From: owner-ammf-digest@smoe.org (alt.music.moxy-fruvous digest) To: ammf-digest@smoe.org Subject: alt.music.moxy-fruvous digest V14 #4077 Reply-To: ammf@fruvous.com Sender: owner-ammf-digest@smoe.org Errors-To: owner-ammf-digest@smoe.org Precedence: bulk alt.music.moxy-fruvous digest Saturday, May 2 2020 Volume 14 : Number 4077 Today's Subjects: ----------------- Remove wrinkles and sagging skin in 60 seconds ["South Beach Skin" Subject: Remove wrinkles and sagging skin in 60 seconds Remove wrinkles and sagging skin in 60 seconds http://woodthe.guru/WmffJfB9b7er6hGT5jOqmD7lJWOst7c5tje-5i4kqGXSISI6 http://woodthe.guru/3YcGmiHVDvdlwZecJ69gTiCP0n8-VNR005pZhbubl4b4Ew gives the state of the system for only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many timesbeach advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, given an initial point it is possible to determine all its future positions, a collection of points known as a trajectory or orbit. Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: The systems studied may only be known approximatelybthe parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability. The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood ------------------------------ End of alt.music.moxy-fruvous digest V14 #4077 **********************************************