These range from simplified forms of the first-principles equations that are easier or faster to solve, to approximations limiting the size of the system ( for example, periodic boundary conditions ), to fundamental approximations to the underlying equations that are required to achieve any solution to them at all.

These algorithms are numerical approximations to the rendering equation.

These equations are notoriously difficult to solve, but in many instances approximations suffice for a good description of lifting airfoils.

These approximations take two forms:

These solutions yield good rational approximations of the form x / y to the square root of n. In Cartesian coordinates, the equation has the form of a hyperbola ; it can be seen that solutions occur where the curve has integral ( x, y ) coordinates.

These fractions can be found by the method of continued fractions: this arithmetical technique provides a series of progressively better approximations of any real numeric value by proper fractions.

These approximations are only valid for antennas at heights that are small compared to the radius of the Earth.

These are German terms for which there are no exact English equivalents, however we have approximations:

These species are of interest as molecular approximations of the cubic diamond framework, terminated with C-H bonds.

These asymptotic approximations are called straight line Bode plots or uncorrected Bode plots and are useful because they can be drawn by hand following a few simple rules.

These approximations are used in a variety of practical flow problems of scientific and engineering interest.

These bandwidth requirements are important in digital electronics, where finite-bandwidth analog approximations to square-wave-like waveforms are used.

These approximations, however, must be taken with caution, as no actual recording exists and it is known that Tupi had at least seven dialects.

These are commonly referred to as post-Newtonian approximations.

These ideas are inspired from Auslander's work in approximations.

For the four largest Heegner numbers, the approximations one obtains < ref > These can be checked by computing on a calculator, and

These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier – Stokes equations.

These suggest an iterative scheme, qualitatively similar in structure to a Gibbs sampler, to evolve successively improved approximations to and.

These methods are called differential since they are based on local Taylor series approximations of the image signal ; that is, they use partial derivatives with respect to the spatial and temporal coordinates.

These approximations are possible because the mixing angle θ < sub > 13 </ sub > is very small and because two of the mass states are very close in mass compared to the third.

These approximations can be derived from the continued fraction expansion of:

These numbers are all even ; each such number is twice the numerator in one of the rational approximations to discussed above.

These approximations are only valid under restricted conditions.

These remarkable ships and weapons, ranging the oceans, will be capable of accurate fire on targets virtually anywhere on earth.

These principles, however, will not be served by violence in any form.

These chatty merchants made amusing and instructive traveling companions, for their business took them to all four corners of the globe, and Florentine gossip had already reached a high stage of development as even a cursory glance at the Inferno will prove.

These cases in which light is already visible at the other end of the tunnel are ones which over the next few years will absorb the bulk of our capital assistance.

These data are not of the precision obtainable by the methods previously mentioned, but the vast number of approximate values available will be useful in many areas.

These additional modern bombers will replace some of the older B-47 medium bombers ; ;

These cut-down plants will bud and blossom in record time and will behave just as they did in early spring.

These will be drilled and tapped later on.

These will serve as lifting pads for the electrical contacts.

These will be available at cost from our Plans Department.

These shifts will continue in the next 10 yrs..

These are only halfway measures, and the answer will come when some way is found to allow the technical man in industry to progress without limit in salary and prestige ''.

These area reports will be followed, according to present plans, by a summary report, which will include a detailed statement on methods.

These findings, and others which will in time be developed, will affect the method of glottochronological inquiry.

These are arranged approximately in the order of the vociferousness of their proponents but will be discussed in the reverse order in the hope that the true order of importance will result.

These will be numbered in the direction opposite to the flow of the process stream, so that stage R is the T stage from the end.

These scientists are convinced that a telescope unclouded by the earth's gases will go a long way toward bolstering or destroying cosmic theories.

These arrangements were imperiled by the ill will of Alfonso XI of Castile, who was, at the time, publicly mistreating his wife.

These will go away into eternal punishment, but the righteous into eternal life.

These realities form a " composite faithfulness " and are ( i ) " perseverance in the apostolic doctrine "; ( ii ) " the will to proclaim God's word "; ( iii ) " communion in the fundamental continuity of the Church, the Body of Christ, the faithful celebration of Baptism and the Eucharist "; ( iv ) " succession in the laying on of hands, the sign of ministerial continuity ".

These buses often need to undergo a degree of vehicle restoration to restore them to their original condition, and will have their livery and other details such as internal notices and rollsigns restored to be authentic to a specific time in the bus's actual history.