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Static problem For a set of N numbers find the maximal one. The problem may be solved in O(N) time. Dynamic problem For an initial set of N numbers, dynamically maintain the maximal one when insertion and deletions are allowed. A well-known solution for this problem is using a self-balancing binary search tree. It takes space O(N), may be ...
The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. [3]: ND22, ND23
Overlapping sub-problems means that the space of sub-problems must be small, that is, any recursive algorithm solving the problem should solve the same sub-problems over and over, rather than generating new sub-problems. For example, consider the recursive formulation for generating the Fibonacci sequence: F i = F i−1 + F i−2, with base ...
Originally introduced by Richard E. Bellman in (Bellman 1957), stochastic dynamic programming is a technique for modelling and solving problems of decision making under uncertainty. Closely related to stochastic programming and dynamic programming, stochastic dynamic programming represents the problem under scrutiny in the form of a Bellman ...
Dynamics (mechanics) Classical mechanics; Isolated physical system. Lagrangian mechanics; Hamiltonian mechanics; Routhian mechanics; Hamilton-Jacobi theory; Appell's equation of motion; Udwadia–Kalaba equation; Celestial mechanics; Orbit; Lagrange point. Kolmogorov-Arnold-Moser theorem; N-body problem, many-body problem; Ballistics
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. [1] [2] Nonlinear problems are of interest to engineers, biologists, [3] [4] [5] physicists, [6] [7] mathematicians, and many other scientists since most systems are inherently nonlinear in nature. [8]
The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms (PID controllers, state-space representation, neural networks, fuzzy control, genetic algorithms, etc.). Variations on this problem include multiple links, allowing the motion of the cart to be commanded ...
As a simple example, we investigate the properties of the one-dimensional Riemann problem in gas dynamics (Toro, Eleuterio F. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics, Pg 44, Example 2.5) The initial conditions are given by