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 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 equations explain quantitatively how quarks can bind together into protons and neutrons ( and all the other hadrons ).

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 continuing pressures, social, economic and military, are doing much to keep China in a heightening state of tension.

These large amphibians retain several larval characteristics in their adult state ; gills slits are present and the eyes are unlidded.

These are naturally in a ground state but can be excited

These states are labeled by a set of quantum numbers summarized in the term symbol and usually associated with particular electron configurations, i. e., by occupation schemes of atomic orbitals ( e. g., 1s < sup > 2 </ sup > 2s < sup > 2 </ sup > 2p < sup > 6 </ sup > for the ground state of neon -- term symbol: < sup > 1 </ sup > S < sub > 0 </ sub >).

These include four national forests and one national preserve within state borders that provide over 25 % of the state's public recreation land.

These processes are important in the vacuum state and renormalization of a quantum field theory.

These eventually created a state within the state.

In the center of the helmet was a center black stripe surrounded by 2 gold stripes and 2 white stripes, These colors represented the two college rival schools in the state of Georgia ; rival schools Georgia Tech Yellow Jackets ( White and Gold ) and the Georgia Bulldogs ( Red and Black ) Although the gold was later taken out, the white remains to this day.

These services would operate in rural telephone exchanges, primarily in small towns in the state of Iowa.

These Sunday laws enacted at the state and local levels would sometimes carry penalties for doing non-religious activities on Sunday as part of an effort to enforce religious observance and church attendance.

These attributes may be implemented, in a variety of systems, by means of a two state device.

These extraordinary performances led Muttiah Muralitharan to state that Lara was the most dangerous batsman he had ever faced.

These changes came about partly as the result of the urgings of Edwin Samuel Montagu, an influential anti-Zionist Jew and secretary of state for India, who was concerned that the declaration without those changes could result in increased anti-Semitic persecution.

: These views were made clear in a disclaimer printed in the Factbook: " Serbia and Montenegro have asserted the formation of a joint independent state, but this entity has not been recognized as a state by the United States.

These professions state that Jesus suffered, died, was buried, and was resurrected from the dead in order to grant eternal life to those who believe in him and trust him for the remission of their sins ( salvation ).

These policies stimulated rapid growth of productivity, while boosting the army without much burden on the state treasury.

Because his vision of personal and social perfections was framed as a revival of the ordered society of earlier times, Confucius is often considered a great proponent of conservatism, but a closer look at what he proposes often shows that he used ( and perhaps twisted ) past institutions and rites to push a new political agenda of his own: a revival of a unified royal state, whose rulers would succeed to power on the basis of their moral merits instead of lineage. These would be rulers devoted to their people, striving for personal and social perfection, and such a ruler would spread his own virtues to the people instead of imposing proper behavior with laws and rules.

These are intended to ensure basic political, social and economic standards that a nation state, or intergovernmental body is obliged to provide to its citizens but many do include its governments.

These languages are exactly all languages that can be decided by a finite state automaton.

These perturbations result in different arrangement of electrons in the excited state of the involved molecules and therefore in different effects.

These critics state that Mary Baker Eddy's interpretation of Christian scripture diverges too greatly from basic tenets of Christianity.

These storms can be severe, and the state usually averages one tornado per year.