we call the year 3053 A. D. ( the year the novel begins ) is in the novel the year 1108 A. E.
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, 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 and is
Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.
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 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, 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 Span of its ossification was shortened and the center's ability to `` catch up '' in ossification is demonstrated.
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 Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.