Monotonic function in the context of "Coefficient of thermal expansion"

Thermal expansion is the tendency of matter to increase in length, area, or volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions).In simple words, the change in size of a body due to heating is called thermal expansion. Substances usually contract with decreasing temperature (thermal contraction), with rare exceptions within limited temperature ranges (negative thermal expansion). The SI unit of thermal expansion is inverse Kelvin (1/K).

Temperature is a monotonic function of the average molecular kinetic energy of a substance. In simple words, temperature is the measure of kinetic energy of a body or the measure of hotness or coldness of a body. As energy in particles increases, they start moving faster and faster, weakening the intermolecular forces between them and therefore expanding the substance.When a substance is heated, molecules begin to vibrate and move more, usually creating more distance between themselves.

In fluid dynamics, wave shoaling is the effect by which surface waves, entering shallower water, increase in wave height. It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, decreases with water depth. Under stationary conditions, a decrease in transport speed must be compensated by an increase in energy density in order to maintain a constant energy flux. Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.

In other words, as the waves approach the shore and the water gets shallower, the waves get taller, slow down, and get closer together.

In mathematics, the replicator equation is a type of dynamical system used in evolutionary game theory to model how the frequency of strategies in a population changes over time. It is a deterministic, monotone, non-linear, and non-innovative dynamic that captures the principle of natural selection in strategic interactions.

The replicator equation describes how strategies with higher-than-average fitness increase in frequency, while less successful strategies decline. Unlike other models of replication—such as the quasispecies model—the replicator equation allows the fitness of each type to depend dynamically on the distribution of population types, making the fitness function an endogenous component of the system. This allows it to model frequency-dependent selection, where the success of a strategy depends on its prevalence relative to others.

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable ${\displaystyle X}$, or just distribution function of ${\displaystyle X}$, evaluated at ${\displaystyle x}$, is the probability that ${\displaystyle X}$ will take a value less than or equal to ${\displaystyle x}$.

Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) ${\displaystyle F\colon \mathbb {R} \rightarrow [0,1]}$ satisfying ${\displaystyle \lim _{x\rightarrow -\infty }F(x)=0}$ and ${\displaystyle \lim _{x\rightarrow \infty }F(x)=1}$.

In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material.Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material. Spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.

In chemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: the common logarithm of the ratio of incident to transmitted radiant power through a material. It is the optical depth divided by log_e(10), because of the different logarithm bases used.