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 and ( approximately 3. 45 ), from almost all initial conditions the population will approach permanent oscillations between two values.

* With r between 3. 45 and 3. 54 ( approximately ), from almost all initial conditions the population will approach permanent oscillations among four values.

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, and ∆ r 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 increasing advance of the knife into the coating the shear plane extends to the coatings surface and the shear angle rapidly decreases.

With the increasing production of talking pictures in the early 1930s, film narrators like Heigo began to lose work, and Akira moved back in with his parents.

With increasing cold, the former gave rise to water, earth and stones.

With the outsourcing of maintenance staff and facilities, the increase in company health and safety regulations, and the increasing curb weights of buses, many operators now contract their towing needs to a professional vehicle recovery company.

With the increasing prevalence of " combination products ," the lines are now blurring among healthcare products such as drugs, biologics, and various types of devices.

With the changes of dynasty historically, many people escaped the war and crossed the Central Plains, the increasing integration of the two communities.

With increasing wealth, Chinese diets have become richer over time, consuming more meats, fats, and sugar ( with the major exception of late 1950s famine ).

With the increasing sophistication of computer graphics since the 1970s, it has become a more distinct subject.

With shifts in competition and the increasing reliance by corporations on CRM systems, development of software has become more important than ever.

* " With feeling profound concern, grief, indignation, am compelled disclose Bahá ' í world recent developments Holy Land furnishing further incontestable proof relationship established old and new Covenant-breakers demonstrating increasing boldness, marked, tragic decline in character and spiritual condition grandchildren ` Abdu ' l-Bahá.

With the increasing use of computers the usage of and need for digital signal processing has increased.

With the increasing complexities of the information age, consultants and executives have found the term useful to describe the design of business processes as well as manufacturing processes.

With FEM / BEM increasing popularity comes the incentive to improve automatic meshing algorithms.

With the rise of Islam in the 7th century the power of Aksum declined and the Kingdom became isolated, the Dahlak archipelago, northern and western Eritrea, came under increasing control of Islamic powers based in Yemen and Beja lands in Sudan. The Beja were often in alliance with the Umayyads of Arabia who themselves established footholds along stretches of the Eritrean coastline and the Dahlak archipelago while the Funj of Sudan exacted tribute from the adjacent western lowlands of Eritrea.

With increasing concern over both the methods and products of modern industrial agriculture, there has been a growing trend toward sustainable agricultural practices.

With the increasing availability of the Internet in the 21st and late 20th century, the traditional paper zine has begun to give way to the webzine ( or " e-zine ") that is easier to produce and uses the potential of the Internet to reach an ever larger, possibly global, audience.

With the increasing use of processed foods since the 19th century, there has been a great increase in the use of food additives of varying levels of safety.

With the sharp growth of economies in the 20th century, and increasing foreign exchange, the world's gold reserves and their trading market have become a small fraction of all markets and fixed exchange rates of currencies to gold were no longer sustained.

With its increasing rarity, specimens of the Great Auk and its eggs became collectible and highly prized by rich Europeans, and the loss of a large number of its eggs to collection contributed to the demise of the species.

With the increasing popularity of multi-core processors, programmers now exploit thread-level parallel Gaussian elimination algorithms to increase the speed of computing.

With increasing amounts of acid rain in parts of the world, granite has begun to supplant marble as a monument material, since it is much more durable.

With the promise of political emancipation many Jews saw no reason to continue to observe Jewish law and increasing numbers of Jews assimilated into Christian Europe.

With increasing beam power the net gain ( gain minus loss ) reduces to unity and the gain medium is said to be saturated.

With tensions increasing with Great Britain, in 1809 Fort Bellefontaine was converted to a US military fort, and used for that purpose until 1826.