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The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. [1] It has applications in geophysics, seismic imaging, photonics and more recently in neural networks. [2] The adjoint state space is chosen to simplify the physical interpretation of equation ...
Definition. A self-adjoint operator A has uniform multiplicity n where n is such that 1 ≤ n ≤ ω if and only if A is unitarily equivalent to the operator M f of multiplication by the function f (λ) = λ on. where Hn is a Hilbert space of dimension n. The domain of M f consists of vector-valued functions ψ on R such that.
The set of self-adjoint elements is referred to as . A subset that is closed under the involution *, i.e. , is called self-adjoint.[2] A special case of particular importance is the case where is a complete normed *-algebra, that satisfies the C*-identity ( ), which is called a C*-algebra.
Thus, for example, a vector x in the algebra generates a vector field X in the group G. Similarly, the adjoint map ad x y = [ x , y ] of vectors in g {\displaystyle {\mathfrak {g}}} is homomorphic [ clarification needed ] to the Lie derivative L X Y = [ X , Y ] of vector fields on the group G considered as a manifold .
Adjoint functors. In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint.
The One Definition Rule (ODR) is an important rule of the C++ programming language that prescribes that classes/structs and non-inline functions cannot have more than one definition in the entire program and templates and types cannot have more than one definition by translation unit.
QA76.73.C153 A42 2001. Modern C++ Design: Generic Programming and Design Patterns Applied is a book written by Andrei Alexandrescu, published in 2001 by Addison-Wesley. It has been regarded as "one of the most important C++ books" by Scott Meyers. [1] The book makes use of and explores a C++ programming technique called template metaprogramming.
This method usually calls the clone() method of its parent class to obtain a copy, and then does any custom copying procedures. Eventually, this gets to the clone() method of the uppermost object ( Object ), which creates a new instance of the same class as the object and copies all the fields to the new instance (a shallow copy).