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 are commonly referred to as post-Newtonian approximations.

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

These approximations will fail when dealing with relatively strong acids or bases ( pKa more than a couple units away from 7 ), dilute or very concentrated solutions ( less than 1 mM or greater than 1M ), or heavily skewed acid / base ratios ( more than 100 to 1 ).

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 are not, however, differences in federal structure.

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

These little songs, however, were sweet nothings from the heart, tender memories of his childhood, little melodies that anyone could hum and that would make one want to weep.

These became `` strays '', the term bein' restricted to cattle, however, as hosses, under like circumstances, were spoken of as `` stray hosses '', not merely `` strays ''.

These qualities alone, however, would not account for their success, and it took me a while to discover the crowning virtue that completes this company's collective personality.

These, however, are exceptions.

These applications are declining, however.

These ordinances proved, however, generally ineffectual to secure strictness of diet, and contemporaneous literature abounds with satirical remarks and complaints concerning the inordinate extravagance of the tables of the abbots.

These authors proved, however, that no optical system can justify these suppositions, since they are contradictory to the fundamental laws of reflexion and refraction.

These conclusions were, however, questioned by another review, which found that clozapine, amisulpride, and olanzapine and risperidone were more effective Clozapine has appeared to be more effective than other atypical antipsychotics, although it has previously been banned due to its potentially lethal side effects.

These fish can still, however, find their way around by means of their lateral lines, which are highly sensitive to fluctuating water pressure.

These tests are created ideally through collaboration between business customers, business analysts, testers and developers, however the business customers ( product owners ) are the primary owners of these tests.

These elections constitutionally must be held no longer than every five years apart, however elections can be called whenever the Government so chooses to seek a new mandate or loses a vote of no confidence in Parliament.

( These are actually referred to in the translation as " balls "; however, they were clearly intended to be bubbles, since they pop, and are taken from Bubble Bobble.

These were however used as abbreviations in diagrams or procedures ; there was no concept of atoms combining to form molecules.

These programs, however, were never supported by the landholding families, because they favored the peasants.

These names do not use the name " Confucius " at all, but instead center on the figure or ideal of the Confucian scholar ; however, the suffixes of jiā, jiào, and xué carry different implications as to the nature of Confucianism itself.

These frequencies are at the upper range of human hearing and are inaudible to many people ; however, some people ( especially children ) will perceive a high-pitched tone near an operating television CRT.

These " proofs ", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type.

These claims, however, were unsubstantiated and neither the Rockies nor Paciolan have sought investigation into the matter.

These machines were, however, very heavy and expensive.

These games, however, do not allow the participants to choose how much they drink, as this is dependent on the act of skill.

These terms, however, do not apply to other beneficiaries of government spending such as tax incentives and military spending.

These however may have been a granuloma, or even a post-traumatic neuroma.

These more recent versions, however, maintain that only the subjective, qualitative aspects of mental states are epiphenomenal.