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72-408: Traditionally, XML APIs are either: Both have advantages: DOM , for example, allows for random access to the document, and event driven algorithm like SAX has a small memory footprint and is typically much faster. These two access metaphors can be thought of as polar opposites. A tree based API allows unlimited, random access and manipulation, while an event based API is a 'one shot' pass through

144-438: A n ) n = − ∞ ∞ {\textstyle {(a_{n})}_{n=-\infty }^{\infty }} is a bi-infinite sequence , and can also be written as ( … , a − 1 , a 0 , a 1 , a 2 , … ) {\textstyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )} . In cases where

216-464: A n ) . {\textstyle (a_{n}).} Here A is the domain, or index set, of the sequence. Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of nets . A net is a function from a (possibly uncountable ) directed set to a topological space. The notational conventions for sequences normally apply to nets as well. The length of

288-413: A sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set , it contains members (also called elements , or terms ). The number of elements (possibly infinite ) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally,

360-420: A DOM structure from an XML or JSON data, using JavaScript methods to parse the data and create the nodes accordingly. Creating a DOM structure does not necessarily mean that it will be displayed in the web page, it only exists in memory and should be appended to the document body or a specific container to be rendered. In summary, creating a DOM structure involves creating individual nodes and organizing them in

432-456: A DOM structure is using the innerHTML property to insert HTML code as a string, creating the elements and children in the process. For example: Another method is to use a JavaScript library or framework such as jQuery , AngularJS , React , Vue.js , etc. These libraries provide a more convenient, eloquent and efficient way to create, manipulate and interact with the DOM. It is also possible to create

504-663: A distance from L {\displaystyle L} less than d {\displaystyle d} . For example, the sequence a n = n + 1 2 n 2 {\textstyle a_{n}={\frac {n+1}{2n^{2}}}} shown to the right converges to the value 0. On the other hand, the sequences b n = n 3 {\textstyle b_{n}=n^{3}} (which begins 1, 8, 27, ...) and c n = ( − 1 ) n {\displaystyle c_{n}=(-1)^{n}} (which begins −1, 1, −1, 1, ...) are both divergent. If

576-499: A document. Nodes can have event handlers (also known as event listeners) attached to them. Once an event is triggered, the event handlers get executed. The principal standardization of the DOM was handled by the World Wide Web Consortium (W3C), which last developed a recommendation in 2004. WHATWG took over the development of the standard, publishing it as a living document . The W3C now publishes stable snapshots of

648-599: A function of n . Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions. Not all sequences can be specified by a recurrence relation. An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, ...). There are many different notions of sequences in mathematics, some of which ( e.g. , exact sequence ) are not covered by

720-451: A hierarchical structure using JavaScript or other programming languages, and it can be done using several methods depending on the use case and the developer's preference. Because the DOM supports navigation in any direction (e.g., parent and previous sibling) and allows for arbitrary modifications, implementations typically buffer the document. However, a DOM need not originate in a serialized document at all, but can be created in place with

792-416: A limit if the elements of the sequence become closer and closer to some value L {\displaystyle L} (called the limit of the sequence), and they become and remain arbitrarily close to L {\displaystyle L} , meaning that given a real number d {\displaystyle d} greater than zero, all but a finite number of the elements of the sequence have

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864-521: A loaded HTML document. DHTML required extensions to the rudimentary document object that was available in the Legacy DOM implementations. Although the Legacy DOM implementations were largely compatible since JScript was based on JavaScript, the DHTML DOM extensions were developed in parallel by each browser maker and remained incompatible. These versions of the DOM became known as the "Intermediate DOM". After

936-483: A natural number N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} we have If ( a n ) {\displaystyle (a_{n})} is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that | ⋅ | {\displaystyle |\cdot |} denotes

1008-405: A particular browser, such as Internet Explorer. Others, including Blink , WebKit , and Gecko , are shared by a number of browsers, such as Google Chrome , Opera , Safari , and Firefox . The different layout engines implement the DOM standards to varying degrees of compliance. DOM implementations: APIs that expose DOM implementations: Inspection tools: Sequence In mathematics ,

1080-415: A recurrence relation is Recamán's sequence , defined by the recurrence relation with initial term a 0 = 0. {\displaystyle a_{0}=0.} A linear recurrence with constant coefficients is a recurrence relation of the form where c 0 , … , c k {\displaystyle c_{0},\dots ,c_{k}} are constants . There

1152-400: A sequence are discussed after the examples. The prime numbers are the natural numbers greater than 1 that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics , particularly in number theory where many results related to them exist. The Fibonacci numbers comprise

1224-403: A sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family , defined as a function from an arbitrary index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also,

1296-440: A sequence converges, then the value it converges to is unique. This value is called the limit of the sequence. The limit of a convergent sequence ( a n ) {\displaystyle (a_{n})} is normally denoted lim n → ∞ a n {\textstyle \lim _{n\to \infty }a_{n}} . If ( a n ) {\displaystyle (a_{n})}

1368-404: A sequence is defined as the number of terms in the sequence. A sequence of a finite length n is also called an n -tuple . Finite sequences include the empty sequence  ( ) that has no elements. Normally, the term infinite sequence refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence

1440-463: A sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting

1512-409: A sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways. Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be

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1584-450: A sequence of sequences: ( ( a m , n ) n ∈ N ) m ∈ N {\textstyle ((a_{m,n})_{n\in \mathbb {N} })_{m\in \mathbb {N} }} denotes a sequence whose m th term is the sequence ( a m , n ) n ∈ N {\textstyle (a_{m,n})_{n\in \mathbb {N} }} . An alternative to writing

1656-565: Is monotonically decreasing if each consecutive term is less than or equal to the previous one, and is strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function . The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing , respectively. If

1728-426: Is a cross-platform and language-independent interface that treats an HTML or XML document as a tree structure wherein each node is an object representing a part of the document. The DOM represents a document with a logical tree. Each branch of the tree ends in a node, and each node contains objects. DOM methods allow programmatic access to the tree; with them one can change the structure, style or content of

1800-474: Is a divergent sequence, then the expression lim n → ∞ a n {\textstyle \lim _{n\to \infty }a_{n}} is meaningless. A sequence of real numbers ( a n ) {\displaystyle (a_{n})} converges to a real number L {\displaystyle L} if, for all ε > 0 {\displaystyle \varepsilon >0} , there exists

1872-463: Is a general method for expressing the general term a n {\displaystyle a_{n}} of such a sequence as a function of n ; see Linear recurrence . In the case of the Fibonacci sequence, one has c 0 = 0 , c 1 = c 2 = 1 , {\displaystyle c_{0}=0,c_{1}=c_{2}=1,} and the resulting function of n

1944-411: Is a simple classical example, defined by the recurrence relation with initial terms a 0 = 0 {\displaystyle a_{0}=0} and a 1 = 1 {\displaystyle a_{1}=1} . From this, a simple computation shows that the first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. A complicated example of a sequence defined by

2016-401: Is a strictly increasing sequence of positive integers. Some other types of sequences that are easy to define include: An important property of a sequence is convergence . If a sequence converges, it converges to a particular value known as the limit . If a sequence converges to some limit, then it is convergent . A sequence that does not converge is divergent . Informally, a sequence has

2088-464: Is bi-infinite. This sequence could be denoted ( 2 n ) n = − ∞ ∞ {\textstyle {(2n)}_{n=-\infty }^{\infty }} . A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For example, the sequence ( a n ) n = 1 ∞ {\textstyle {(a_{n})}_{n=1}^{\infty }}

2160-409: Is called a lower bound . If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded . A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of

2232-465: Is called a singly infinite sequence or a one-sided infinite sequence when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence , two-way infinite sequence , or doubly infinite sequence . A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ),

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2304-418: Is called an index , and the set of values that it can take is called the index set . It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} , which denotes a sequence whose n th element

2376-416: Is easily discernible by inspection. Other examples are sequences of functions , whose elements are functions instead of numbers. The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences. Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of π . One such notation

2448-406: Is given by Binet's formula . A holonomic sequence is a sequence defined by a recurrence relation of the form where c 1 , … , c k {\displaystyle c_{1},\dots ,c_{k}} are polynomials in n . For most holonomic sequences, there is no explicit formula for expressing a n {\displaystyle a_{n}} as

2520-503: Is given by the variable a n {\displaystyle a_{n}} . For example: One can consider multiple sequences at the same time by using different variables; e.g. ( b n ) n ∈ N {\textstyle (b_{n})_{n\in \mathbb {N} }} could be a different sequence than ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} . One can even consider

2592-431: Is in contrast to the definition of sequences of elements as functions of their positions. To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation. The Fibonacci sequence

2664-404: Is monotonically increasing if and only if a n + 1 ≥ a n {\textstyle a_{n+1}\geq a_{n}} for all n ∈ N . {\displaystyle n\in \mathbf {N} .} If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing . A sequence

2736-502: Is rendered in browsers, the browser downloads the HTML into local memory and automatically parses it to display the page on screen. However, the DOM does not necessarily need to be represented as a tree, and some browsers have used other internal models. When a web page is loaded, the browser creates a Document Object Model of the page, which is an object oriented representation of an HTML document that acts as an interface between JavaScript and

2808-500: Is replaced by the expression dist ⁡ ( a n , L ) {\displaystyle \operatorname {dist} (a_{n},L)} , which denotes the distance between a n {\displaystyle a_{n}} and L {\displaystyle L} . If ( a n ) {\displaystyle (a_{n})} and ( b n ) {\displaystyle (b_{n})} are convergent sequences, then

2880-568: Is to write down a general formula for computing the n th term as a function of n , enclose it in parentheses, and include a subscript indicating the set of values that n can take. For example, in this notation the sequence of even numbers could be written as ( 2 n ) n ∈ N {\textstyle (2n)_{n\in \mathbb {N} }} . The sequence of squares could be written as ( n 2 ) n ∈ N {\textstyle (n^{2})_{n\in \mathbb {N} }} . The variable n

2952-459: The Fibonacci sequence F {\displaystyle F} is generally denoted as F n {\displaystyle F_{n}} . In computing and computer science , finite sequences are usually called strings , words or lists , with the specific technical term chosen depending on the type of object the sequence enumerates and the different ways to represent

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3024-668: The WHATWG , which continues to maintain a living standard. In 2009, the Web Applications group reorganized DOM activities at the W3C. In 2013, due to a lack of progress and the impending release of HTML5 , the DOM Level 4 specification was reassigned to the HTML Working Group to expedite its completion. Meanwhile, in 2015, the Web Applications group was disbanded and DOM stewardship passed to

3096-441: The codomain of the sequence is fixed by context, for example by requiring it to be the set R of real numbers, the set C of complex numbers, or a topological space . Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, that is, a n rather than a ( n ) . There are terminological differences as well:

3168-427: The convergence properties of sequences. In particular, sequences are the basis for series , which are important in differential equations and analysis . Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of prime numbers . There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify

3240-411: The limit of a sequence of rational numbers (e.g. via its decimal expansion , also see completeness of the real numbers ). As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of π , that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that

3312-420: The natural numbers . In the second and third bullets, there is a well-defined sequence ( a k ) k = 1 ∞ {\textstyle {(a_{k})}_{k=1}^{\infty }} , but it is not the same as the sequence denoted by the expression. Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion . This

3384-452: The DOM API. And even before the idea of the DOM originated, there were implementations of equivalent structure with persistent disk representation and rapid access, for example DynaText 's model disclosed in and various database approaches. Web browsers rely on layout engines to parse HTML into a DOM. Some layout engines, such as Trident/MSHTML , are associated primarily or exclusively with

3456-417: The DOM tree as: The DOM tree can be manipulated using JavaScript or other programming languages. Common tasks include navigating the tree, adding, removing, and modifying nodes, and getting and setting the properties of nodes. The DOM API provides a set of methods and properties to perform these operations, such as getElementById , createElement , appendChild , and innerHTML . Another way to create

3528-423: The DOM tree. Text nodes do not have attributes or child nodes, and are always leaf nodes in the tree. For example, the text content "My Website" in the title element and "Welcome" in the h1 element in the above example are both represented as text nodes. Attributes of an element are represented as properties of the element node in the DOM tree. For example, an element with the following HTML: will be represented in

3600-602: The WHATWG standard. In HTML DOM (Document Object Model), every element is a node: The history of the Document Object Model is intertwined with the history of the " browser wars " of the late 1990s between Netscape Navigator and Microsoft Internet Explorer , as well as with that of JavaScript and JScript , the first scripting languages to be widely implemented in the JavaScript engines of web browsers . JavaScript

3672-520: The Web Platform group. Beginning with the publication of DOM Level 4 in 2015, the W3C creates new recommendations based on snapshots of the WHATWG standard. To render a document such as a HTML page, most web browsers use an internal model similar to the DOM. The nodes of every document are organized in a tree structure , called the DOM tree , with the topmost node named as "Document object". When an HTML page

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3744-441: The complex modulus, i.e. | z | = z ∗ z {\displaystyle |z|={\sqrt {z^{*}z}}} . If ( a n ) {\displaystyle (a_{n})} is a sequence of points in a metric space , then the formula can be used to define convergence, if the expression | a n − L | {\displaystyle |a_{n}-L|}

3816-417: The context or a specific convention. In mathematical analysis , a sequence is often denoted by letters in the form of a n {\displaystyle a_{n}} , b n {\displaystyle b_{n}} and c n {\displaystyle c_{n}} , where the subscript n refers to the n th element of the sequence; for example, the n th element of

3888-432: The definitions and notations introduced below. In this article, a sequence is formally defined as a function whose domain is an interval of integers . This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring

3960-421: The document itself. This allows the creation of dynamic web pages , because within a page JavaScript can: A Document Object Model (DOM) tree is a hierarchical representation of an HTML or XML document. It consists of a root node, which is the document itself, and a series of child nodes that represent the elements, attributes, and text content of the document. Each node in the tree has a parent node, except for

4032-570: The document. StAX has its roots in a number of incompatible pull APIs for XML, most notably XMLPULL , the authors of which (Stefan Haustein and Aleksander Slominski) collaborated with, amongst others, BEA Systems , Oracle , Sun and James Clark . From JSR-173 Specification• Final, V1.0 (used under fair use). Quote: Competing and complementary ways to process XML in Java (the order is loosely based on initial date of introduction): Document Object Model The Document Object Model ( DOM )

4104-697: The domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation ( k 2 ) ) k = 1 10 {\textstyle (k^{2}){\vphantom {)}}_{k=1}^{10}} denotes the ten-term sequence of squares ( 1 , 4 , 9 , … , 100 ) {\displaystyle (1,4,9,\ldots ,100)} . The limits ∞ {\displaystyle \infty } and − ∞ {\displaystyle -\infty } are allowed, but they do not represent valid values for

4176-433: The domain of a sequence to be the set of natural numbers . This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. In some contexts, to shorten exposition,

4248-458: The first generation of these languages eventually became known as "DOM Level 0" or "Legacy DOM." No independent standard was developed for DOM Level 0, but it was partly described in the specifications for HTML 4 . Legacy DOM was limited in the kinds of elements that could be accessed. Form , link and image elements could be referenced with a hierarchical name that began with the root document object. A hierarchical name could make use of either

4320-474: The index, only the supremum or infimum of such values, respectively. For example, the sequence ( a n ) n = 1 ∞ {\textstyle {(a_{n})}_{n=1}^{\infty }} is the same as the sequence ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} , and does not contain an additional term "at infinity". The sequence (

4392-434: The integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). Other examples of sequences include those made up of rational numbers , real numbers and complex numbers . The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as

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4464-495: The names or the sequential index of the traversed elements. For example, a form input element could be accessed as either document.myForm.myInput or document.forms[0].elements[0] . The Legacy DOM enabled client-side form validation and simple interface interactivity like creating tooltips . In 1997, Netscape and Microsoft released version 4.0 of Netscape Navigator and Internet Explorer respectively, adding support for Dynamic HTML (DHTML) functionality enabling changes to

4536-628: The positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved. Formally, a subsequence of the sequence ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} is any sequence of the form ( a n k ) k ∈ N {\textstyle (a_{n_{k}})_{k\in \mathbb {N} }} , where ( n k ) k ∈ N {\displaystyle (n_{k})_{k\in \mathbb {N} }}

4608-408: The root node, and can have multiple child nodes. Elements in an HTML or XML document are represented as nodes in the DOM tree. Each element node has a tag name and attributes, and can contain other element nodes or text nodes as children. For example, an HTML document with the following structure: will be represented in the DOM tree as: Text content within an element is represented as a text node in

4680-427: The sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite , as in these examples, or infinite , such as the sequence of all even positive integers (2, 4, 6, ...). The position of an element in a sequence is its rank or index ; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on

4752-415: The sequence in computer memory . Infinite sequences are called streams . The empty sequence ( ) is included in most notions of sequence. It may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions , spaces , and other mathematical structures using

4824-426: The sequence of real numbers ( a n ) is such that all the terms are less than some real number M , then the sequence is said to be bounded from above . In other words, this means that there exists M such that for all n , a n ≤ M . Any such M is called an upper bound . Likewise, if, for some real m , a n ≥ m for all n greater than some N , then the sequence is bounded from below and any such m

4896-457: The set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes ( a k ) {\textstyle (a_{k})} for an arbitrary sequence. Often, the index k is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in In some cases, the elements of the sequence are related naturally to

4968-538: The source document. StAX was designed as a median between these two opposites. In the StAX metaphor, the programmatic entry point is a cursor that represents a point within the document. The application moves the cursor forward - 'pulling' the information from the parser as it needs. This is different from an event based API - such as SAX - which 'pushes' data to the application - requiring the application to maintain state between events as necessary to keep track of location within

5040-636: The standardization of ECMAScript , the W3C DOM Working Group began drafting a standard DOM specification. The completed specification, known as "DOM Level 1", became a W3C Recommendation in late 1998. By 2005, large parts of W3C DOM were well-supported by common ECMAScript-enabled browsers, including Internet Explorer 6 (from 2001), Opera , Safari and Gecko -based browsers (like Mozilla , Firefox , SeaMonkey and Camino ). The W3C DOM Working Group published its final recommendation and subsequently disbanded in 2004. Development efforts migrated to

5112-543: The value of a sequence at the lowest input (often 1) is called the "first element" of the sequence, the value at the second smallest input (often 2) is called the "second element", etc. Also, while a function abstracted from its input is usually denoted by a single letter, e.g. f , a sequence abstracted from its input is usually written by a notation such as ( a n ) n ∈ A {\textstyle (a_{n})_{n\in A}} , or just as (

5184-474: Was released by Netscape Communications in 1995 within Netscape Navigator 2.0. Netscape's competitor, Microsoft , released Internet Explorer 3.0 the following year with a reimplementation of JavaScript called JScript. JavaScript and JScript let web developers create web pages with client-side interactivity. The limited facilities for detecting user-generated events and modifying the HTML document in

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