These assumptions and the resulting forms of the momentum equations are outlined below.

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 two equations can be differentiated and combined in various ways to produce the following data:

These equations of state were also previously published by the Soviet physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter.

These equations can be used as the starting point in the analysis of a flexible chain acting under any external force.

These equations express mathematically that, in the case of an object that moves along a circular path with a changing speed, the acceleration of the body may be decomposed into a perpendicular component that changes the direction of motion ( the centripetal acceleration ), and a parallel, or tangential component, that changes the speed.

These equations are usually combined into a single equation

These texts deal with solving algebraic equations, many of which are now lost.

These interactions are described by Maxwell's equations and the Lorentz force law.

These equations are derived from Maxwell's equations.

These equations specify how the geometry of space and time is influenced by whatever matter is present, and form the core of Einstein's general theory of relativity.

These equations are often referred to as the " equations of the straight line.

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

These equations are valid for all types of ( one-dimensional ) linear prediction.

These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.

These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.

These may be combined into so-called coupled equations, of which the strain-charge form is:

These equations of motion are then supplemented by commutation relations derived from the canonical quantization procedure described below, thereby incorporating quantum mechanical effects into the behavior of the field.

These observations corroborated Alexander Friedman's 1922 work, in which he derived the famous Friedmann equations.

These equations demonstrate that the simple harmonic motion is isochronous ( the period and frequency are independent of the amplitude and the initial phase of the motion ).

These equations were verified by experiment by Heinrich Hertz in 1887, and the wave theory became widely accepted.

These keys are the working principles of physics, mathematics and astronomy, principles which are then extrapolated, or projected, to explain phenomena of which we have little or no direct knowledge.

These circumstances explain an apparent contradiction in Strabo, who in one sentence says that the Aedui lived between the Saône and the Doubs, and in the next, that the Sequani lived across the Saône ( eastward ).

These genetic factors are believed to, in part, explain the differing rates of alcohol dependence among racial groups.

These actions served the purposes of both drawing audience and causing that audience to ask questions, giving a prophet the opportunity to explain the meaning of the behavior.

These findings help explain the categories used in Holdridge ’ s bioclassification scheme, which were then later simplified in Whittaker ’ s.

These definitions are archaic, their relevance having dissipated with the development of the English legal system over the centuries, but they do explain the origin of the term as used today.

These continual borrowings from other genres and cultural products create a familiar access point for a western audience and must in some part explain Cowboy Bebop ’ s popularity.

These secondary works ( e. g. Deconstruction for Beginners and Deconstructions: A User's Guide ) have attempted to explain deconstruction while being academically criticized as too far removed from the original texts and Derrida's actual position.

These assumptions are then used to explain why players choose Nash equilibrium strategies.

These studies explain the chemical changes associated with the major seismic discontinuities in the mantle, and show the crystallographic structures expected in the inner core of the Earth.

These claims are contradicted by experimental data and fail to explain evidence in the Luneau video of white dorsal plumage, distinctive flight behavior, and a perched woodpecker with white upper parts.

These " laws " are synthesized from a multitude of belief systems from around the world, and were collected in order to explain and categorize magical beliefs within a cohesive framework.

These teachings are thus held by followers in Judaism to define the inner meaning of both the Hebrew Bible and traditional Rabbinic literature, their formerly concealed transmitted dimension, as well as to explain the significance of Jewish religious observances.

These mean, respectively, how people understand themselves to be able to be in a right relationship with supernatural powers, and how to explain evil-or why bad things seem to happen to those who seem to be good people ".

These theories were later expanded and modified to explain a wider range of phenomena, especially conversions to new religious movements ( NRMs ).

These contrasting attitudes partly explain why the USSR has often been charged with playing a double game in 1939: carrying on open negotiations for an alliance with Britain and France while secretly considering propositions from Germany.

These laws have been employed to produce cosmological models that successfully explain the structure and the evolution of the universe we can observe.

These models can successfully explain nuclide properties, for example, whether or not a certain nuclide undergoes radioactive decay.

These experiments unequivocally exposed discrepancies which the theory was unable to explain.

These notions would be enough to explain the origin and the nature of laws that we call revolutionary ...

These commentaries could be read with the text of the Talmud and would help explain the meaning of the text.

These natural philosophers saw God as the first cause, and sought secondary causes to explain design in nature: the leading figure Sir John Herschel wrote in 1836 that by analogy with other intermediate causes " the origination of fresh species, could it ever come under our cognizance, would be found to be a natural in contradistinction to a miraculous process ".

These travels explain as well how the vinviðir ( wine wood ) the Norse were cutting down in the sagas is actually referring to the vines of Vitis riparia a species of wild grape that grows on trees.

These findings are also difficult to explain for the decay hypothesis, because decay of memory representations should depend only on how long the processing task delays rehearsal or recall, not on the content of the processing task.