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.

* 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, 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 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 no loss of generality, we can pick a polar coordinate system ( r, θ ) such

* With τ, h, φ, θ, e, and f, it composed the Arabic asterism Sarīr Banāt al-Na ' sh, the Throne of the daughters of Na ' sh, and al -< u > H </ u > au < u > d </ u >, the Pond,

With φ, h, υ, θ, e, and f, it composed the Arabic asterism Sarīr Banāt al-Na ' sh, the Throne of the daughters of Na ' sh, and al -< u > H </ u > au < u > d </ u >, the Pond,

With φ, υ, θ, τ, e, and f, it composed the Arabic asterism Sarīr Banāt al-Na ' sh, the Throne of the daughters of Na ' sh, and al -< u > H </ u > au < u > d </ u >, the Pond,

With the advent of z / TPF, all programs will reside in core-eventually-the only question is when they get fetched the first time.

With reasonable arithmetic types, the result for < tt > z </ tt > should be zero.

The band has participated in a Slovenian contest EMA 2007 for the Eurovision Song Contest 2007 with a song Kdo hoče plesati z menoj ( Who Wants To Dance With Me ).

With these definitions, is an even function and its derivative with respect to z, &# 8472 ;&# 8242 ;, an odd function.

With respect to one of the complex variables ( conventionally called z ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function.

He wrote dozens of hits for the revue-theatres and films including Miłość ci wszystko wybaczy ( Love Will Forgive You Everything ) sung by Hanka Ordonówna, Umówiłem się z nią na dziewiatą ( I Have A Date With Her At Nine ) and Sex Appeal sung by Eugeniusz Bodo, Ach, jak przyjemnie ( Ah, How Lovely!

With σ and the relation IP in hand, we can define the interior of x, as the mereological sum of all interior parts z of x, or:

With non-nested models and iid exogenous variables, model 1 ( 2 ) is preferred with significance level α, if the z statistic

More recently, he returned to Poland, and to film making as a writer and director after a 17 year hiatus with Cztery noce z Anną ( Four Nights With Anna ) in 2008.

With this choice of bending moment sign convention, in order to have, it is necessary that the shear force acting on the right side of the section be positive in the z direction so as to achieve static equilibrium of moments.

In 1997 he has published a book Rozhovory s poučením od mých přátel z vesmíru ( Interviews With Instructions From My Friends From the Universe ).

With z / OS Release 9 SDSF supports a REXX interface, allowing batch programs to use SDSF facilities.

With world peace constantly being threatened, most of us regard the future skeptically, and even with fear.

With dealer stocks of new equipment averaging about 25% below a year ago, the affects of the rural recovery are being felt almost immediately by the country's farm equipment manufacturers.

With civil disobedience resulting in coercive and intolerable acts, and armed conflict resulting in dissidents being proclaimed rebels and outside the King's protection, any loyalty remaining shifted toward independence and how to achieve it.

With German being a pluricentric language, Austrian dialects should not be confused with the variety of Standard German spoken by most Austrians, which is distinct from that of Germany or Switzerland.

With Naram-Sin, Sargon's grandson, this went further than with Sargon, with the king not only being called " Lord of the Four Quarters ( of the Earth )", but also elevated to the ranks of the dingir (= gods ), with his own temple establishment.

The Falcons wore the all black combination against the New Orleans Saints for 4 straight seasons starting in 2004, With the last time being in 2007, losing 34 – 14.

With two of his singles Spraakwater and Kaal of Kammen being major hits in the mid-1990s, Extince was the first Dutch rapper making the MegaCharts.

With these innings he became the second player to score two Test triple centuries, the second player to score two career quadruple centuries, the only player to achieve both these milestones, and regained the distinction of being the holder of both the record first-class individual innings and the record Test individual innings.

With the mosque being located in the area, more Muslim immigrants began to move into private lodgings in Balsall Heath.

With the continuing advances being made in the production of synthetic diamonds, future applications are beginning to become feasible.

With this philosophy in mind, many grassroots efforts such as The Project for Public Spaces are being started to create this " Third Place " in communities.

With " Tertiary " being discouraged as a formal time or rock unit by the International Commission on Stratigraphy, the K – T extinction event is now called the Cretaceous – Paleogene ( or K – Pg ) extinction event by many researchers.

With innate talent, passion, diligence, discipline, self-motivation and tenacity, being the core factors of achieving success to becoming a celebrity, fame and fortune sometimes occurs spontaneously with relatively little effort due to sheer luck, being fortunate with connections, or simply being at the right place during the right time.

With prime time competition, Corrie was again seen as being old fashioned, with the introduction of the ' normal ' Clayton family in 1985 being a failure with viewers.

With the first version of VAC, a ban took hold almost instantly after being detected and the cheater had to wait 2 years to have the account unbanned.

With the second version, Valve instituted a policy of ' delayed bans ,' the theory being that if a new hack is developed which circumvents the VAC system, it will spread amongst the ' cheating ' community.

With the development of the internet, by the late 1990s and early 2000, much of that regulation had been replaced where newer industry technologies developed, offering viewers alternate choices for local events and programming leading to what is today, that being Digital Cable, Internet, and Phone being offered to consumers, bundled, by 2010.

With regard to ( 1 ), Hume argues that the uniformity principle cannot be demonstrated, as it is " consistent and conceivable " that nature might stop being regular.

With forest biotopes being irreplaceable source of new drugs ( such as taxol ), deforestation can destroy genetic variations ( such as crop resistance ) irretrievably.

With focus on performance-oriented keyboards and digital computer technology, manufacturers of commercial electronic instruments created some of the earliest digital synthesizers for studio and experimental use with computers being able to handle built-in sound synthesis algorithms.

With data being transferred 64 bits at a time, DDR SDRAM gives a transfer rate of ( memory bus clock rate ) × 2 ( for dual rate ) × 64 ( number of bits transferred ) / 8 ( number of bits / byte ).