In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval 1 parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution.

* If α is a PV number then so are its powers α < sup > k </ sup >, for all natural exponents.

Heun's equation has four regular singular points: 0, 1, a and ∞ with exponents ( 0, 1 − γ ), ( 0, 1 − δ ), ( 0, 1 − ϵ ), and ( α, β ).

The regular singular points are a, b, and c. The pairs of exponents for each are respectively α ; α ', β ; β ', and γ ; γ '.

The following variant holds for arbitrary complex β, but is especially useful for handling negative integer exponents in ( 1 ):

In fact, the antipathy to outward ceremonies hailed by modern exponents as so uniquely characteristic of the `` direct thinking '' Zennist was a feature of Taoism.

This work resumed however with the development of the so-called " Second Quest ", among whose notable exponents was Rudolf Bultmann's student Ernst Käsemann.

It was during this time that he mentored Fernand Braudel, who would become one of the best-known exponents of this school.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

Kadanoff, Widom and Michael Fisher developed the ideas of critical exponents and scaling.

Theoretical models have also been developed to study the physics of phase transitions, such as the Landau-Ginzburg theory, Critical exponents and the use of mathematical techniques of quantum field theory and the renormalization group.

Primary exponents of the cyberpunk field include William Gibson, Neal Stephenson, Bruce Sterling, Pat Cadigan, Rudy Rucker, and John Shirley.

American counterculture exponents like William S. Burroughs ( whose literary influence on Gibson and cyberpunk in general is widely acknowledged ) and Timothy Leary were among the first to extoll the potential of computers and computer networks for individual empowerment.

Leading modern exponents of the instrument include Derek Adlam, Christopher Hogwood, Richard Troeger, and Miklos Spányi.

The normalization factor multiplying the DFT and IDFT ( here 1 and 1 / N ) and the signs of the exponents are merely conventions, and differ in some treatments.

The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1 / N.

If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation.

His two most helpful exponents are the late William W Bartley and David Miller, recently retired from the University of Warwick.

In electronic engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions ( see Fourier analysis ), and these are more conveniently expressed as the real part of exponential functions with imaginary exponents, using Euler's formula.

A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.

However, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.

For example, exponents are usually formatted using superscripts, e. g..

* Multiplied terms are simplified using exponents.

Something that becomes clear through these published compilations is that Satie did not so much reject Romanticism and its exponents like Wagner, but that he rejected certain aspects of it.

John Locke, George Berkeley, and David Hume were the primary exponents of empiricism in the 18th century Enlightenment, with Locke being the person who is normally known as the founder of empiricism as such.

In his writings, Leibniz used the term " fractional exponents ", but lamented that " Geometry " did not yet know of them.

Structurally they can be classified according to the functional groups ' locations as alpha-( α -), beta-( β -), gamma-( γ -) or delta-( δ -) amino acids ; other categories relate to polarity, acid / base / neutral, and side chain group type ( including: aliphatic, acyclic, hydroxyl or sulphur-containing, aromatic ).

Carina has the α, β and ε, Vela has γ and δ, Puppis has ζ, and so on.

This name was applied by the Arabs to the asterism of α, β, and γ Aquilae and probably goes back to the ancient Babylonians and Sumerians, who called α Aquilae the eagle star.

In Chinese, the asterism consisting of α, β, and γ Aquilae is known as (, ).

* In Chinese, (), meaning Northern Dipper, refers to an asterism consisting of ε Ursae Majoris, α Ursae Majoris, β Ursae Majoris, γ Ursae Majoris, δ Ursae Majoris, ζ Ursae Majoris and η Ursae Majoris.

It contains both α and β ' phase ; the β '- phase is body-centered cubic and is harder and stronger than α. Alpha-beta brasses are usually worked hot.

It exists in four major polymorphs: α, β, γ and T. Whereas α, β and T phases are based on B < sub > 12 </ sub > icosahedra, the γ-phase can be described as a rocksalt-type arrangement of the icosahedra and B < sub > 2 </ sub > atomic pairs.

As to the α and β phases, they might both coexist at ambient conditions with the β phase being more stable.

Bayer assigned a lower-case Greek letter, such as alpha ( α ), beta ( β ), gamma ( γ ), etc., to each star he catalogued, combined with the Latin name of the star ’ s parent constellation in genitive ( possessive ) form.

Polaris ( α UMi ), Kochab ( β UMi ), Pherkad ( γ UMi ), Yildun ( δ UMi ), Urodelus ( ε UMi ), Ahfa al Farkadain ( ζ UMi ) and Anwar al Farkadain ( η UMi ).

* Convex cone, a subset C of a vector space V is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C

In the following chemical equation with arrows pointing both ways to indicate equilibrium, A and B are reactant chemical species, S and T are product species, and α, β, σ, and τ are the stoichiometric coefficients of the respective reactants and products:

The goat's tail consists of stars β Cap and α < sup > 2 </ sup > Cap: star β Cap being of the third magnitude.

The α and β anomers of glucose.

Note the position of the hydroxyl group ( red or green ) on the anomeric carbon relative to the CH < sub > 2 </ sub > OH group bound to carbon 5: they are either on the opposite sides ( α ), or the same side ( β ).

where A ∈ N ( i. e., A is a single nonterminal ), α, β ∈ ( N U Σ )* ( i. e., α and β are strings of nonterminals and terminals ) and γ ∈ ( N U Σ )< sup >+</ sup > ( i. e., γ is a nonempty string of nonterminals and terminals ).