If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

A still weaker example is the axiom of countable choice ( AC < sub > ω </ sub > or CC ), which states that a choice function exists for any countable set of nonempty sets.

where ω is implicitly a function of k. We assume that the wave packet α is almost monochromatic, so that A ( k ) is nonzero only in the vicinity of a central wavenumber k < sub > 0 </ sub >.

The dispersion relation is often expressed as a function, ω ( k ).

For this reason, permittivity is often treated as a complex function of the ( angular ) frequency of the applied field ω: ( since complex numbers allow specification of magnitude and phase ).

The complex permittivity is usually a complicated function of frequency ω, since it is a superimposed description of dispersion phenomena occurring at multiple frequencies.

The dielectric function ε ( ω ) must have poles only for frequencies with positive imaginary parts, and therefore satisfies the Kramers – Kronig relations.

The primary quantity that characterizes the electronic structure of any crystalline material is the probability of photon absorption, which is directly related to the imaginary part of the optical dielectric function ε ( ω ).

Let f be the function which maps database entries to 0 or 1, where f ( ω )= 1 if and only if ω satisfies the search criterion.

where ε < sub > r </ sub >( ω ) is now a complex function, with an imaginary part related to absorption of energy from the field by the medium.

The initial point is found by putting the initial angular frequency ω into the function and finding | H ( jω )|.

The output function ω represents the mapping of ordered pairs of input symbols and states onto output symbols, denoted mathematically as ω: Σ × Q → Z.

f < sub > n </ sub >( n ) > 2 ↑< sup > n-1 </ sup > n > 2 ↑< sup > n − 2 </ sup > ( n + 3 ) − 3 = A ( n, n ) for n ≥ 2, where A is the Ackermann function ( of which f < sub > ω </ sub > is a unary version ).

It is also worthy of note that any continuous function from ω < sub > 1 </ sub > to R ( the real line ) is eventually constant: so the Stone-Čech compactification of ω < sub > 1 </ sub > is ω < sub > 1 </ sub >+ 1, just as its one-point compactification ( in sharp contrast to ω, whose Stone-Čech compactification is much larger than ω ).

( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number k, every family of k nonempty sets has a choice function.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

Positive h and k values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.

Above, h is enthalpy, k is the thermal conductivity of the fluid, T is temperature, and is the viscous dissipation function.

A real number a is said to be computable if it can be approximated by some computable function in the following manner: given any integer, the function produces an integer k such that:

In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,

Every contraction mapping is Lipschitz continuous and hence uniformly continuous ( for a Lipschitz continuous function, the constant k is no longer necessarily less than 1 ).

The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valued function:

where a is the radius of the circular aperture, k is equal to 2π / λ and J < sub > 1 </ sub > is a Bessel function.

is an entire function in the variable b for all positive integers a where c < sub > d </ sub >( k ) is Ramanujan's sum.

Using this information, the expected value of the greatest common divisor function can be seen ( informally ) to not exist when k = 2.

where means the gradient of the angular frequency as a function of the wave vector, and is the unit vector in direction k.

For such data, a function that extracts the numeric part k of the file name and returns k mod n would be nearly optimal.

The committee debated the possibility of a shift key function ( like the Baudot code ), which would allow more than 64 codes to be represented by six bits.

The function of Apollo as a " healer " is connected with Paean ( Παιών-Παιήων ), the physician of the Gods in the Iliad, who seems to come from a more primitive religion.

Animals sacred to Apollo included wolves, dolphins, roe deer, swans, cicadas ( symbolizing music and song ), hawks, ravens, crows, snakes ( referencing Apollo's function as the god of prophecy ), mice and griffins, mythical eagle – lion hybrids of Eastern origin.

* Argument ( complex analysis ), a function which returns the polar angle of a complex number

They differ from other related families by often being pachycauline ( i. e. with a thickened trunk, usually wider at the base, which has a water storage function ), by usually having succulent leaves, and by possessing a trimerous flower with a superior ovary and seeds with an aryl.

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

Axons are distinguished from dendrites by several features, including shape ( dendrites often taper while axons usually maintain a constant radius ), length ( dendrites are restricted to a small region around the cell body while axons can be much longer ), and function ( dendrites usually receive signals while axons usually transmit them ).

They may not administer Penance ( Reconciliation ), Anointing of the Sick ( Extreme Unction ), or function as an ordained celebrant or concelebrant of the Mass ( by virtue of their office and their training and institution, they may act, if the need arises, as altar servers, lectors, ushers, porters, or Eucharistic ministers of the Cup, and if need be, the Host ).

The original song Advance Australia Fair was composed by Peter Dodds McCormick under the pen-name ' Amicus ' ( which means ' friend ' in Latin ), in the late 19th century, and first performed by Andrew Fairfax at a Highland Society function in Sydney on 30 November 1878.

It is has been argued that Luke may be writing: a letter of apology ( traditionally a defense for one ’ s beliefs ), a letter of legitimation for Christian beliefs, a letter to equip the church to function amidst the Roman Empire, or a letter that is apolitical.

* Despite its heritage, Applesoft lacked commands common to most other Microsoft BASIC interpreters, such as INSTR ( which searched for a substring in a given string ; this had to be done manually with loops and the MID $ function ), PRINT USING ( which formatted numbers with commas and currency signs according to a format string ), and INKEY $ ( which checked for a keypress without stopping the program as Applesoft's GET command — analogous to the INPUT $ function — did, although a PEEK location did provide this functionality ).

Functionalists like Monroe Beardsley argue that whether or not a piece counts as art depends on what function it plays in a particular context ; the same Greek vase may play a non-artistic function in one context ( carrying wine ), and an artistic function in another context ( helping us to appreciate the beauty of the human figure ).

This is the primary function of the imagination operating in the absence of the original experiential stimulus by which the images were first appropriated.

But because it is the function of the mind to turn the one into the other by means of the capacities with which words endow it, we do not unwisely examine the type of distinction, in the sphere of politics, on which decisions hang.

It was this fear which explained the development of a priestly caste whose function in society was to mollify and appease the angry deities.

A primary function is the operation of a Government Bid Center, which receives bids daily from the Federal Government's principal purchasing agencies.

Furthermore, one can find a neighborhood of Q in which the difference function is monotone, for since it is analytic it can have only a finite number of extrema in any interval.

The values Af are the ordinary values at Af of a multi-valued function g{t} which has components corresponding to those of f{t}.

Closely related to this function is the fact that the religious system provides a body of ultimate ends for the society, which are compatible with the supreme eternal ends.

Tone languages use for linguistic contrasts speech parameters which also function heavily in nonlinguistic use.

The form of the industry demand function is one which makes quantity demanded vary inversely with the product price, and vary directly with the level of Aj.

The extent to which the public-limit price is raised by a given increase in the basic wage rate is itself a function of three things: the passage of time, the level of GNP, and the size of the wage increase.

We are abstracting from the fact of strikes here, but it should be obvious that the extent to which the public-limit price is raised by a given increase in the basic wage rate is also a function of the show of resistance put up by the industry.

Moreover, whereas in Interstate Commerce Commission parlance `` variable cost '' means a cost deemed to vary in direct proportion to changes in rate of output, in the type of analysis now under review `` variable cost '' has been used more broadly, so as to cover costs which, while a function of some one variable ( such as output of energy, or number of customers ), are not necessarily a linear function.

271 - 307 ) indicate, that the concept of function in sociology has been built up from physiological and biological models, in which the notions of teleology, i.e., metaphysical purpose, are central.

In a properly ordered society the massive force of public law performs the function which in natural law theory ineptly is left altogether to a small voice so often still.

Extreme caution should be used, however, to avoid the conflicting usage of an index word or electronic switch which may result from the assignment of more than one name or function to the same address.

The specification of the objective function of which the optimization is desired.

The objective function, which is to be maximized, is some function, usually piecewise continuous, of the product state.

An optimal policy is an admissible policy Af which maximizes the objective function P.

We thus have an optimal policy which maximizes the expected value of the objective function for a given Af.

As retinal images are conceded to be an integral function of the brain it seems logical to suppose that the nerves, between the inner brain and the eyes, carry the direct drive for cooperation from the various brain centers -- rather than to theorize on the transmission of an image which is already in required location.

A local church which conceives its function to be entirely that of ministering to the conscious desires and concerns of its members tends to look on everything ecumenical as an extra, not as a normal aspect of its own life as a church.

Exceptions are Lambertian surfaces, which scatter radiation in all directions according to a cosine function, so their albedo does not depend on the incident distribution.