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A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing. When f {\displaystyle f} is a strictly monotonic function, then f {\displaystyle f} is injective on its domain, and if T {\displaystyle T} is the range of f {\displaystyle f} , then there is an inverse function on T ...
With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function. If both arguments and values of a function are in the logarithmic scale (i.e., when log ( y ) is a linear function of log ( x ) ), then the straight line represents a power law :
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
A benefit of isotonic regression is that it is not constrained by any functional form, such as the linearity imposed by linear regression, as long as the function is monotonic increasing. Another application is nonmetric multidimensional scaling , [ 1 ] where a low-dimensional embedding for data points is sought such that order of distances ...
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
A famous example is the recurrence for the Fibonacci numbers, = + where the order is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients , because the coefficients of the linear function (1 and 1) are constants that do not depend on n . {\displaystyle n.}
A family of isoquants can be represented by an isoquant map, a graph combining a number of isoquants, each representing a different quantity of output.An isoquant map can indicate decreasing or increasing returns to scale based on increasing or decreasing distances between the isoquant pairs of fixed output increment, as output increases. [7]