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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.,
The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the hidden subgroup problem. On a quantum computer, to factor an integer N {\displaystyle N} , Shor's algorithm runs in polynomial time , meaning the time taken is polynomial in log N {\displaystyle \log ...
Himmelblau's function-- Hindley–Milner type system-- Hindmarsh–Rose model-- Hindu–Arabic numeral system-- Hindu units of time-- Hindustani numerals-- Hinge theorem-- Hinged dissection-- Hippopede-- Hiptmair–Xu preconditioner-- Hironaka decomposition-- Hironaka's example-- Hiroshima Mathematical Journal-- Hiroyuki Goto-- Hirsch ...
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 b 2 n = a n b a − n {\displaystyle b^{2^{n}}=a^{n}ba^{-n}} is distance 2 n from the origin in ℤ , but distance 2 n + 1 from the origin in BS(1, 2) .
The newly released Guinness World Records book isn't limited to records that show off years of dedicated practice or unique traits you were born with — it's also a place where people with very ...
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, any subgroup of the group of integers (, +) is generated by some integer . If = then the subgroup takes up 0 proportion. Otherwise, it takes up / of the whole group. Even though both the group and the subgroup has infinitely many elements, there is a well-defined sense of proportion.