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With and r
With the reduced state variables, i. e. V < sub > r </ sub >= V < sub > m </ sub >/ V < sub > c </ sub >, P < sub > r </ sub >= P / P < sub > c </ sub > and T < sub > r </ sub >= T / T < sub > c </ sub >, the reduced form of the Van der Waals equation can be formulated:
* With r between 0 and 1, the population will eventually die, independent of the initial population.
* With r between 1 and 2, the population will quickly approach the value
* With r between 2 and 3, the population will also eventually approach the same value
* With r between 3. 45 and 3. 54 ( approximately ), from almost all initial conditions the population will approach permanent oscillations among four values.
* With r increasing beyond 3. 54, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc.
Consider now the acceleration due to the sphere of mass M experienced by a particle in the vicinity of the body of mass m. With R as the distance from the center of M to the center of m, let ∆ r be the ( relatively small ) distance of the particle from the center of the body of mass m. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass M. If the body of mass m is itself a sphere of radius ∆ r, then the new particle considered may be located on its surface, at a distance ( R ± ∆ r ) from the centre of the sphere of mass M, andr may be taken as positive where the particle's distance from M is greater than R. Leaving aside whatever gravitational acceleration may be experienced by the particle towards m on account of ms own mass, we have the acceleration on the particle due to gravitational force towards M as:
With the aid of the eunuch Zheng Zhong ( d. 107 CE ), Emperor He ( r. 88 – 105 CE ) had Empress Dowager Dou ( d. 97 CE ) put under house arrest and her clan stripped of power.
With the beginning of the Byzantine offensives in the East and the Balkans in the 10th century, especially under the warrior-emperors Nikephoros II ( r. 963 – 969 ), John I Tzimiskes ( r. 969 – 976 ) and Basil II ( r. 976 – 1025 ), newly gained territories were also incorporated into themes, although these were generally smaller than the original themes established in the 7th and 8th centuries.
As derived above, the ordinary generating function for the sequence of squares is With r = 1, α = 0, β = 3, A ( x ) = 0, and B ( x ) = x ( x + 1 ), we can verify that the squares grow as expected, like the squares:
With r = 1 / 4, α = 1, β = − 1 / 2, A ( x ) = 1 / 2, and B ( x ) = − 1 / 2, we can conclude that, for the Catalan numbers,
* Eric Uyttewaal: < cite > Dynamic Scheduling With Microsoft ( r ) Project 2000: The Book By and For Professionals </ cite >, ISBN 0-9708276-0-1
With r fixed, the versors
With a large-area emitter, which contains many individual emission sites, A < sub > r </ sub > will nearly always be very very much less than the " macroscopic " geometrical area ( A < sub > M </ sub >) of the emitter as observed visually ( see below ).
With the death of Theophilos, the young Michael III ( r. 842 – 867 ) ascended the throne.
With r, θ, and z being constant over time, this simplifies to
With some change of symbols ( and units ): r → J, R → − E and the material constant E < sup >− 2 </ sup > → 4π ε < sub > r </ sub > ε < sub > 0 </ sub > these equations take the familiar form:

With and between
With new mechanization the modern farmer must perform the work of six men: a machine stands between the agrarian and his soil.
With this conviction, the partition between the sacred and the secular disappears.
With tips, the girls average between $150 and $200 a week, depending on basic salary.
With completion of filling, net-like anastomoses were noted to be present between these separately derived branches.
With each vertex we associate certain numerical values, namely the set of positive differences in the parameter T between the vertex and its corresponding forward corner points.
With the vertex at Af in the C-plane we assume that Af is the parametric location on C of an ordinary intersection Q between C and Af.
With the exception of the Roman Catholic and the Orthodox Catholic Churches, most churches make no moral distinction between rhythm and mechanical or chemical contraceptives, allowing the couple free choice.
With a cop patrolling the road Muller would have to be inside a building -- if he was here at all, and not waiting for the prime minister somewhere between this street and the terminal building at La Guardia Airport.
With a small bomb, I tuck it between Cadillacs.
With the Academies in Europe ( second half of 16th century ) the gap between fine and applied arts was definitely set.
With the lack of international competition, representative matches between state teams were regarded with great importance.
With some exceptions, the knowledge of hypersonic aerodynamics has matured between the 1960s and the present decade.
With the coming to power of Thomas Sankara in Burkina Faso in 1983, relations between Ghana and Burkina became both warm and close.
With its strategic position, Constantinople controlled the route between Asia and Europe, as well as the passage from the Mediterranean Sea to the Black Sea.
With the internationalism of modern architecture and design, there were many exchanges between the Vkhutemas and the Bauhaus.
With both sides seeking battle, they soon stumbled upon one other on the dry ground between the Mehaigne and Petite Gheete rivers, close to the small village of Ramillies.
With the initial encouragement of Russian agents, a series of agreements was concluded between Serbia and Bulgaria in March 1912.
With this treaty, a mutual border was agreed between the two countries, together with an agreement for mutual military and diplomatic support in case of a Bulgarian or / and Austro-Hungarian attack.
With the fall of the Venetian Republic in 1797, its possessions in eastern Adriatic became subject to a dispute between France and Austria.
With such conflicting perspectives, it can be argued that a certain polarization between both supporters and opponents of President Uribe seems to be forming both inside and outside the country.
With what can crudely be summed up as a clash of ideologies between an expansion of ITV's commercial ethos and a public service approach more akin to the BBC, it was ultimately something of a compromise that eventually led to the formation of Channel 4 as launched in 1982.
With large spaces of land at a premium in south-west London, Fulham appear to be committed to a gradual increase of the ground's capacity often during the summer between seasons.
With Constantine ’ s death in 337, Constans and his two brothers, Constantine II and Constantius II divided the Roman world between themselves, after first deposing of virtually all of the relatives of their father who could possibly have a claim on the throne.
With no obstructions, these low mass particles were accelerated to high velocities by the voltage between the electrodes.
With consistent and high prices, area and production doubled between 1903 and 1917 and prices fell.

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