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Fixed point (mathematics) Element mapped to itself by a mathematical function. The function (shown in red) has the fixed points 0, 1, and 2. In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point ...
The fixed point iteration xn+1 = cos xn with initial value x1 = −1. An attracting fixed point of a function f is a fixed point xfix of f with a neighborhood U of "close enough" points around xfix such that for any value of x in U, the fixed-point iteration sequence is contained in U and converges to xfix. The basin of attraction of xfix is ...
Time series. In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily ...
The point in the complex plane directly below the peak frequency is the fixed point attractor. A fixed point of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains ...
A continuous signal or a continuous-time signal is a varying quantity (a signal) whose domain, which is often time, is a continuum (e.g., a connected interval of the reals). That is, the function's domain is an uncountable set. The function itself need not to be continuous. To contrast, a discrete-time signal has a countable domain, like the ...
Decomposition of time series. The decomposition of time series is a statistical task that deconstructs a time series into several components, each representing one of the underlying categories of patterns. [1] There are two principal types of decomposition, which are outlined below.
If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable, i.e. a value of x close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at x = 0.6 (since 1.5(1 − 0.6) = 0.6) but starting at x = 0.61 we get
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...