distribution
n. 1. The act of distributing or dispensing; the act of dividing
or apportioning among several or many; apportionment; as, the distribution
of an estate among heirs or children.
The phenomena of geological distribution are exactly analogous to
those of geography.
- A. R. Wallace.
2. Separation into parts or classes; arrangement of anything into
parts; disposition; classification.
3. That which is distributed.
4. (Logic) A resolving a whole into its parts.
5. (Print.) The sorting of types and placing them in their proper
boxes in the cases.
6. (Steam Engine) The steps or operations by which steam is supplied
to and withdrawn from the cylinder at each stroke of the piston;
viz., admission, suppression or cutting off, release or exhaust,
and compression of exhaust steam prior to the next admission.Geographical
distribution
the natural arrangements of animals and plants in particular regions
or districts.
Noun 1. distribution - (statistics) an arrangement of values of
a variable showing their observed or theoretical frequency of occurrence
Synonyms: statistical distribution
2. distribution - the spatial property of being scattered about
over an area or volume
Synonyms: dispersion
Antonyms:
compactness, concentration, denseness, density - the spatial property
of being crowded together
3. distribution - the act of distributing or spreading or apportioning
4. distribution - the commercial activity of transporting and selling
goods from a producer to a consumer
DISTRIBUTION. By this term is understood the division of an intestate's
estate according to law.
2. The English statute of 22 and 23 Car. II. c. 10, which was itself
probably borrowed from the 118th Novel of Justinian, is the foundation
of, perhaps, most acts of distribution in the several states. Vide
2 Kent, Com. 342, note; 8 Com. Dig. 522; 11 Vin. Ab. 189, 202; Com.
Dig. Administration, H.
Dictionary of
Computing
1. (software) distribution - A software source tree packaged for
distribution; but see kit.
2. (messaging) distribution - A vague term encompassing mailing
lists and Usenet newsgroups (but not BBS fora); any topic-oriented
message channel with multiple recipients.
3. (messaging) distribution - An information-space domain (usually
loosely correlated with geography) to which propagation of a Usenet
message is restricted; a much-underused feature.
Wikipedia
This page deals with mathematical distributions. For other meanings
of distribution, see distribution (disambiguation). This article
is not about probability distributions.
In mathematical analysis, distributions (also known as generalized
functions) are objects which generalize functions and probability
distributions. They extend the concept of derivative to all continuous
functions and beyond and are used to formulate generalized solutions
of partial differential equations. They are important in physics
and engineering where many non-continuous problems naturally lead
to differential equations whose solutions are distributions, such
as the Dirac delta distribution.
"Generalized functions" were introduced by Sergei Sobolev
in 1935. They were independently discovered in late 1940s by Laurent
Schwartz, who developed a comprehensive theory of distributions.
Sometimes, people
talk of "probability distribution" when they just mean
"probability measure", especially if it is obtained by
taking the product of the Lebesgue measure by a positive, real-valued
measurable function of integral equal to 1.
Basic idea The basic idea is as follows. If f : R ? R is an integrable
function, and f : R ? R is a smooth ( = infinitely often differentiable)
function with compact support ( = it is identically zero except
on some bounded set), then ?ffdx is a real number which linearly
and continuously depends on f. One can therefore think of the function
f as a continuous linear functional on the space which consists
of all the "test functions" f. Similarly, if P is a probability
distribution on the reals and f is a test function, then ?fdP is
a real number that continuously and linearly depends on f: probability
distributions can thus also be viewed as continuous linear functionals
on the space of test functions. This notion of "continuous
linear functional on the space of test functions" is therefore
used as the definition of a distribution.
Such distributions
may be multiplied with real numbers and can by added together, so
they form a real vector space. In general it is not possible to
define a multiplication for distributions, but distributions may
be multiplied with infinitely often differentiable functions.
To define the
derivative of a distribution, we first consider the case of a differentiable
and integrable function f : R ? R. If is a test function, then we
have using integration by parts (note that f is zero outside of
a bounded set and that therefore no boundary values have to be taken
into account). This suggests that if S is a distribution, we should
define its derivative S' as the linear functional which sends the
test function f to -S(f'). It turns out that this is the proper
definition; it extends the ordinary definition of derivative, every
distribution becomes infinitely often differentiable and the usual
properties of derivatives hold.
The Dirac delta
(so-called Dirac delta function) is the distribution which sends
the test function f to f(0). It is the derivative of the Heaviside
step function H(x) = 0 if x < 0 and H(x) = 1 if x = 0. The derivative
of the Dirac delta is the distribution which sends the test function
f to -f'(0). This latter distribution is our first example of a
distribution which is neither a function nor a probability distribution.
An
alternate definition is the limit of a sequence of functions. For
instance the delta function is given by
where da(x)
is 1/(2a) if x is between -a and a, and is 0 otherwise.
Formal definition In the sequel, real-valued distributions on an
open subset U of Rn will be formally defined. (With minor modifications,
one can also define complex-valued distributions, and one can replace
Rn by any smooth manifold.) First, the space D(U) of test functions
on U needs to be explained. A function f : U ? R is said to have
compact support if there exists a compact subset K of U such that
f(x) = 0 for all x in U \ K. The elements of D(U) are the infinitely
often differentiable functions f : U ? R with compact support. This
is a real vector space. We turn it into a topological vector space
by requiring that a sequence (or net) (fk) converges to 0 if and
only if there exists a compact subset K of U such that all fk are
identically zero outside K, and for every e > 0 and natural number
d = 0 there exists a natural number k0 such that for all k = k0
the absolute value of all d-th derivatives of fk is smaller than
e. With this definition, D(U) becomes a complete topological vector
space (in fact, a so-called LF-space).
The dual space
of the topological vector space D(U), consisting of all continuous
linear functionals S : D(U) ? R, is the space of all distributions
on U; it is a vector space and is denoted by D'(U).
The function
f : U ? R is called locally integrable if it is Lebesgue integrable
over every compact subset K of U. This is a large class of functions
which includes all continuous functions. The topology on D(U) is
defined in such a fashion that any locally integrable function f
yields a continuous linear functional on D(U) whose value on the
test function f is given by the Lebesgue integral ?U ff dx. Two
locally integrable functions f and g yield the same element of D(U)
if and only if they are equal almost everywhere. Similarly, every
Radon measure µ on U (which includes the probability distributions)
defines an element of D'(U) whose value on the test function f is
?f dµ.
As mentioned
above, integration by parts suggests that the derivative dS/dx of
the distribution S in direction x should be defined using the formula
dS / dx (f) = - S (df / dx) for all test functions f. In this way,
every distribution is infinitely often differentiable, and the derivative
in direction x is a linear operator on D'(U).
The space D'(U)
is turned into a locally convex topological vector space by defining
that the sequence (Sk) converges towards 0 if and only if Sk(f)
? 0 for all test functions f. This is the case if and only if Sk
converges uniformly to 0 on all bounded subsets of D(U). (A subset
of E of D(U) is bounded if there exists a compact subset K of U
and numbers dn such that every f in E has its support in K and has
its n-th derivatives bounded by dn.) With respect to this topology,
differentiation of distributions is a continuous operator; this
is an important and desirable property that is not shared by most
other notions of differentiation. Furthermore, the test functions
(which can itself be viewed as distributions) are dense in D'(U)
with respect to this topology.
If ? : U ? R
is an infinitely often differentiable function and S is a distribution
on U, we define the product S? by (S?)(f) = S(?f) for all test functions
f. The ordinary product rule of calculus remains valid.
Compact support and convolution We say that a distribution S has
compact support if there is a compact subset K of U such that for
every test function f whose support is completely outside of K,
we have S(f) = 0. Alternatively, one may define distributions with
compact support as continuous linear functionals on the space C8(U);
the topology on C8(U) is defined such that fk converges to 0 if
and only if all derivatives of fk converge uniformly to 0 on every
compact subset of U.
If both S and
T are distributions on Rn and one of them has compact support, then
one can define a new distribution, the convolution S*T of S and
T, as follows: if f is a test function in D(Rn) and x, y elements
of Rn, write fx(y) = x + y, ?(x) = T(fx) and (S*T)(f) = S(?). This
generalizes the classical notion of convolution of functions and
is compatible with differentiation in the following sense: d/dx
(S * T) = (d/dx S) * T = S * (d/dx T).
Tempered distributions and Fourier transform By using a larger space
of test functions, one can define the tempered distributions, a
subspace of D'(Rn). These distributions are useful if one studies
the Fourier transform in generality: all tempered distributions
have a Fourier transform, but not all distributions have one.
The space of
test functions employed here, the so-called Schwartz-space, is the
space of all infinitely differentiable rapidly decreasing functions,
where f : Rn ? R is called rapidly decreasing if any derivative
of f, multiplied with any power of |x|, converges towards 0 for
|x| ? 8. These functions form a complete topological vector space
with a suitably defined family of seminorms. More precisely, let
for a, ß multi-indices of size n. Then f is rapidly-decreasing
if all the values The family of seminorms pa, ß defines a
locally convex topology on the Schwartz-space. It is metrizable
and complete.
The derivative
of a tempered distribution is again a tempered distribution. Tempered
distributions generalize the bounded (or slow-growing) locally integrable
functions; all distributions with compact support and all square-integrable
functions can be viewed as tempered distributions.
To study the
Fourier transform, it is best to consider complex-valued test functions
and complex-linear distributions. The ordinary continuous Fourier
transform F yields then an automorphism of Schwartz-space, and we
can define the Fourier transform of the tempered distribution S
by (FS)(f) = S(Ff) for every test function f. FS is thus again a
tempered distribution. The Fourier transform is a continuous, linear,
bijective operator from the space of tempered distributions to itself.
This operation is compatible with differentiation in the sense that
F (d/dx S) = ix FS and also with convolution: if S is a tempered
distribution and ? is a slowly increasing infinitely often differentiable
function on Rn (meaning that all derivatives of ? grow at most as
fast as polynomials), then S? is again a tempered distribution and
F(S?) = FS * F?.
Using holomorphic functions as test functions The success of the
theory led to investigation of the idea of hyperfunction, in which
spaces of holomorphic functions are used as test functions. A refined
theory has been developed, in particular by Mikio Sato, using sheaf
theory and several complex variables. This extends the range of
symbolic methods that can be made into rigorous mathematics, for
example Feynman integrals.
See also Colombeau
algebra.
This article
is a copy of the article on Wikipedia.org - the free encyclopedia.
It is distributed under the terms of GNU Free Documentation License.
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volatilization More Related Words and Usage
Noun 1. distribution - (statistics) an arrangement of values of
a variable showing their observed or theoretical frequency of occurrence
Synonyms: statistical
2. - the spatial property of being scattered about over an area
or volume
Synonyms: dispersion
Antonyms:
compactness, concentration, denseness, density - the spatial property
of being crowded together
3. - the act of distributing or spreading or apportioning
4. - the commercial activity of transp Samples
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