ຊຸດຈຳນວນແມ່ນຜົນບວກລວມຂອງອັນດັບຈຳນວນໃດໜຶ່ງ. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.

In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using the summation symbol ∑. An example is the famous series from Zeno's dichotomy:

The terms of the series are often produced according to a certain rule, such as by a formula, or by an algorithm. As there are an infinite number of terms, this notion is often called an infinite series.  Unlike finite summations, infinite series need tools from mathematical analysis to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics and computer science.

Basic properties

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For any sequence  of rational numbers, real numbers, complex numbers, functions thereof, etc., the associated series is defined as the ordered formal sum
.
The sequence of partial sums  associated to a sequence  is defined for each  as the sum of the sequence  from  to
.
By definition the series  converges to a limit  if and only if the associated sequence of partial sums  converges to . This definition is usually written as
.

More generally, if  is a function from an index set I to a set G, then the series associated to  is the formal sum of the elements  over the index elements  denoted by
.

When the index set is the natural numbers , the function  is a sequence denoted by . A series indexed on the natural numbers is an ordered formal sum and so we rewrite  as  in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers
.

When the set  is a semigroup, the sequence of partial sums  associated to a sequence  is defined for each  as the sum of the sequence   from  to
.

When the semigroup  is also a topological space, then the series  converges to an element  if and only if the associated sequence of partial sums  converges to . This definition is usually written as
.

Convergent series
A series  ∑an  is said to 'converge' or to 'be convergent' when the sequence SN of partial sums has a finite limit.  If the limit of SN is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the sum of the series

An easy way that an infinite series can converge is if all the an are zero for n sufficiently large.  Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

Working out the properties of the series that converge even if infinitely many terms are non-zero is the essence of the study of series. Consider the example

It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼.  This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2.  In other words, the series has an upper bound. Proving that the series is equal to 2 requires only elementary algebra, however. If the series is denoted S, it can be seen that

Therefore,

Mathematicians extend the idiom discussed earlier to other, equivalent notions of series.  For instance, when we talk about a recurring decimal, as in

we are talking, in fact, just about the series

But since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand.  In particular, it should offend no sensibilities if we make no distinction between 0.111… and 1/9.  Less clear is the argument that , but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations.  See 0.999... for more.

Examples
 A geometric series is one where each successive term is produced by multiplying the previous term by a constant number.  Example:

In general, the geometric series

converges if and only if |z| < 1.

 The harmonic series is the series

The harmonic series is divergent.

 An alternating series  is a series where terms alternate signs.  Example:

The p-series

converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of r, the sum of this series is Riemann's zeta function.

A telescoping series

converges if the sequence bn converges to a limit L as n goes to infinity. The value of the series is then b1 − L.

Calculus and partial summation as an operation on sequences

Observe that partial summation takes as input a sequence, { an }, and gives as output another sequence, { SN } – partial summation is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference operator, Δ. These behave as discrete analogs of integration and differentiation, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence {1, 1, 1, ...} has series {1, 2, 3, 4, ...} as its partial summation, which is analogous to the fact that

In computer science it is known as prefix sum.

Properties of series
Series are classified not only by whether they converge or diverge, but also by the properties of the terms an (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term an (whether it is a real number, arithmetic progression, trigonometric function);  etc.

Non-negative terms

When an is a non-negative real number for every n, the sequence SN of partial sums is non-decreasing. It follows that a series ∑an with non-negative terms converges if and only if the sequence SN of partial sums is bounded.

For example, the series

is convergent, because the inequality

and a telescopic sum argument implies that the partial sums are bounded by 2.

Absolute convergence

A series

is said to converge absolutely if the series of absolute values

converges. It can be proved that this is sufficient to make not only the original series converge to a limit, but also for any reordering of it to converge to the same limit.

Conditional convergence

A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series

which is convergent (and its sum is equal to ln 2), but the series formed by taking the absolute value of each term is the divergent harmonic series. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the an are real and S is any real number, that one can find a reordering so that the reordered series converges with sum equal to S.

Abel's test is an important tool for handling semi-convergent series. If a series has the form

where the partial sums BN =   are bounded, λn has bounded variation, and  exists:

then the series  is convergent.  This applies to the pointwise convergence of many trigonometric series, as in

with 0 < x < 2π.  Abel's method consists in writing bn+1 = Bn+1 − Bn, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series  to the absolutely convergent series

Convergence tests

 n-th term test: If limn→∞ an ≠ 0 then the series diverges.
Comparison test 1: If ∑bn  is an absolutely convergent series such that |an | ≤ C |bn | for some number C  and for sufficiently large n , then ∑an  converges absolutely as well. If ∑|bn | diverges, and  |an | ≥ |bn | for all sufficiently large n , then ∑an  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the an  alternate in sign).
Comparison test 2: If ∑bn  is an absolutely convergent series such that |an+1 /an | ≤ |bn+1 /bn | for sufficiently large n , then ∑an  converges absolutely as well. If ∑|bn | diverges, and  |an+1 /an | ≥ |bn+1 /bn | for all sufficiently large n , then ∑an  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the an  alternate in sign).
Ratio test: If there exists a constant C < 1 such that |an+1/an|<C for all sufficiently large n, then ∑an converges absolutely. When the ratio is less than 1, but not less than a constant less than 1, convergence is possible but this test does not establish it.
Root test: If there exists a constant C < 1 such that |an|1/n ≤ C for all sufficiently large n, then ∑an converges absolutely.
Integral test: if ƒ(x) is a positive monotone decreasing function defined on the interval [1, ∞) with ƒ(n) = an for all n, then ∑an converges if and only if the integral  ∫1∞ ƒ(x) dx is finite.
Cauchy's condensation test: If an is non-negative and non-increasing, then the two series  ∑an  and  ∑2ka(2k) are of the same nature: both convergent, or both divergent.
Alternating series test: A series of the form ∑(−1)n an (with an ≥ 0) is called alternating. Such a series converges if the sequence an is monotone decreasing and converges to 0. The converse is in general not true.
For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

Series of functions

A series of real- or complex-valued functions

converges pointwise on a set E, if the series converges for each x in E as an ordinary series of real or complex numbers.  Equivalently, the partial sums

converge to ƒ(x) as N → ∞ for each x ∈ E.

A stronger notion of convergence of a series of functions is called uniform convergence.  The series converges uniformly if it converges pointwise to the function ƒ(x), and the error in approximating the limit by the Nth partial sum,

can be made minimal independently of x by choosing a sufficiently large N.

Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit.  For example, if a series of continuous functions converges uniformly, then the limit function is also continuous.  Similarly, if the ƒn are integrable on a closed and bounded interval I and converge uniformly, then the series is also integrable on I and can be integrated term-by-term. Tests for uniform convergence include the Weierstrass' M-test, Abel's uniform convergence test, Dini's test.

More sophisticated types of convergence of a series of functions can also be defined.  In measure theory, for instance, a series of functions converges almost everywhere if it converges pointwise except on a certain set of measure zero.  Other modes of convergence depend on a different metric space structure on the space of functions under consideration.  For instance, a series of functions converges in mean on a set E to a limit function ƒ provided

as N → ∞.

Power series

Many functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. For example, the series

converges to  for all x.

In general, a power series is any series of the form

Unless it converges only at x=c, such a series converges on a certain open disc of convergence centered at the point c in the complex plane, and may also converge at some of the points of the boundary of the disc.  The radius of this disc is known as the radius of convergence, and can in principle be determined from the asymptotics of the coefficients an.  The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.

Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent.
When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.
However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.

Laurent series

Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents.  A Laurent series is thus any series of the form

If such a series converges, then in general it does so in an annulus rather than a disc, and possibly some boundary points.  The series converges uniformly on compact subsets of the interior of the annulus of convergence.

Dirichlet series

A Dirichlet series is one of the form

where s is a complex number.  For example, if all an are equal to 1, then the Dirichlet series is the Riemann zeta function

Like the zeta function, Dirichlet series in general play an important role in analytic number theory.  Generally a Dirichlet series converges if the real part of s is greater than a number called the abscissa of convergence.  In many cases, a Dirichlet series can be extended to an analytic function outside the domain of convergence by analytic continuation.  For example, the Dirichlet series for the zeta function converges absolutely when Re s > 1, but the zeta function can be extended to a holomorphic function defined on    with a simple pole at 1.

This series can be directly generalized to general Dirichlet series.

Trigonometric series

A series of functions in which the terms are trigonometric functions is called a trigonometric series:

The most important example of a trigonometric series is the Fourier series of a function.

History of the theory of infinite series

Development of infinite series
Greek mathematician Archimedes produced the first known summation of an infinite series with a
method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π.

The idea of an infinite series expansion of a function was conceived in India by Madhava of the Kerala school of astronomy and mathematics in the 14th century, who also developed precursors to the modern concepts of the power series, the Taylor series, the Maclaurin series, rational approximations of infinite series, and infinite continued fractions. He discovered a number of infinite series, including the Taylor series of the trigonometric functions of sine, cosine, tangent and arctangent, the Taylor series approximations of the sine and cosine functions, and the power series of the radius, diameter, circumference, angle θ, π and π/4. His students and followers in the Kerala School further expanded his works with various other series expansions and approximations, until the 16th century.

In the 17th century, James Gregory worked in the new decimal system on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of hypergeometric series and q-series.

Convergence criteria
The study of the convergence criteria of a series began with Madhava in the 14th century, who developed tests of convergence of infinite series, which his followers further developed at the Kerala school.

In Europe, however, the investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Euler had already considered the hypergeometric series

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long before by Gregory (1668). Leonhard Euler and Gauss had given various criteria, and Colin Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form.

Abel (1826) in his memoir on the binomial series

corrected certain of Cauchy's conclusions, and gave a completely
scientific summation of the series for complex values of  and . He showed the necessity of considering the subject of continuity in questions of convergence.

Cauchy's methods led to special rather than general criteria, and
the same may be said of Raabe (1832), who made the first elaborate
investigation of the subject, of De Morgan (from 1842), whose
logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have
shown to fail within a certain region; of Bertrand (1842), Bonnet
(1843), Malmsten (1846, 1847, the latter without integration);
Stokes (1847), Paucker (1852), Chebyshev (1852), and Arndt
(1853).

General criteria began with Kummer (1835), and have been
studied by Eisenstein (1847), Weierstrass in his various
contributions to the theory of functions, Dini (1867),
DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.

Uniform convergence

The theory of uniform convergence was treated by Cauchy (1821), his
limitations being pointed out by Abel, but the first to attack it
successfully were Seidel and Stokes (1847-48). Cauchy took up the
problem again (1853), acknowledging Abel's criticism, and reaching
the same conclusions which Stokes had already found. Thomae used the
doctrine (1866), but there was great delay in recognizing the
importance of distinguishing between uniform and non-uniform
convergence, in spite of the demands of the theory of functions.

Semi-convergence

A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not absolutely convergent.

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834),
who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847).  Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function

Genocchi (1852) has further contributed to the theory.

Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into
prominence.

Fourier series

Fourier series were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
Jacob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still
earlier by Vieta. Euler and Lagrange simplified the subject,
as did Poinsot, Schröter, Glaisher, and Kummer.

Fourier (1807) set for himself a different problem, to
expand a given function of x in terms of the sines or cosines of
multiples of x, a problem which he embodied in his Théorie analytique de la chaleur (1822). Euler had already given the
formulas for determining the coefficients in the series;
Fourier was the first to assert and attempt to prove the general
theorem. Poisson (1820-23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for Cauchy (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see convergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine, Lipschitz, Schläfli, and
du Bois-Reymond. Among other prominent contributors to the theory of
trigonometric and Fourier series were Dini, Hermite, Halphen,
Krause, Byerly and Appell.

Generalizations

Asymptotic series
Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain.  In general they do not converge. But they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.

Divergent series

Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense.  A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence.  Summability methods include Cesàro summation, (C,k) summation, Abel summation, and Borel summation, in increasing order of generality (and hence applicable to increasingly divergent series).

A variety of general results concerning possible summability methods are known.  The Silverman–Toeplitz theorem characterizes matrix summability methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients.  The most general method for summing a divergent series is non-constructive, and concerns Banach limits.

Series in Banach spaces
The notion of series can be easily extended to the case of a Banach space.  If xn is a sequence of elements of a Banach space X, then the series Σxn converges to x ∈ X if the sequence of partial sums of the series tends to x; to wit,

as N → ∞.

More generally, convergence of series can be defined in any abelian Hausdorff topological group.  Specifically, in this case, Σxn converges to x if the sequence of partial sums converges to x.

Summations over arbitrary index sets

Definitions may be given for sums over an arbitrary index set I.  There are two main differences with the usual notion of series: first, there is no specific order given on the set I; second, this set I may be uncountable.

Families of non-negative numbers

When summing a family {ai}, i ∈ I, of non-negative numbers, one may define

When the sum is finite, the set of i ∈ I such that ai > 0 is countable.  Indeed for every n ≥ 1, the set  is finite, because

If I  is countably infinite and enumerated as I = {i0,  i1,...} then the above defined sum satisfies

provided the value ∞ is allowed for the sum of the series.

Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the counting measure, which accounts for the many similarities between the two constructions.

Abelian topological groups

Let a : I → X, where I  is any set and X  is an abelian Hausdorff topological group.  Let F  be the collection of all finite subsets of I.  Note that F  is a directed set ordered under inclusion with union as join.  Define the sum S  of the family a  as the limit

if it exists and say that the family a is unconditionally summable.  Saying that  the sum S  is the limit of finite partial sums means that for every neighborhood V  of 0 in X, there is a finite subset A0 of I  such that

Because F  is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.

For every W, neighborhood of 0 in X, there is a smaller neighborhood V  such that V − V ⊂ W. It follows that the finite partial sums of an unconditionally summable family  ai, i ∈ I, form a Cauchy net, that is: for every W, neighborhood of 0 in X, there is a finite subset A0 of I  such that

When X  is complete, a family a is unconditionally summable in X  if and only if the finite sums satisfy the latter Cauchy net condition. When X  is complete and ai, i ∈ I, is unconditionally summable in X, then for every subset J ⊂ I, the corresponding subfamily  aj, j ∈ J, is also unconditionally summable in X.

When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group X = R.

If a family a in X  is unconditionally summable, then for every W, neighborhood of 0 in X, there is a finite subset A0 of I  such that ai ∈ W  for every i not in A0.  If X  is  first-countable, it follows that the set of i ∈ I  such that ai ≠ 0 is countable.  This need not be true in a general abelian topological group (see examples below).

Unconditionally convergent series

Suppose that I = N.  If a family an, n ∈ N, is unconditionally summable in an abelian Hausdorff topological group X, then the series in the usual sense converges and has the same sum,

By nature, the definition of unconditional summability is insensitive to the order of the summation.  When ∑an is unconditionally summable, then the series remains convergent after any permutation σ of the set N of indices, with the same sum,

It can be proved that the converse holds: is a series ∑an converges after any permutation, then it is unconditionally convergent.  When X  is complete, then unconditional convergence is also equivalent to the fact that all subseries are convergent;  if  X  is a Banach space, this is equivalent to say that for every sequence of signs εn = 1 or −1, the series

converges in X.  If X  is a Banach space, then one may define the notion of absolute convergence.  A series ∑an of vectors in X  converges absolutely if

If a series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite dimensional Banach spaces (theorem of ).

Well-ordered sums

Conditionally convergent series can be considered if I is a well-ordered set, for example an ordinal number α0.  One may define by transfinite recursion:

and for a limit ordinal α,

if this limit exists.  If all limits exist up to α0, then the series converges.

Examples

Given a function f : X→Y, with Y an abelian topological group, define for every a ∈ X

a function whose support is a singleton {a}.  Then

in the topology of pointwise convergence (that is, the sum is taken in the infinite product group YX ).
In the definition of partitions of unity, one constructs sums of functions over arbitrary index set I,

While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given x, only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise.  Actually, one usually assumes more: the family of functions is locally finite, i.e., for every x there is a neighborhood of x in which all but a finite number of functions vanish.  Any regularity property of the φi,  such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.
On the first uncountable ordinal ω1 viewed as a topological space in the order topology, the constant function f: [0,ω1) → [0,ω1] given by f(α) = 1 satisfies

(in other words, ω1 copies of 1 is ω1) only if one takes a limit over all countable partial sums, rather than finite partial sums.  This space is not separable.

See also

Convergent series
Convergence tests
Sequence transformation
Infinite product
Infinite expression
Continued fraction
Iterated binary operation
List of mathematical series
Prefix sum
Series expansion

Notes

References
 Bromwich, T.J. An Introduction to the Theory of Infinite Series MacMillan & Co. 1908, revised 1926, reprinted 1939, 1942, 1949, 1955, 1959, 1965.

External links

 Graphical simulation of series convergence
 Many example problems on series, with solutions
 Infinite Series Tutorial

Calculus

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The Hesperides () are nymphs in Greek mythology. They look after a peaceful garden in the far west of the world. The garden is located near Cyrene (or Benghazi) in Libya or the Atlas Mountains further west. In the Ancient Greek view of the world, this is at the edge of Oceanus, the world-ocean. The nymphs are said to be the daughters of Hesperus.

Nymphs

Hesperides


In Hinduism, a murti (also spelled murthi or murthy) usually means an image in which the Divine Spirit is expressed ('murta'). Hindus call the presence of God into the image so that they can communicate with him and receive his blessings.Hindus don't worship the murti or statue itself but the god who is present in their minds and souls. It can also be set up any where

Role of murtis in worship
Murtis sometimes are abstract, but almost always stone or metal images of God in a human-like form like Shiva or Ganesha, Rama or Krishna, Saraswati or Kali. Murtis are made according to strict prescriptions and then installed by highly trained priests through a ceremony. The priests can then call on God in the image daily.

Murtis in Hindu  temples and shrines are a mystical form of communication with God and devas. This is similar to our ability to communicate with others through the telephone. One does not talk to the telephone; rather the telephone is a way to interact with another person. Without the telephone, one could not have a conversation across long distances; and without the sanctified image in the temple, one cannot easily talk with the Deity.

Going beyond murtis in Hinduism
The image or murti of worship is a focus for prayers, although Hindus can see God is in all things, in stone and water, fire, air and ether, in the enlightened person of a satguru and inside their own soul. Some temples do not have any murti in the sanctum but a symbolic diagram. Some Hindu branches reject the worship of images.

In Hinduism, one of the highest achievements is when one goes beyond the need of all form and symbol. This is the yogi's goal obtained through meditation.  Hinduism is also one of the religions that uses more symbols to represent God in preparation for getting beyond them.

Sacred images in other religions
Image worship appears to be an intelligent, mystical practice shared by all of the world's great religions. All religions have their symbols of holiness: the Christian cross, or statues of Mother Mary and Saint Theresa, the holy Kaaba in Mecca, the Sikh Adi Granth enshrined in the Golden Temple in Amritsar, the Arc and Torah of the Jews, the image of a meditating Buddha, the totems of indigenous and Pagan faiths, and the artifacts of the holy men and women of all religions. Any Christian respects the Bible and considers it sacred. His book and the Hindu's murtis are similar in this way.

Critics of image worship consider this practice "idolatry". People who practice idolatry believe that God is the material object itself. Instead, Hindus worship murtis to call on the presence of the spiritual God and then communicate with him.

References

Hinduism
Religious objects

Murti


In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.

In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate.

If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor (see strange attractor below). If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.

A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time.  The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).

Motivation of attractors
A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamical system specify its behavior over any given short period of time.  To determine the system's behavior for a longer period, it is often necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers.

Dynamical systems in the physical world tend to arise from dissipative systems: if it were not for some driving force, the motion would cease.  (Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of the phase space of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.

Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity.  Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.

For example, the damped pendulum has two invariant points: the point  of minimum height and the point  of maximum height.  The point  is also a limit set, as trajectories converge to it; the point  is not a limit set.  Because of the dissipation due to air resistance, the point  is also an attractor.  If there was no dissipation,  would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.

Some attractors are known to be chaotic (see strange attractor), in which case the evolution of any two distinct points of the attractor result in exponentially diverging trajectories, which complicates prediction when even the smallest noise is present in the system.

Mathematical definition
Let  represent time and let  be a function which specifies the dynamics of the system.  That is, if  is a point in an -dimensional phase space, representing the initial state of the system, then  and, for a positive value of ,  is the result of the evolution of this state after  units of time.  For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane  with coordinates , where  is the position of the particle,  is its velocity, , and the evolution is given by

An attractor is a subset  of the phase space characterized by the following three conditions:
  is forward invariant under : if  is an element of  then so is , for all .
 There exists a neighborhood of , called the basin of attraction for  and denoted , which consists of all points  that "enter"  in the limit . More formally,  is the set of all points  in the phase space with the following property:
 For any open neighborhood  of , there is a positive constant  such that  for all real .
 There is no proper (non-empty) subset of  having the first two properties.

Since the basin of attraction contains an open set containing , every point that is sufficiently close to  is attracted to .  The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of , the Euclidean norm is typically used.

Many other definitions of attractor occur in the literature.  For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that  be a neighborhood.

Types of attractors
Attractors are portions or subsets of the phase space of a dynamical system.  Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space.  More complex attractors that cannot be categorized as simple geometric subsets, such as topologically wild sets, were known of at the time but were thought to be fragile anomalies.  Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

Two simple attractors are a fixed point and the limit cycle.  Attractors can take on many other geometric shapes (phase space subsets).  But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. intersection and union) of fundamental geometric objects (e.g. lines, surfaces, spheres, toroids, manifolds), then the attractor is called a strange attractor.

Fixed point

A fixed point of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation.  The final state that a dynamical system evolves towards corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of a damped pendulum, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl containing a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor.  This is equivalent to the difference between stable and unstable equilibria. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (unstable equilibrium).

In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the nonlinear dynamics of stiction, friction, surface roughness, deformation (both elastic and plasticity), and even quantum mechanics. In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble's spherical shape, are both much more complex surfaces when examined under a microscope, and their shapes change or deform during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains. There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are considered stationary or fixed points, some of which are categorized as attractors.

Finite number of points

In a discrete-time system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called a periodic point. This is illustrated by the logistic map, which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2n points, 3 points, 3×2n points, 4 points, 5 points, or any given positive integer number of points.

Limit cycle

A limit cycle is a periodic orbit of a continuous dynamical system that is isolated. It concerns a cyclic attractor.  Examples include the swings of a pendulum clock, and the heartbeat while resting. The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting. For a physical pendulum under friction, the resting state will be a fixed-point attractor. The difference with the clock pendulum is that there, energy is injected by the escapement mechanism to maintain the cycle.

Limit torus
There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. This kind of attractor is called an  -torus if there are  incommensurate frequencies. For example, here is a 2-torus:

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of  periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.

Strange attractor

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.  If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.

The term strange attractor was coined by David Ruelle and Floris Takens to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.

Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor.

Attractors characterize the evolution of a system

The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters.  An example is the well-studied logistic map,  , whose basins of attraction for various values of the parameter  are shown in the figure. If , all starting  values of  will rapidly lead to function values that go to negative infinity; starting  values of  will also go to negative infinity. But for  the  values rapidly converge to , i.e. at this value of , a single value of  is an attractor for the function's behaviour. For other values of , more than one value of  may be visited: if  is 3.2, starting values of  will lead to function values that alternate between  and . At some values of , the attractor is a single point (a "fixed point"), at other values of  two values of  are visited in turn (a period-doubling bifurcation), or, as a result of further doubling, any number  values of ; at yet other values of , any given number of values of  are visited in turn; finally, for some values of , an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.

Basins of attraction

An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor. For a stable linear system, every point in the phase space is in the basin of attraction. However, in nonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.

Linear equation or system

An univariate linear homogeneous difference equation  diverges to infinity if  from all initial points except 0; there is no attractor and therefore no basin of attraction. But if  all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.

Likewise, a linear matrix difference equation in a dynamic vector , of the homogeneous form  in terms of square matrix  will have all elements of the dynamic vector diverge to infinity if the largest eigenvalues of  is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire -dimensional space of potential initial vectors is the basin of attraction.

Similar features apply to linear differential equations. The scalar equation  causes all initial values of  except zero to diverge to infinity if  but to converge to an attractor at the value 0 if , making the entire number line the basin of attraction for 0. And the matrix system  gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix  is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.

Nonlinear equation or system

Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example, for the function , the following initial conditions are in successive basins of attraction:

2.35287527 converges to 4;
2.35284172 converges to −3;
2.35283735 converges to 4;
2.352836327 converges to −3;
2.352836323 converges to 1.

Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.

Partial differential equations
Parabolic partial differential equations may have finite-dimensional attractors.  The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global attractors of finite dimension.

For the three-dimensional, incompressible Navier–Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimensions.

See also

 Cycle detection
 Hyperbolic set
 Stable manifold
 Steady state
 Wada basin
 Hidden oscillation
 Rössler attractor
 Stable distribution
 Convergent evolution

References

Further reading

 Edward N. Lorenz (1996) The Essence of Chaos
 James Gleick (1988) Chaos: Making a New Science

External links
 Basin of attraction on Scholarpedia
 A gallery of trigonometric strange attractors
 Double scroll attractor Chua's circuit simulation
 A gallery of polynomial strange attractors
 Chaoscope, a 3D Strange Attractor rendering freeware
 Research abstract and software laboratory
 Online strange attractors generator
 Interactive trigonometric attractors generator
 Economic attractor

Limit sets

Attractor


The North Sea is a sea that is part of the Atlantic Ocean in northern Europe. The North Sea is between Norway and Denmark in the east, Scotland and England in the west, Germany, the Netherlands, Belgium and France in the south.

Borders
The Skagerrak connects the North Sea to the Baltic Sea. In the south, the North Sea becomes the English Channel, a sea between England and France. This is called the Dover Straits and is very busy with ships.

The border between the North Sea and the Skagerrak is at an imagined line between Lindesnes in Norway, and Hanstholm in Denmark. In the North, the North sea is open towards the Atlantic. The border between the two is an imagined line from Northern Scotland, to Shetland, and then to Ålesund in Norway. According to the Oslo-Paris Treaty of 1962 it is a bit more to the west and the north though. The treaty puts it at 5° East longitude, and 62° North latitude. That is at the parallel of the Geirangerfjord in Norway.

Other facts
On average, the North Sea has a depth of only 94 meters. About 80 million people live near the North Sea, at most 150 km away from the coast. Together with the English Channel in the south, the southern North Sea is the busiest body of water in the world.

Rivers that drain into it
Well-known rivers that drain into the North Sea include the Tay (at Dundee), the Forth (at Edinburgh), the Tyne (South Shields), the Wear (at Sunderland),  the Tees (near Middlesbrough),  the Elbe (at Cuxhaven), the Weser (at Bremerhaven), the Rhine and Meuse or Maas (at Rotterdam), the Scheldt (at Flushing or Vlissingen), the Thames, and the Humber (at Hull), and the river Nairn (at Nairn)

The Kiel Canal, one of the world's busiest artificial waterways, connects the North Sea with the Baltic.

Name
Its name comes from its relationship to the land of the Frisians (see Frisia). They live directly to the south of the North Sea, and to the west of the East Sea (Oostzee, the Baltic Sea), the former South Sea (Zuiderzee, today's IJsselmeer) and the today reclaimed Middle Sea (Middelzee). But the spread of the name could also be from the view of the cities of the Hanseatic League. Some of its main cities, like Lübeck, Bremen or Hamburg had the same view.

In classical times this body of water was also called the Oceanum Germanicum or Mare Germanicum, meaning German Ocean or Sea. This name was commonly used in English and other languages along with the name North Sea, until the early eighteenth century. By the late nineteenth century, German Sea was a rare, scholarly usage even in Germany. In Danish the North Sea is also named Vesterhavet (besides Nordsøen), meaning Western Ocean because it is west of Denmark.

Geographic divisions

Most of the North sea is on the European Continental shelf. On average, the depth is about 93 to 94 meters only. In the south it is very shallow, only 25 to 35 meters. In the north in the bathyal zone north of Shetland, this depth increases to between 100 and 200 metres. In the south, the depth is at most 50 metres. An exception to this is the Norwegian Trench. It is deepest there, with a depth of 725 metres. The most shallow part of it is a sand bank called Dogger Bank. In the southern part, there are many sand banks.

Looking at the satellite picture it is easy to see the geographic divisions of the North Sea:
a generally shallow southern North Sea
the central North Sea
 the northern North Sea, with the Norwegian Trench, near the Skagerrak.

The southern north sea is composed of the Southern Bight, before the coast of Belgium and the Netherlands and the German Bight before the coastline of Germany. The Dogger Bank is the limit between the southern and central parts. The Waddenzee runs all the way from Den Helder in the Netherlands to Esbjerg in Denmark.

The Dogger Bank covers an area about half the size of the Netherlands. There, the North Sea has a depth of between 13 and 20 metres only. The area is very famous for fishing. With some storms there are even waves breaking there.

The Norwegian Trench has an average depth of around 250 to 300 metres; at the entrance to the Skagerrak, the depth increases up to 725 meters. Along the trench is the Norwegian Current, which brings most of the waters of the North Sea into the Atlantic Ocean. Also, most of the waters of the Baltic Sea flow northwards here.

About 200 km east of the Scottish city of Dundee there are more trenches, known collectively as the Devil's hole. Generally, the water is about 90 meters deep there. The trenches very often are only a few kilometers in length. In these trenches, the depth increases to up to 230 meters.

In the Dover Strait the water is about 30 meters deep. At the end of the English Channel, this depth increases to about 100 meters.

History
In the last ice age the North Sea was covered by large areas of ice called glaciers. About 20,000 years ago the ice melted and the North Sea was formed (made).

North Sea oil
In the 1960s, geologists found large areas of oil and natural gas under the North Sea. Most of the oil fields are owned by the United Kingdom and Norway but some belong to Denmark, the Netherlands and Germany. Drilling began in the 1960s and led to a famous argument between the UK Government and Scottish Nationalists how the revenue (money) from the oil should be spent.

Animal life

People have been fishing in the North Sea for thousands of years. However, so many fish are now caught there that new ones may not be able to grow fast enough to keep the fishery going.

Terns, Atlantic puffins, razorbills, kittiwakes and other seabirds live on the North Sea coast. Many coastal areas are protected nature reserves.

Other websites

North Sea

North Sea