Distribution (mathematics) in the context of Particle-size distribution

In mathematical analysis, the Dirac delta function (or δ distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as

$${\displaystyle \delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}}$$

In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom is 1s 2s 2p, meaning that the 1s, 2s, and 2p subshells are occupied by two, two, and six electrons, respectively.

Electronic configurations describe each electron as moving independently in an orbital, in an average field created by the nuclei and all the other electrons. Mathematically, configurations are described by Slater determinants or configuration state functions.

Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in parametric statistics. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated.

In mathematical analysis, a bump function (also called a test function) is a function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ on a Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain ${\displaystyle \mathbb {R} ^{n}}$ forms a vector space, denoted ${\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})}$ or ${\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).}$ The dual space of this space endowed with a suitable topology is the space of distributions.

In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species (N_i) and the molar mass (M_i) of that species. In linear polymers, the individual polymer chains rarely have exactly the same degree of polymerization and molar mass, and there is always a distribution around an average value. The molar mass distribution of a polymer may be modified by polymer fractionation.

In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for treating discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. Important motivations have been the technical requirements of theories of partial differential equations and group representations.

A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and some contemporary developments are closely related to Mikio Sato's algebraic analysis.

In complex analysis, the Hardy spaces (or Hardy classes) ${\displaystyle H^{p}}$ are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz (Riesz 1923), who named them after G. H. Hardy, because of the paper (Hardy 1915). In real analysis Hardy spaces are spaces of distributions on the real n-space ${\displaystyle \mathbb {R} ^{n}}$, defined (in the sense of distributions) as boundary values of the holomorphic functions. Hardy spaces are related to the L spaces. For ${\displaystyle 1\leq p<\infty }$ these Hardy spaces are subsets of ${\displaystyle L^{p}}$ spaces, while for ${\displaystyle 0<p<1}$ the ${\displaystyle L^{p}}$ spaces have some undesirable properties, and the Hardy spaces are much better behaved. Hence, ${\displaystyle H^{p}}$ spaces can be considered extensions of ${\displaystyle L^{p}}$ spaces.

Hardy spaces have a number of applications, both in mathematical analysis itself as well as in interdisciplinary areas such as control theory (e.g. ${\displaystyle H^{\infty }}$ methods) and scattering theory.