Search results
Results From The WOW.Com Content Network
Order of operations. In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression . These rules are formalized with a ranking of the operations. The rank of an operation is called its precedence, and ...
A creation operator (usually denoted ^ †) increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.
Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that. If S is a density operator, Tr (Φ ( S )) ≤ 1. Φ is completely positive, that is for any natural number n, and any square matrix of size n whose entries are trace-class operators and which is non-negative, then ...
Here is a composition example in Perl. subcompare($$){my($a,$b)=@_;return$a->{unit}cmp$b->{unit}||$a->{rank}<=>$b->{rank}||$a->{name}cmp$b->{name};} Note that cmp, in Perl, is for strings, since <=> is for numbers. Two-way equivalents tend to be less compact but not necessarily less legible.
Trace class. In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra.
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ∇ ⋅ ∇ {\displaystyle abla \cdot abla } , ∇ 2 {\displaystyle abla ^{2}} (where ∇ {\displaystyle abla } is the nabla operator ), or Δ ...
By the recursive formula, a[n]0 = a[n − 1] (a[n] (−1)) ⇒ 1 = a[n − 1]x. One solution is x = 0, because a[n − 1]0 = 1 by definition when n ≥ 4. This solution is unique because a[n − 1]b > 1 for all a > 1, b > 0 (proof by recursion). ^ a b Ordinal addition is not commutative; see ordinal arithmetic for more information.
The commutator of two elements, g and h, of a group G, is the element. [g, h] = g−1h−1gh. This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of G generated by all ...
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian.
Theorem — Let X be a Banach space, C be a compact operator acting on X, and σ(C) be the spectrum of C. Every nonzero λ ∈ σ ( C ) is an eigenvalue of C . For all nonzero λ ∈ σ ( C ), there exist m such that Ker (( λ − C ) m ) = Ker (( λ − C ) m +1 ), and this subspace is finite-dimensional.