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Upper and lower methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively. The values of the sums converge as the subintervals halve from top-left to bottom-right. In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum.
Dividing by ‖ ‖ and taking the infimum over all possible proves that the Galerkin approximation is the best approximation in the energy norm within the subspace , i.e. is nothing but the orthogonal, with respect to the scalar product (,), projection of the solution to the subspace .
The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality , the left-hand side of the approximation is greater than or equal to the right-hand side whenever x > − 1 {\displaystyle x>-1} and α ≥ 1 {\displaystyle \alpha \geq 1} .
Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and not, for example, derivatives of the density or the Kohn–Sham orbitals). Many approaches can yield ...
Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row is an example of a state which is not a stationary state.
The first term on the right-hand side is for an ideal gas. The remaining terms quantify the departure from the ideal gas law with changing pressure, P {\displaystyle P} . It can be shown by statistical mechanics that the second virial coefficient arises from the intermolecular forces between pairs of molecules, the third virial coefficient ...
Computing the free energy is an intractable problem for all but the simplest models in statistical physics. A powerful approximation method is mean-field theory, which is a variational method based on the Bogoliubov inequality. This inequality can be formulated as follows.
The proof consists of transforming the left-hand side (in the statement of the theorem) to the right-hand side by three approximations. First, according to Stirling's formula, the factorial of a large number n can be replaced with the approximation