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Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.
from Brown Corpus
Some Related Sentences
Since and Af
Since Af is ferromagnetic, we felt that any results obtained from the magnetically contaminated Af would be suspect.
Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).
Since D and D' are both diagonalizable and they commute, they are simultaneously diagonalizable, and Af is diagonalizable.
Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.
A final class of exceptional lines is identifiable from the following considerations: Since no two generators of Af can intersect, it follows that their image curves can have no free intersections.
Since two curves of symbol Af on Q intersect in Af points, it follows that there are Af lines of Af which are tangent to Aj.
Since by the principle of optimality the remaining stages use an optimal Af-stage policy with respect to Af, we may enter section Af of the table at this state Af and read off the optimal policy for stage Af and the resulting state Af.
Since and are
Since the hazards of poor communication are so great, p can be justified as a habitable site only on the basis of unusual productivity such as is made available by a waterfall for milling purposes, a mine, or a sugar maple camp.
Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.
Since the recognition of Israel as a nation state, claims are made in many cases which identify the claimant as a member of the new body politic.
Since accurate base maps are necessary for any planning program, the first step taken by the planning division to implement the long-range state plan has been to prepare two series of base maps -- one at a scale of 1 inch to a mile, and the second a series of 26 sheets at a scale of 1 inch to 2000 feet, covering the entire state.
Since conventional methods are insensitive at the low frequencies of these molecular transitions, the paramagnetic resonance method is being used instead.
Since broadcast frequencies are very limited in number, these objectives are to some extent inconsistent in that not all of them can be fully realized, and to the extent that each is realized, there is a corresponding reduction of the possibilities for fullest achievement of the others.
Since the provision of skywave service requires adequate freedom from interference, only class 1, stations are capable of rendering skywave service.
Since fly cutters are one sided and not balanced, they should be used at the slowest speed available, and fed very slowly to avoid binding.
Since there are almost no schools in the country offering graduate work in interior design this rule cannot at present be observed.
Since charges are relatively the same, reserving a car before you leave for Europe will assure you of having one on tap when you want it.
Since practical washing procedures are both lengthy and expensive, a number of laboratory tests have been developed for the numerical evaluation of detergents.
Since the circulating thyroid hormones are the amino acids thyroxine and tri-iodothyronine ( cf. Section C ), it is clear that some mechanism must exist in the thyroid gland for their release from proteins before secretion.
Since emotional reactions in the higher vertebrates depend on individual experience and are aroused in man, in addition, by complex symbols, one would expect that the hypothalamus could be excited from the cortex.
Since and linearly
Since in oral languages the elements of sound are for the most part produced linearly in time ( that is, in a word like cat the a sound comes after the c sound, and the t sound comes after that ), they can generally be easily written in a linear ( one-dimensional ) writing system such as an alphabet.
Since this is a second-order differential equation, there must be two linearly independent solutions.
Since the amplitude at these reflections grows linearly with the number N of scatterers, the observed intensity of these spots should grow quadratically, like N < sup > 2 </ sup >.
Since the field inside a parallel plate capacitor depends linearly on the potential, the index of refraction depends linearly on the field ( for crystals where Pockels effect dominates ), and the phase depends linearly on the index of refraction, the phase modulation must depend linearly on the potential applied to the EOM.
Since the power dissipation, P = I < sup > 2 </ sup > R, increases with the square of the current but only increases approximately linearly with the number of windings, the power lost in the windings can be minimized by reducing I and increasing the number of turns N proportionally.
Since the recording is imposed linearly on the medium, and the medium is easily cut and glued, sounds recorded can be easily re-sequenced and separated onto separate tracks, allowing more control in mixing.
Since, for a given bullet shape, frontal surface increases as the square of the calibre, and mass increases as the cube of the diameter, then sectional density grows linearly with bore diameter.
Since barycentric coordinates are a linear transformation of Cartesian coordinates, it follows that they vary linearly along the edges and over the area of the triangle.
Since illumination and reflectance combine multiplicatively, the components are made additive by taking the logarithm of the image intensity, so that these multiplicative components of the image can be separated linearly in the frequency domain.
Since calculations show that the Kuzyk limit increases faster than linearly, it is best to link together as many molecules as possible.
Since the space of meromorphic 1-forms is a 1-dimensional vector space over the field of meromorphic functions, any two meromorphic 1-forms yield linearly equivalent divisors.
Since the total flight-time difference between " fast " and " slow " lightsignals would be expected to scale linearly with distance in simple emission theory, and the study would ( statistically ) have included stars with a reasonable spread of distances and orbital speeds and orientations, deSitter concluded that the effect should have been seen if the model was correct, and its absence meant that the emission theory was almost certainly wrong.
Since electric field components parallel to the wires are reflected, the transmitted wave has an electric field purely in the direction perpendicular to the wires, and is thus linearly polarized.