monotonic function

(noun)

a function that either never decreases or never increases as its independent variable increases

Examples of monotonic function in the following topics:

  • Comparing Two Populations: Independent Samples

    • It assesses how well the relationship between two variables can be described using a monotonic function.
    • If there are no repeated data values, a perfect Spearman correlation of $1$ or $-1$ occurs when each of the variables is a perfect monotone function of the other.

  • Tips for Testing Series

    • Integral test: For a positive, monotone decreasing function $f(x)$ such that $f(n)=a_n$, if $\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty$ then the series converges.

  • Distribution-Free Tests

    • non-parametric statistics (in the sense of a statistic over data, which is defined to be a function on a sample that has no dependency on a parameter), whose interpretation does not depend on the population fitting any parameterized distributions.
    • Spearman's rank correlation coefficient: measures statistical dependence between two variables using a monotonic function.

  • Consumer Purchasing Behavior

    • Examples of Social Functions: Decisiveness, neutrality, anonymity, monotonicity, unanimity, homogeneity and weak and strong Pareto optimality.
    • Social functions can be categorized into social choice and welfare functions.Each method for vote counting is assumed as social function but if Arrow's possibility theorem is used for a social function, social welfare function is achieved.
    • Some specifications of the social functions are decisiveness, neutrality, anonymity, monotonicity, unanimity, homogeneity and weak and strong Pareto optimality.
    • No social choice function meets these requirements in an ordinal scale simultaneously.
    • Consumers evaluate alternatives in terms of the functional and psychological benefits that they offer.

  • The Mean Value Theorem, Rolle's Theorem, and Monotonicity

    • The MVT states that for a function continuous on an interval, the mean value of the function on the interval is a value of the function.
    • The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
    • More precisely, if a function $f$ is continuous on the closed interval $[a, b]$, where $a < b$, and differentiable on the open interval $(a, b)$, then there exists a point $c$ in $(a, b)$ such that
    • For any function that is continuous on $[a, b]$ and differentiable on $(a, b)$ there exists some $c$ in the interval $(a, b)$ such that the secant joining the endpoints of the interval $[a, b]$ is parallel to the tangent at $c$.
    • Use the Mean Value Theorem and Rolle's Theorem to reach conclusions about points on continuous and differentiable functions

  • The Integral Test and Estimates of Sums

    • Consider an integer $N$ and a non-negative function $f$ defined on the unbounded interval $[N, \infty )$, on which it is monotonically decreasing.
    • The above examples involving the harmonic series raise the question of whether there are monotone sequences such that $f(n)$ decreases to $0$ faster than $\frac{1}{n}$but slower than $\frac{1}{n^{1 + \varepsilon}}$ in the sense that:

  • Postmodern

    • The functional and formal spaces of the Modernist style were replaced by diverse aesthetics: styles collide, form is adopted for its own sake, and new ways of viewing familiar styles and space abound.
    • New trends became evident in the last quarter of the 20th century as some architects started to turn away from modern Functionalism, which they viewed as boring, or even unwelcoming and unpleasant.
    • Modernism's preoccupation with Functionalism and economical building meant that ornaments were done away with and that buildings appeared stark and merely functional.
    • Monotonous apartment blocks were seen as drab and undesirable.
    • With Postmodern architecture, form was no longer defined solely by functional requirements or a minimalist appearance.

  • Alternating Series

    • The theorem known as the "Leibniz Test," or the alternating series test, tells us that an alternating series will converge if the terms $a_n$ converge to $0$ monotonically.
    • Proof: Suppose the sequence $a_n$ converges to $0$ and is monotone decreasing.
    • Since $a_n$ is monotonically decreasing, the terms are negative.
    • $a_n = \frac1n$ converges to 0 monotonically.

  • The Role of Color

    • Some presenters change up their color schemes regularly to prevent their presentations from becoming too monotonous.

  • General Case

    • Thereafter, there is a monotonic decrease in the observed frequency as it gets closer to the observer, through equality when it is closest to the observer, and a continued monotonic decrease as it recedes from the observer.