In particular, when a function f: R → R is said to be Lebesgue measurable what is actually meant is that is a measurable function — that is, the domain and range represent different σ-algebras on the same underlying set ( here is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on R ).

In probability theory, the sigma algebra often represents the set of available information, and a function ( in this context a random variable ) is measurable if and only if it represents an outcome that is knowable based on the available information.

* A Lebesgue measurable function is a measurable function, where is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers C. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.

When applying the Riemann constant in sigma mathematics, or when using integrals, it is a common practice to apply the quadratic function when ascertaining the equation's alpha variable.

It uses geopotential, specific heat, the exner function π, and change in sigma coordinate.

In mathematics, by sigma function one can mean one of the following:

* Weierstrass sigma function, related to elliptic functions

* Kronecker's sigma function

The sum of positive divisors function σ < sub > x </ sub >( n ), for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as

When x is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ ( n ) is equivalent to σ < sub > 1 </ sub >( n ) ().

* For the trivial sigma algebra the conditional probability is a constant function,

There are also anti-sigma factors that inhibit the function of sigma factors and anti-anti-sigma factors that restore sigma factor function.

where denotes the conditional expectation of the indicator function of the event,, given the sigma algebra.

Included are those involved in chemotaxis, outer membrane channel function, degradation of aromatic ring compounds, and the breakdown of plant-derived carbon sources, in addition to many extracytoplasmic function sigma factors, providing the organism with the ability to respond to a wide range of environmental fluctuations.

In mathematics, additivity and sigma additivity ( also called countable additivity ) of a function defined on subsets of a given set are abstractions of the intuitive properties of size ( length, area, volume ) of a set.

In this equation, represents the wavelength of the sinusoidal factor, represents the orientation of the normal to the parallel stripes of a Gabor function, is the phase offset, is the sigma of the Gaussian envelope and is the spatial aspect ratio, and specifies the ellipticity of the support of the Gabor function.

In the case of normally distributed data, roughly 1 in 22 observations will differ by twice the standard deviation or more from the mean, and 1 in 370 will deviate by three times the standard deviation ; see three sigma rule for details.

For the sigma N-transcription factor activator protein in prokaryotes, see NtrC

The term can also refer to an establishment which has long ceased to function as an abbey, in some cases for centuries ( for example, see Westminster Abbey below ).

For alternative ( equivalent ) formulations of the definition in terms of a bijective function or a surjective function, see the section Formal definition and properties below.

To see this, write the function f ( x )/ g ( x ) as the product.

To see that this is true, compute the characteristic function of the sample mean:

The delta function has only radial dependence, so the Laplace operator ( aka scalar Laplacian ) in the spherical coordinate system simplifies to ( see del in cylindrical and spherical coordinates )

The final stage of the audition process in some orchestras is a test week, in which the performer plays with the orchestra for a week or two, which allows the conductor and principal players to see if the individual can function well in an actual performance setting.

In the original 1902 definition by Bayliss and Starling ( see below ), they specified that, to be classified as a hormone, a chemical must be produced by an organ, be released ( in small amounts ) into the blood, and be transported by the blood to a distant organ to exert its specific function.

It shares this function with several other letters ( ι, υ ) and digraphs ( ει, οι ), which are all pronounced alike ( see iotacism ).

The original proof is based on the Taylor series expansions of the exponential function e < sup > z </ sup > ( where z is a complex number ) and of sin x and cos x for real numbers x ( see below ).

The exponential function e < sup > x </ sup > for real values of x may be defined in a few different equivalent ways ( see Characterizations of the exponential function ).

where ζ denotes the Riemann zeta function ( see Lehmer ; one approach to prove the inequality is to obtain the Fourier series for the polynomials B < sub > n </ sub >).

Many Methodist churches ( see The United Methodist Church, among others ) retain the form and function of episcopal polity, although in a modified form, called connexionalism.

A transcendental entire function is an entire function that is not a polynomial ( see transcendental function ).

One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand drawn images of a similar function, which is now called the Koch curve ( see Figure 2 ).

( see Sampling the DTFT ) In most cases of practical interest, the s sequence represents a longer sequence that was truncated by the application of a finite-length window function or FIR filter array.

There is in fact no complex number z for which, and hence the reciprocal gamma function is an entire function, with zeros at z = 0, − 1, − 2 ,.... We see that the gamma function has a local minimum at where it attains the value.

( In an ideal " perfect hash function ", no bucket should have more than one record ; but a small number of collisions is virtually inevitable, even if n is much larger than m – see the birthday paradox ).

However, the expression sin < sup >− 1 </ sup > x usually does not represent the multiplicative inverse to sin x, but the inverse of the sine function applied to x ( actually a partial inverse ; see below ).

This process is able to adapt complex structures from simpler beginnings, or convert complex structures from one function to another ( see spandrel ).

The magnitude of the CCH is an instantaneous value subject to change as a function of time ( i. e. from moment to moment ), and is subject to study including load curve and busy hour as other measures of traffic are.

Loading characteristic: In multichannel telephone systems, a plot, for the busy hour, of the equivalent mean power and the peak power as a function of the number of voice channels.

A busy beaver function quantifies these upper limits on a given type of " operational busyness ", and is a noncomputable function.

In fact, a busy beaver function can be shown to grow faster asymptotically than does any computable function.

Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

The city is unified by a complicated system of shared religious responsibilities, called cargas, which function mostly to support a very busy calendar of saint ’ s days and other festivals which occur in one part or another almost all year round.