In recent years there was a high demand for massively distributed databases with high partition tolerance but according to the CAP theorem it is impossible for a distributed system to simultaneously provide consistency, availability and partition tolerance guarantees.

A fundamental theorem states that every spline function of a given degree, smoothness, and domain partition, can be uniquely represented as a linear combination of B-splines of that same degree and smoothness, and over that same partition.

Many theorems, which are modeled after Ramsey's theorem itself, assert that in every partition of a large structured object, one of the classes necessarily contains a large structured subobject, but give no information about which class this is.

Between 1886 and 1892 he published a series of papers on the foundations of the kinetic theory of gases, the fourth of which contained what was, according to Lord Kelvin, the first proof ever given of the Waterston-Maxwell theorem ( equipartition theorem ) of the average equal partition of energy in a mixture of two gases.

Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah-Singer index theorem.

This result is given by Grimmett and Welsh as the partition theorem, a name that they also give to the related law of total expectation.

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.

In this case the equipartition theorem may be derived from the partition function Z ( β ), where β = 1 /( k < sub > B </ sub > T ) is the canonical inverse temperature.

For instance, Dilworth's theorem states that the minimum number of chains in a partition of a partially ordered set into chains equals the maximum size of an antichain, and can be rephrased as stating that the complements of comparability graphs are perfect.

Mirsky's theorem states that the minimum number of antichains into a partition into antichains equals the maximum size of a chain, and corresponds in the same way to the perfection of comparability graphs.

By Whitney's theorem, if G is not a triangle, there can be only one partition of this type.

In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains.

Dilworth's theorem states that there exists an antichain A, and a partition of the order into a family P of chains, such that the number of chains in the partition equals the cardinality of A.

An equivalent way of stating Dilworth's theorem is that, in any finite partially ordered set, the maximum number of elements in any antichain equals the minimum number of chains in any partition of the set into chains.

( U, V, E ), form a partial order on the vertices of G in which u < v exactly when u is in U, v is in V, and there exists an edge in E from u to v. By Dilworth's theorem, there exists an antichain A and a partition into chains P both of which have the same size.

Therefore, by the De Bruijn – Erdős theorem, P itself also has a w-colorable incomparability graph, and thus has the desired partition into chains.

But the high-speed read / write access results in reduced consistency, as it is not possible to have both consistency, availability, and partition tolerance of the network, as it has been proven by the CAP theorem.

The proof of Batchelor's theorem relies in an essential way on the existence of a partition of unity, so it does not hold for complex or real-analytic supermanifolds.

A civil war against his brother Rudolf due to new disputes on the partition of their lands was ended in 1313, when peace was made at Munich.

* Crisis of the Third Century: An economic crisis strikes the Roman empire ; due to the partition of the empire, invasions and usurpations and the sacking of the countryside and cities by invaders, agricultural and industrial production are significantly decreased, and mines lie unused.

On April 7, 1806, Genesee ’ s area was reduced to due to a partition that created Allegany County.

On March 8, 1808, Genesee ’ s area was again reduced, this time to due to a partition that created Cattaraugus, Chautauqua, and Niagara Counties.

On February 23, 1821, Genesee ’ s area was again reduced, this time to due a complex partition that produced Livingston and Monroe Counties.

In the time of the partition, unemployment arose because the Austrians brought modern equipment which caused a cease of production throughout the city and surrounding areas, due to low wages, forcing were dismiss en masse of Polish miners.

Despite the increase in tensions due to the Second Moroccan Crisis of 1911, Bethmann Hollweg did improve relations with Britain to some extent, working with British foreign secretary Sir Edward Grey to alleviate tensions during the Balkan Crises of 1912-1913, and negotiating treaties over an eventual partition of the Portuguese colonies and the Berlin-Baghdad railway.

Recently, partition chromatography has become popular again with the development of HILIC bonded phases which demonstrate improved reproducibility, and due to a better understanding of the range of usefulness of the technique.

This, due to the fact that after the partition of the Kingdom of Armenia by Persia and Byzantium in 387 AD, the Armenian provinces of Artsakh and Utik were disassociated from Armenia proper and included by Persians into a single province ( marzpanate ) called Aghvank ( Arran ).

This was due to the perceived failure to have Home Rule implemented when the IPP resisted the partition of Ireland demanded by Ulster Unionists in 1914, 1916 and 1917, but also popular antagonism towards the British authorities created by the execution of most of the leaders of the 1916 rebels and by their botched attempt to introduce Home Rule linked with military conscription in Ireland ( see Conscription Crisis of 1918 ).

The nature of Pakistani literature soon after independence aroused controversy among writers due to its being centered heavily on the negative events related to the India-Pakistan partition.

In ANCOVA, we basically perform a regression analysis within each cell to partition out the variance component due to the CV.

The thick mangroves in between the pipeline and railway lines were cleared to make way for the construction of residential houses for Government servants who were displaced due to the partition.

In fact, newer volatile agents ( e. g., sevoflurane, desflurane ) have been popular not due to their potency ( minimum alveolar concentration ), but due to their versatility for a faster emergence from anesthesia, thanks to their lower blood – gas partition coefficient.

The Bhattacharyya coefficient will be 0 if there is no overlap at all due to the multiplication by zero in every partition.

The theatres screenings of the film were disturbed with violent demonstration due to the raw reconstruction of the partition riots and sloganeering featured in it.

As to how long the period of partition was to last, due to the ambiguities of the wording of the final document purposely intrigued by Walter Long to jeopardise Home Rule, Redmond, understanding it would be temporary, broke off negotiations when he realised this was not so.

At the time of partition, ruler of Jodhpur Hanwant Singh did not want to join India, but finally due to the effective persuasion of Sardar Vallab Patel, the then Home Minister at the centre, the princely state of Jodhpur was included in Indian Republic.

In 1793, due to the second partition of Poland, Trzemeszno became a part of Prussia and after Napoleon I, in 1815 it was incorporated into the Grand Duchy of Posen.

The silver jubilee year of the university in 1947 coincided with India's independence, and the national flag was hoisted in the main building for the first time by VKRV Rao, the convocation ceremony for the year, however could not be held due to partition disturbances, thus a special ceremony was held in 1948, which was attended by prime minister Jawaharlal Nehru, Lord Mountbatten, Lady Mountbatten, Abul Kalam Azad, Zakir Hussain and S. S. Bhatnagar.

Then in 1801, due to the territorial partition carried out by Toussaint-Louverture during his control of the Spanish part of the island, it became a district of the department of the Ozama.

But these successes were followed by a series of misfortunes, due in the first instance to the unfortunate partition which Alfonso had made of his possessions, and the consequent rivalry which ensued between the Castilian and Leonese branches of his dynasty.

The partition function is a function of the temperature T and the microstate energies E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, etc.

Given a connected graph G =( V, E ) with V the set of vertices and E the set of edges, and with a root vertex r, the level structure is a partition of the vertices into subsets L < sub > i </ sub > called levels, consisting of the vertices at distance i from r. Equivalently, this set may be defined by setting L < sub > 0 </ sub > =

We will start with a set of states S. We can then define an event E as a subset of the set of states S. For each agent i, define a partition on S, P < sub > i </ sub >.

Specifically, L is the gas-liquid partition constant on hexadecane at 298 K ; E the excess molar refraction ; S the ability of a solute to stabilize a neighboring dipole by virtue of its capacity for orientation and induction interactions ; A the solute ’ s effective hydrogen bond acidity ; and B the solute ’ s effective hydrogen-bond basicity.

Before he established himself as a lyrics writer for Hindi film songs, he served for 2 years in the Royal Indian Navy, ( Karachi ) and after partition he served 6 years in the Indian Army, based at Jabalpur, with The Corps of Signals, and later with the E. M. E., Infantry.