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The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. The framework captures problems such as factoring , discrete logarithm , graph isomorphism , and the shortest vector problem .
Simon's problem considers access to a function : {,} {,}, as implemented by a black box or an oracle. This function is promised to be either a one-to-one function, or a two-to-one function; if is two-to-one, it is furthermore promised that two inputs and ′ evaluate to the same value if and only if and ′ differ in a fixed set of bits. I.e.,
An important example in the theory of Lie groups arises when is taken to be (;), the group of invertible matrices with complex entries. In that case, a basic result is the following: [ 5 ] Theorem : Suppose φ : R → G L ( n ; C ) {\displaystyle \varphi :\mathbb {R} \rightarrow \mathrm {GL} (n;\mathbb {C} )} is a one-parameter group.
For example, consider the infinite cyclic group ℤ = b , embedded as a normal subgroup of the Baumslag–Solitar group BS(1, 2) = a, b . With respect to the chosen generating sets, the element = is distance 2 n from the origin in ℤ, but distance 2n + 1 from the origin in BS(1, 2).
The subgroup method is an algorithm used in the mathematical field of group theory. It is used to find the word of an element. It doesn't always return the minimal word, but it can return optimal words based on the series of subgroups that is used. The code looks like this:
The examples here will use CIA cryptonyms as a naming convention used to identify members of the cell system. Cryptonyms begin with a two-letter country or subject name (e.g., AL), followed by more letters so as to form an arbitrary word, such as "BERRY", "BEN", and "BEATLE" in the example below.
A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G. Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian. Other properties:
The 2/4th Commando Squadron was one of 12 independent companies and commando squadrons raised by the Australian Army during the Second World War.Raised in August 1941, it was disbanded not long after due to conceptual problems, but it was quickly reformed following the entry of Japan into the war in December 1941.