Function

Function may refer to:

Math, computing and engineering

Music

Other uses

See also

Minkowski's question mark function

In mathematics, the Minkowski question mark function (or the slippery devil's staircase), denoted by ?(x), is a function possessing various unusual fractal properties, defined by Hermann Minkowski (1904, pages 171–172). It maps quadratic irrationals to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. In addition, it maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.

Definition

If is the continued fraction representation of an irrational number x, then

whereas:

If is a continued fraction representation of a rational number x, then

Intuitive explanation

To get some intuition for the definition above, consider the different ways of interpreting an infinite string of bits beginning with 0 as a real number in [0,1]. One obvious way to interpret such a string is to place a binary point after the first 0 and read the string as a binary expansion: thus, for instance, the string 001001001001001001001001... represents the binary number 0.010010010010..., or 2/7. Another interpretation views a string as the continued fraction [0; a₁, a₂, … ], where the integers a_i are the run lengths in a run-length encoding of the string. The same example string 001001001001001001001001... then corresponds to [0; 2, 1, 2, 1, 2, 1, …] = √3 − 1/2. If the string ends in an infinitely long run of the same bit, we ignore it and terminate the representation; this is suggested by the formal "identity":

Functional theories of grammar

Functional theories of grammar are those approaches to the study of language that see the functions of language and its elements to be the key to understanding linguistic processes and structures. Functional theories of language propose that since language is fundamentally a tool, it is reasonable to assume that its structures are best analyzed and understood with reference to the functions they carry out. Functional theories of grammar differ from formal theories of grammar, in that the latter seeks to define the different elements of language and describe the way they relate to each other as systems of formal rules or operations, whereas the former defines the functions performed by language and then relates these functions to the linguistic elements that carry them out. This means that functional theories of grammar tend to pay attention to the way language is actually used in communicative context, and not just to the formal relations between linguistic elements.

Index

An index is an indirect shortcut derived from and pointing into a greater volume of values, data, information or knowledge. Index may refer to:

Business

Publishing

Index (statistics)

In statistics and research design, an index is a composite statistic – a measure of changes in a representative group of individual data points, or in other words, a compound measure that aggregates multiple indicators. Indexes summarize and rank specific observations.

Much data in the field of social sciences is represented in various indices such as Gender Gap Index, Human Development Index or the Dow Jones Industrial Average.

Item in indexes are usually weighted equally, unless there are some reasons against it (for example, if two items reflect essentially the same aspect of a variable, they could have a weight of 0.5 each).

Constructing the items involves four steps. First, items should be selected based on their face validity, unidimensionality, the degree of specificity in which a dimension is to be measured, and their amount of variance. Items should be empirically related to one another, which leads to the second step of examining their multivariate relationships. Third, indexes scores are designed, which involves determining their score ranges and weights for the items. Finally, indexes should be validateds, which involves testing whether they can predict indicators related to the measured variable not used in their construction.

Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H. The index of H in G is usually denoted |G : H| or [G : H] or (G:H).

Formally, the index of H in G is defined as the number of cosets of H in G. (The number of left cosets of H in G is always equal to the number of right cosets.) For example, let Z be the group of integers under addition, and let 2Z be the subgroup of Z consisting of the even integers. Then 2Z has two cosets in Z (namely the even integers and the odd integers), so the index of 2Z in Z is two. To generalize,

for any positive integer n.

If N is a normal subgroup of G, then the index of N in G is also equal to the order of the quotient group G / N, since this is defined in terms of a group structure on the set of cosets of N in G.

If G is infinite, the index of a subgroup H will in general be a non-zero cardinal number. It may be finite - that is, a positive integer - as the example above shows.

Soul (The Kentucky Headhunters album)

Soul is the sixth studio album released by American country rock & southern rock band The Kentucky Headhunters. It was released in 2003 on Audium Entertainment. No singles were released from the album, although one of the tracks, "Have You Ever Loved a Woman?", was first a single for Freddie King in 1960.

Track listing

All songs written and composed by The Kentucky Headhunters except where noted. 

Personnel

The Kentucky Headhunters

Guest musicians