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In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected structures. The exponential formula is a power series version of a special case of Faà di Bruno's ...
Exponential Random Graph Models (ERGMs) are a family of statistical models for analyzing data from social and other networks. [ 1 ] [ 2 ] Examples of networks examined using ERGM include knowledge networks, [ 3 ] organizational networks, [ 4 ] colleague networks, [ 5 ] social media networks, networks of scientific development, [ 6 ] and others.
to solve for Ω recursively in terms of A "in a continuous analog of the BCH expansion", as outlined in a subsequent section. The equation above constitutes the Magnus expansion, or Magnus series, for the solution of matrix linear initial-value problem. The first four terms of this series read
A much simpler way of solving this equation (and power series solution in general) using the Taylor series form of the expansion. Here we assume the answer is of the form f = ∑ k = 0 ∞ A k z k k ! {\displaystyle f=\sum _{k=0}^{\infty }{A_{k}z^{k} \over {k!}}}
The step size is =. The same illustration for = The midpoint method converges faster than the Euler method, as .. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs).
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.
Complex replacement is used for solving differential equations when the non-homogeneous term is expressed in terms of a sinusoidal function or an exponential function, which can be converted into a complex exponential function differentiation and integration. Such complex exponential function is easier to manipulate than the original function.
Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have exponential running time. In other words, any problem in EXPTIME is solvable by a deterministic Turing machine in O (2 p ( n ) ) time, where p ( n ) is a polynomial function of n .