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These and formulas
These formulas can even serve as the definition of the trigonometric functions for complex arguments x.
These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT ( i. e. the trigonometric function values ), and it is not unusual for incautious FFT implementations to have much worse accuracy, e. g. if they use inaccurate trigonometric recurrence formulas.
These simple formulas will not do if the input values are not equally likely, or are not independent.
These two researchers developed several formulas for mineral nutrient solutions, known as Hoagland solution.
These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.
These formulas provide the solution for the initial-value problem for the wave equation.
These changes can be represented in formulas by inserting appropriate electrons into each half-reaction:
These formulas however only provide the calculation for the τ = τ < sub > 0 </ sub > case.
These formulas may be used when h is much smaller than the radius of the Earth ( 6371 km ), including all views from any mountaintops, aeroplanes, or high-altitude balloons.
These formulas characterize the periapsis and apoapsis of an orbit:
These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians.
These formulas are the basis for comparing and investigating the nature of non-random patterns in the structure of food web networks among many different types of ecosystems.
These formulas are generalized by the Faà di Bruno's formula.
These formulas can also be written using Lagrange's notation.
These formulas contain the " slope " of the line connecting P and Q, hence involve divisions between residue classes modulo n, which can be performed using the extended Euclidean algorithm.
These operators are direct extensions of their complex analogs: if and are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.
These ideas were regarded as dynamic forces that Herbart attempted to explain by means of mathematical formulas.
These formulas for denatured alcohol are found in 27 CFR part 21 of the Code of Federal Regulations.
These formulas are for single-frequency sine waves or narrowband signals.
These formulas only state in another form the assertions in the table of derivatives.
These formulas all provide a different view into the processes that generate longshore drift.
These five formulas are:
These formulas give the explicit numbers of rational curves of all degrees in a large class of Calabi – Yau manifolds, in terms of the Picard – Fuchs equations of the corresponding mirror manifolds.
These two formulas, added together, produce

These and are
These are just another couple of characters to roll.
These men are not callous.
These things are important to almost all Persians and perhaps most important to the most ordinary.
These are traversed by another line of vaults, and thus rooms, arched on all four sides, are formed.
These songs ( practically all Persian music, for that matter ) are limited to a range of two octaves.
These are not, however, differences in federal structure.
These my grandmother left in their places ( they are still there, more persistent and longer-lived than the generations of man ) and planted others like them, that flourished without careful tending.
These are like the initial ways in which the world forces itself upon the self and thrusts the self into decision and choice.
These are suggestive of Selma Lagerlof.
These assumptions lead to an organization with one man at the top, six directly under him, six under each of these, and so on until there are six levels of personnel.
These ways are absolutely irreconcilable because they offer two different recipes for man's redemption from chaos.
These responses are explicable in terms of characteristics inherent in the crisis.
These discoveries are of present interest chiefly to the scientific community ; ;
These are personality factors ; ;
These are, if the research is done with subtlety and skill, researchable topics, but the research is missing.
These moments are historical events in the lives of individual authors with which the student of comparative literature must be frequently concerned.
These lives are in themselves outside of the moral order and are unburdened with moral responsibility.
These conceptions and the manner in which they were transposed into poetry or engendered by poetic form are intrinsic to western life from the time of Aeschylus to that of Shakespeare.
These biographical analogies are obvious, and far too much time has been spent speculating on their possible implications.
These women whose organization grew out of the old suffrage movement are dedicated to Thomas Jefferson's dictum that one must cherish the people's spirit but `` Keep alive their attention ''.
These are, of course, the same people whose support he has only now rejected to seek the independent vote.
These are New York, Pennsylvania, Michigan, Maryland, Missouri, New Jersey, Illinois and Minnesota.
These men are spenders.

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