Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into k chains then the width of the order must be at most k. Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width.

Any hypothetical partition of natural numbers into interesting and dull sets seems to fail.

Any skew semistandard tableau of shape / with positive integer entries gives rise to a sequence of partitions ( or Young diagrams ), by starting with, and taking for the partition places further in the sequence the one whose diagram is obtained from that of by adding all the boxes that contain a value ≤ in ; this partition eventually becomes equal to.

Any collection of finitely many lines partitions the plane into convex polygons ( possibly unbounded ); this partition is known as an arrangement of lines.

Any partition will have less TL at lower frequencies.

( 1 ) Any person interested in land in Ontario, or the guardian of a minor entitled to the immediate possession of an estate therein, may bring an action or make an application for the partition of such land or for the sale thereof under the directions of the court if such sale is considered by the court to be more advantageous to the parties interested.

Any example of grand or exquisite diction may have been created by the poet who compiled numerous lays into the two works we possess or may be due to one of his completely unknown fellow-craftsmen.

Any citizen could bring a matter before a grand jury directly, from a public work that needed repair, to the delinquent conduct of a public official, to a complaint of a crime, and grand juries could conduct their own investigations.

In terms of the law that " Any two time-lines approximate to the exact degree to which their material contents are alike " the tower is obviously a repeat, on a grand scale, of the Othertimers ' successful but small experiment with a railway shed constructed in the same space as ours.

In his letter, Polding gives Wardell a completely free hand in the design, saying " Any plan, any style, anything that is beautiful and grand, to the extent of our power.

* Any Minority Government composed of a coalition of several ideologically unrelated political parties united only by opposition to one or more dominant parties.

* The partition function under the grand canonical ensemble in statistical mechanics

The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential.

In a manner similar to the definition of the canonical partition function for the canonical ensemble, we can define a grand canonical partition function for a grand canonical ensemble, a system that can exchange both heat and particles with the environment, which has a constant temperature T, and a chemical potential μ.

The grand canonical partition function, although conceptually more involved, simplifies the theoretical handling of quantum systems because it incorporates in a simple way the spin-statistics of the particles ( i. e. whether particles are bosons or fermions ).

The grand canonical partition function for an ideal quantum gas ( a gas of non-interacting particles in a given potential well ) is given by the following expression:

Seeing no hope of prevailing, Charles made a secret agreement to betray his King and offer his services to the Holy Roman Emperor Charles V. The Emperor, the Constable, and King Henry VIII of England devised a grand plan to partition France, which came to nothing ; the plot was discovered and Charles was stripped of his offices and fled into Italy in 1523.

The thermodynamics of an ideal Bose gas is best calculated using the grand partition function.

The grand partition function for a Bose gas is given by:

All thermodynamic quantities may be derived from the grand partition function and we will consider all thermodynamic quantities to be functions of only the three variables z, β ( or T ), and V.

Classically, the partition function of the grand canonical ensemble is given as a weighted sum of canonical partition functions with different number of particles,

The task of summing over states ( calculating the partition function ) appears to be simpler if we do not fix the total number of particles of the system because, for a noninteracting system, the partition function of grand canonical ensemble can be converted to a product of the partition functions of individual quantum states.

Therefore the partition function ( of grand canonical ensemble ) for each state can be written as

The grand partition, the normalizing constant for Tr ( ρ ) to be 1, is

Remember that for the grand partition, the states are states with multiple particles in Fock space, and the trace sums over all of them.

In the special case of a non-interacting system, the grand partition can be simplified and expressed in terms of the eigenvalues of the single-particle Hamiltonian ; after all, the eigenvalues of the multiple-particle Hamiltonian will have the form

Notice that, for the microcanonical ensemble, plays the role of the partition function in the canonical and grand canonical ensembles.

of the grand canonical partition function

The grand partition function can be expanded in a sum of contributions from one-body, two-body, etc.

Knowing Gandhi's deep anguish regarding proposals of partition, Patel engaged him in frank discussion in private meetings over the perceived practical unworkability of any Congress-League coalition, the rising violence and the threat of civil war.

When, on September 14, 1714, he suddenly returned to his dominions, Stralsund and Wismar were all that remained to him of his continental possessions ; while by the end of 1715 Sweden, now fast approaching the last stage of exhaustion, was at open war with Great Britain, Hanover, Russia, Prussia, Saxony and Denmark, who had formed a coalition to partition her continental territory between them.

The partition coalition forced King Stanislaw to abdicate and he retired to St. Petersburg as Catherine II ’ s trophy prisoner, where he died in 1798.

Austria, Russia, and Prussia sought to permanently erase the existence of Poland, even down to the country ’ s name, as proven by a secret and separate article signed by the partition coalition:

Prussia withdrew from the coalition that was working on the impending partition of Poland, and where appropriate, withdrew its troops that were aligned against Austria and Russia.

In the mid 3rd century, the Bastarnae were part of a Gothic-led grand coalition of lower Danube tribes which inflicted immense damage on the Balkan provinces of the Roman empire in a series of massive invasions.

The Goths appear to have established a loose political hegemony over the existing tribes in the region, or at least to have played the leading role in a series of major invasions of the empire launched by a grand coalition of lower Danubian tribes from ca.

However, if none of the larger parties can receive enough votes to form their preferred coalition, a grand coalition might be their only choice for forming a government.

A grand coalition government was subsequently forged between the CDU / CSU and the SPD.

The Government of the 31st Dáil, though a traditional Fine Gael – Labour coalition, resembles a grand coalition, due to the collapse of Fianna Fáil to third place among parties in Dáil Éireann.

The most likely outcome of coalition talks was a so-called " grand coalition " between the Christian Democrats ( CDU / CSU ) and the Social Democrats ( SPD ).

The existence of the grand coalition on federal level helped smaller parties ' electoral prospects in state elections.

Since in 2008, the CSU lost its absolute majority in Bavaria and formed a coalition with the FDP, the grand coalition had no majority in the Bundesrat and depended on FDP votes on important issues.

As a result of that federal election, the grand coalition came to an end.

joined the smaller Popular Orthodox Rally in a grand coalition, pledging their parliamentary support for a government of national unity headed by former European Central Bank vice-president Lucas Papademos.

On 4 November 2007, leader of the main opposition party, Ichiro Ozawa announced his resignation from the post of party president, after controversy over an offer to the DPJ to join the ruling coalition in a grand coalition,

The political process in Malaysia has generally been described as taking the form of " consociationalism " whereby " communal interests are resolved in the framework of a grand coalition " " Malaysia: Developmental State Challenged ".

On 1 October 2006 the SPÖ won a head on head elections and negotiated a grand coalition with the ÖVP.

The grand coalition headed by Alfred Gusenbauer collapsed in the early summer of 2008 over disagreements about the country's EU policy.

After undertaking various journeys she succeeded in 1635 in forming a grand coalition with the states of Matamba and Ndongo, Kongo, Kassanje, Dembos and Kissamas.

Jaruzelski was unsuccessful in convincing Wałęsa to include Solidarity in a " grand coalition " with the Communists, and resigned his position of general secretary of the Polish Communist Party.

His successor was Kurt Georg Kiesinger ( CDU ), who formed a grand coalition with the SPD.

Erhard's government was short-lived, however, and in 1966 a grand coalition between the SPD and CDU was formed, with Brandt as Foreign Minister and Vice-Chancellor.

As the Minister of Foreign Affairs in Kiesinger's grand coalition cabinet, Brandt helped to gain further international approval for Western Germany, and he laid the foundation stones for his future Neue Ostpolitik.

Consequently, a grand coalition with the SPD took over government under CDU Chancellor Kurt Georg Kiesinger.