Interval (mathematics) in the context of Interval estimation

Relativistic speed refers to speed at which relativistic effects become significant to the desired accuracy of measurement of the phenomenon being observed. Relativistic effects are those discrepancies between values calculated by models considering and not considering relativity. Related words are velocity, rapidity, and celerity which is proper velocity. Speed is a scalar, being the magnitude of the velocity vector which in relativity is the four-velocity and in three-dimension Euclidean space a three-velocity. Speed is empirically measured as average speed, although current devices in common use can estimate speed over very small intervals and closely approximate instantaneous speed. Non-relativistic discrepancies include cosine error which occurs in speed detection devices when only one scalar component of the three-velocity is measured and the Doppler effect which may affect observations of wavelength and frequency.

Relativistic effects are highly non-linear and for everyday purposes are insignificant because the Newtonian model closely approximates the relativity model. In special relativity the Lorentz factor is a measure of time dilation, length contraction and the relativistic mass increase of a moving object.

In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extrema, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

A histogram is a visual representation of the distribution of quantitative data. To construct a histogram, the first step is to "bin" (or "bucket") the range of values— divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The bins (intervals) are adjacent and are typically (but not required to be) of equal size.

Histograms give a rough sense of the density of the underlying distribution of the data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the x-axis are all 1, then a histogram is identical to a relative frequency plot.

In mathematics and statistics, a quantitative variable may be continuous or discrete. If it can take on two real values and all the values between them, the variable is continuous in that interval. If it can take on a value such that there is a non-infinitesimal gap on each side of it containing no values that the variable can take on, then it is discrete around that value. In some contexts, a variable can be discrete in some ranges of the number line and continuous in others. In statistics, continuous and discrete variables are distinct statistical data types which are described with different probability distributions.

Continuous optimization is a branch of optimization in applied mathematics.

As opposed to discrete optimization, the variables used in the objective function are required to be continuous variables—that is, to be chosen from a set of real values between which there are no gaps (values from intervals of the real line). Because of this continuity assumption, continuous optimization allows the use of calculus techniques.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers ${\displaystyle \mathbb {R} }$, or a subset of ${\displaystyle \mathbb {R} }$ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of ${\displaystyle \mathbb {R} }$-vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an ${\displaystyle \mathbb {R} }$-algebra, such as the complex numbers or the quaternions. The structure ${\displaystyle \mathbb {R} }$-vector space of the codomain induces a structure of ${\displaystyle \mathbb {R} }$-vector space on the functions. If the codomain has a structure of ${\displaystyle \mathbb {R} }$-algebra, the same is true for the functions.

Spectral bands are regions of a given spectrum, having a specific range of wavelengths or frequencies. Most often, it refers to electromagnetic bands, regions of the electromagnetic spectrum. More generally, spectral bands may also be means in the spectra of other types of signals, e.g., noise spectrum.

A frequency band is an interval in the frequency domain, limited by a lower frequency and an upper frequency. For example, it may refer to a radio band, such as wireless communication standards set by the International Telecommunication Union.

In mathematics, an open set is a generalization of an open interval in the real line.

In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.

Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

In medicine and health-related fields, a reference range or reference interval is the range or the interval of values that is deemed normal for a physiological measurement in healthy persons (for example, the amount of creatinine in the blood, or the partial pressure of oxygen). It is a basis for comparison for a physician or other health professional to interpret a set of test results for a particular patient. Some important reference ranges in medicine are reference ranges for blood tests and reference ranges for urine tests.

The standard definition of a reference range (usually referred to if not otherwise specified) originates in what is most prevalent in a reference group taken from the general (i.e. total) population. This is the general reference range. However, there are also optimal health ranges (ranges that appear to have the optimal health impact) and ranges for particular conditions or statuses (such as pregnancy reference ranges for hormone levels).

Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis, spectral analysis, and neuroscience.

The term "harmonics" originated from the Ancient Greek word harmonikos, meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes. Still, the term has been generalized beyond its original meaning.

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that every infinite sequence of points has limiting values. For example, the real line is not compact since the sequence of natural numbers has no real limiting value. The open interval (0,1) is not compact because it excludes the limiting values 0 and 1, whereas the closed interval [0,1] is compact. Similarly, the space of rational numbers ${\displaystyle \mathbb {Q} }$ is not compact, because every irrational number is the limit of the rational numbers that are lower than it. On the other hand, the extended real number line is compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence ⁠1/2⁠, ⁠4/5⁠, ⁠1/3⁠, ⁠5/6⁠, ⁠1/4⁠, ⁠6/7⁠, ... accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering ${\displaystyle \mathbb {R} ^{1}}$ (the real number line), the sequence of points 0,  1,  2,  3, ... has no subsequence that converges to any real number.