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Page "Equations of motion" ¶ 4
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However and equations
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, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics.
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, linear equations that have non-zero y-intercepts, when written in this manner, produce functions which will have neither property above and hence are not linear functions in this sense.
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, 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 ).

However and quantum
However, this was the temperature of one particular degree of freedom a quantum property called nuclear spin not the overall average thermodynamic temperature for all possible degrees in freedom.
However, in quantum physics, there is another type of angular momentum, called spin angular momentum, represented by the spin operator S. Almost all elementary particles have spin.
However, quantum field theories that are " local " in this sense appear to violate the principle of locality as defined by EPR, but they nevertheless do not violate locality in a more general sense.
However, this work has been subject to much criticism by quantum chemists as it was concluded that the predicted I < sub > h </ sub > symmetric structure was vibrationally unstable and the resulting cage undergoes a spontaneous symmetry break, yielding a puckered cage with rare T < sub > h </ sub > symmetry ( symmetry of a volleyball ).
However, most chemists followed Kekulé's lead in pursuing and developing what some have called " classical " structure theory, which was modified after the discovery of electrons ( 1897 ) and the development of quantum mechanics ( in the 1920s ).
However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.
However, according to quantum physics, individual molecules ( and atoms, electrons, protons etc.
However, the reason for this correction to the partition function remained obscure until the discovery of quantum mechanics.
However, quantum mechanical effects force electrons to take on discrete positions in orbitals.
However in the case of the free electron laser, atomic energy levels are not involved ; it appears that the operation of this rather exotic device can be explained without reference to quantum mechanics.
However, results from quantum mechanical research have suggested that this configuration is incorrect, and is in fact 7s < sup > 2 </ sup > 5f < sup > 14 </ sup > 7p < sup > 1 </ sup >.
However, in quantum physics, organic chemistry, and biochemistry, the term molecule is often used less strictly, also being applied to polyatomic ions.
However, for macroscopic processes classical mechanics is able to solve problems which are unmanageably difficult in quantum mechanics and hence remains useful and well used.
However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement.
However, practical problems have prevented one-time pads from being widely used, except with quantum key distribution.
However, when the well becomes very narrow ( on the scale of a few nanometers ), quantum effects become important.
However, the use of canonical quantization has left its mark on the language and interpretation of quantum field theory.
However, it leads to a fairly simple picture of the vacuum state and is not easily amenable to use in some quantum field theories, such as quantum chromodynamics which is known to have a complicated vacuum characterized by many different condensates.
However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material.
However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum ( since they are conjugate pairs ) or its energy and time ( since they too are conjugate pairs ); rather, it only provides a range of probabilities of where that particle might be given its momentum and momentum probability.
However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity ( the most accurate theory of gravity currently known ) and some of the fundamental assumptions of quantum theory.
However, chaotic systems do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.

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