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t. e. In mathematics, the graph of a function is the set of ordered pairs , where In the common case where and are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve . The graphical representation of the graph of a function is also known as a plot . In the case of functions of two variables – that ...
The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941, [1] and was used by John Nash in his ...
The usual values of interest for the parameter r are those in the interval [0, 4], so that x n remains bounded on [0, 1]. The r = 4 case of the logistic map is a nonlinear transformation of both the bit-shift map and the μ = 2 case of the tent map. If r > 4, this leads to negative population sizes.
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for finding a local minimum of a differentiable multivariate function . The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the ...
Lorenz curve. In economics, the Lorenz curve is a graphical representation of the distribution of income or of wealth. It was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution . The curve is a graph showing the proportion of overall income or wealth assumed by the bottom x % of the people, although this ...
t. e. The likelihood function (often simply called the likelihood) is the joint probability mass (or probability density) of observed data viewed as a function of the parameters of a statistical model. [1] [2] [3] Intuitively, the likelihood function is the probability of observing data assuming is the actual parameter.
The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems. It is also used in numerical analysis as a stand-in for the continuous Laplace operator. Common applications include image processing, [1] where it is known as the Laplace filter, and ...
A linear function is a polynomial function in which the variable x has degree at most one: [2] . Such a function is called linear because its graph, the set of all points in the Cartesian plane, is a line. The coefficient a is called the slope of the function and of the line (see below). If the slope is , this is a constant function defining a ...