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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 ...
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
Linear dynamical systems are dynamical systems whose evolution functions are linear.While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties.
Series of Books in the Mathematical Sciences (1st ed.). New York: W. H. Freeman and Company. ISBN 9780716710455. MR 0519066. OCLC 247570676.. This book is a classic, developing the theory, then cataloguing many NP-Complete problems. Cook, S.A. (1971). "The complexity of theorem proving procedures".
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems.Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does ...
The Soluble Problems of Rigid Dynamics: 7 Theory of Vibrations: 8 Non-Holonomic Systems, Dissipative Systems: 9 The Principles of Least Action and Least Curvature: 10 Hamiltonian Systems and their Integral-Invariants: 11 The Transformation-Theory of Dynamics: 12 The Properties of the Integrals of Dynamic Systems: 13 The Reduction of the Problem ...
One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H ( q , p , t ) {\displaystyle H({\boldsymbol {q}},{\boldsymbol ...
Top-down approach: This is the direct fall-out of the recursive formulation of any problem. If the solution to any problem can be formulated recursively using the solution to its sub-problems, and if its sub-problems are overlapping, then one can easily memoize or store the solutions to the sub-problems in a table (often an array or hashtable ...