However, one must expect that vaporization and ejection of material by hypervelocity impacts would cause a deviation from a linear relationship.

However, the bilinear transformation can also be interpreted as a single linear transformation from the tensor product to Z.

However, Judeo-Christian thought also makes an " innovation of the first importance ", Eliade says, because it embraces the notion of linear, historical time ; in Christianity, " time is no longer the circular Time of the Eternal Return ; it has become linear and irreversible Time ".

However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics.

However, it is more usual to use a linear transformation to transform these absolute values into values which resemble the more familiar Pauling values.

However the general concept of linear filtering is broader, also used in statistics, data analysis, and mechanical engineering among other fields and technologies.

However, for weak fields, a linear approximation can be made.

However, for linear acceleration, Newton expressed the idea of undetectability of straight-line accelerations held in common:

However, iterative methods are often useful even for linear problems involving a large number of variables ( sometimes of the order of millions ), where direct methods would be prohibitively expensive ( and in some cases impossible ) even with the best available computing power.

However, the process of binary fission is very much different from the process of mitosis, because of the non-involvement of nuclear dynamics and lack of linear chromosomes.

However, the delivering of the consolamentum, on which historical Catharism was based, required a linear succession by a bon homme in good standing.

However, using interaction techniques that are not pointer based have proven problematic with both pie and linear menus.

However, Buckhardt has been accused of setting forth a linear Whiggish view of history in seeing the Renaissance as the origin of the modern world.

However, the 1 to 5 scale can be arbitrary and need not be on a linear scale.

However the first result is for positive linear functionals, not bounded linear functionals, so the two facts are not equivalent.

However a section near Madrid has been squatted, becoming a linear city of about 40, 000 people and 15 km in length.

However, angular speed must be in radians, by the assumed direct relationship between linear speed and angular speed at the beginning of the derivation.

However, modern tape drives that used advanced techniques like multilevel forward error correction, shingling, and linear serpentine layout for writing data to tape, along with lower disk drive prices, have made such alternatives obsolete.

However, it may be convenient for notation to consider n-ary functions, as for example multilinear maps ( which are not linear maps on the product space, if n ≠ 1 ).

However, the direct current homopolar linear motor railgun is another high acceleration linear motor design.

However, it is essential that the various mathematical symbols used in the equations be understood so that the mathematical processes can be done properly and in their correct order.

However, the law of mass action is valid only for concerted one-step reactions that proceed through a single transition state and is not valid in general because rate equations do not, in general, follow the stoichiometry of the reaction as Guldberg and Waage had proposed ( see, for example, nucleophilic aliphatic substitution by S < sub > N </ sub > 1 or reaction of hydrogen and bromine to form hydrogen bromide ).

However, he realized that his equations permitted the introduction of a constant term which could counteract the attractive force of gravity on the cosmic scale.

However, equations can also be correct for only certain values of the variables.

However, in the far-field EM radiation which is described by the two source-free Maxwell curl operator equations, a more correct description is that a time-change in one type of field is proportional to a space-change in the other.

However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations.

However, the cost becomes prohibitive for systems with millions of equations.

However, when multiple nucleons are assembled into an atomic nucleus ( nuclide ), these fundamental equations become too difficult to solve directly ( see lattice QCD ).

However in general the Ricci flow equations lead to singularities of the metric after a finite time.

However, this theory needs to be generalized in ( 2 + 1 ) or ( 3 + 1 ) dimensions although, in principle, the field equations are amenable to such generalization as shown with the inclusion of a one-graviton process and yielding the correct Newtonian limit in d dimensions if a dilaton is included.

However, the interpretations leading to the development of the equations were very different, since at the time of development, statistical dynamics had not yet unified the separate studies of thermodynamics and classical dynamics.

However, except for simple geometries these equations are notoriously difficult to solve.

However, soon after Einstein developed his static theory, observations by Edwin Hubble indicated that the universe appears to be expanding ; this was consistent with a cosmological solution to the original general-relativity equations that had been found by the mathematician Friedmann.

However, these equations keep.

However, the equations of quantum mechanics can also be considered equations of motion, since they are differential equations of the wavefunction, which describes how a quantum state behaves analogously using the space and time coordinates of the particles.

However, Newton's original precession equations did not work and were revised considerably by Jean le Rond d ' Alembert and subsequent scientists.

) However some polynomial equations, of arbitrarily high degree, are solvable with such operations.

However, Hamilton's equations usually don't reduce the difficulty of finding explicit solutions.

However, the equations can be further generalized to then be extended to apply to quantum mechanics as well as to classical mechanics, through the deformation of the Poisson algebra over p and q to the algebra of Moyal brackets.

However, classical mechanics still remains useful today ( e. g. for calculating spacecraft trajectories ), because its equations are much simpler and its predictive power is sufficiently accurate in most circumstances ; the difference is that the accuracy of relativity is even greater.

However, even before Limits to Growth was published, Eduard Pestel and Mihajlo Mesarovic of Case Western Reserve University had begun work on a far more elaborate model ( it distinguished ten world regions and involved 200, 000 equations compared with 1000 in the Meadows model ).