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The rank of a partition, shown as its Young diagram Freeman Dyson in 2005 In number theory and combinatorics , the rank of an integer partition is a certain number associated with the partition. In fact at least two different definitions of rank appear in the literature.
A partition is a division of a logical database or its constituent elements into distinct independent parts. Database partitioning refers to intentionally breaking a large database into smaller ones for scalability purposes, distinct from network partitions which are a type of network fault between nodes. [ 1 ]
The rank of a partition is the largest number k such that the partition contains at least k parts of size at least k. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4.
The rank of a partition is the integer obtained by subtracting the number of parts in the partition from the largest part in the partition. For example, the rank of the partition λ = { 4, 2, 1, 1, 1 } of 9 is 4 − 5 = −1. Denoting by N(m, q, n), the number of partitions of n whose ranks are congruent to m modulo q, Dyson considered N(m, 5 ...
A database shard, or simply a shard, is a horizontal partition of data in a database or search engine. Each shard may be held on a separate database server instance, to spread load. Some data in a database remains present in all shards, [a] but some appears only in a single shard. Each shard acts as the single source for this subset of data.
A SELECT statement retrieves zero or more rows from one or more database tables or database views. In most applications, SELECT is the most commonly used data manipulation language (DML) command. As SQL is a declarative programming language, SELECT queries specify a result set, but do not specify how to calculate it.
In 1944, Freeman Dyson defined the rank function for a partition and conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruences modulo 11.
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