izzako/sundanese-pixelgpt huggingface dataset:
Kiwari, biotéhnologi, atawa biotéhnologi modérn, mindeng dipatalikeun jeung dimangpaatkeunana mikroorganisme nu sacara génétik geus dirobah, misalna "E. coli" atawa "kapang" pikeun ngahasilkeun insulin atawa antibiotik
Ogé nujul ka sato atawa tutuwuhan transgenik, misalna sél endog Hamster Cina nu dirobah pikeun ngahasilkeun rupa-rupa ubar
Aya sawatara istilah jargon pikeun sub-widang biotéhnologi
Biotéhnologi beureum nyaéta biotéhnologi nu dilarapkeun kana prosés médis
Contona nyaéta organisme nu dirancang pikeun ngahasilkeun antibiotik, atawa ngarékayasa génétik pikeun nambaan kasakit ku cara manipulasi génomik
Biotéhnologi bodas, ogé katelah biotéhnologi abu, nyaéta biotéhnologi nu dilarapkeun kana prosés industri
Contona nyaéta organisme nu dirancang pikeun ngahasilkeun bahan kimia nu mangpaat
Biotéhnologi bodas umumna bisa ngirit sumberdaya dibandingkeun jeung prosés tradisional pikeun ngahasilkeun barang-barang industri
Biotéhnologi héjo nyaéta biotéhnologi nu dilarapkeun kana prosés tani
Contona nyaéta ngarancang organisme sangkan bisa tumuwuh dina kaayaan nu husus atawa dina kaayaan aya/taya bahan kimia tani nu tangtu
Biotéhnologi héjo umumna boga hasil nu leuwih soméah lingkungan batan tatanén industri
Contona nyaéta ngarancang tutuwuhan sangkan ngéksprésikeun péstisida, antukna ngahapus kabutuh péstisida ti luar
Istilah biotéhnologi bulao ogé geus dipaké pikeun ngagambarkeun larapan biotéhnologi di widang maritim, ngan masih jarang
Dina matematik, variabel random mangrupa diskrit lamun éta probability distribution ogé disckrit; sebaran probibiliti diskrit lamun satemenna dikarakterisasi ku probability mass function
Mangka "X" variabel random diskrit lamun formula_1 salaku "u" ngaliwatan susunan tina sakabéh nilai mungkin variabel random "X"
Poisson distribution, Bernoulli distribution, sebaran binomial, geometric distribution, sarta negative binomial distribution dipikanyaho salaku sebaran probabiliti diskrit
Lamun variabel random variable mangrupa diskrit mangka set sakabéh nilai mungkin bisa dianggap finite sarta countably infinite, sabab jumlah nu teu bisa diitung unggal positive real number positip (nagrupakeun wates handap pangleutikna tina sakabéh susunan jumlah parsial) salawasna kasebar nuju takhingga
Dumasar kasapukan, variabel acak "X" disebut kontinyu mun fungsi sebaran kumulatifna kontinyu
That is equivalent to saying that Pr["X" = "a"] = 0 for all real numbers "a", i.e
: the probability that "X" attains the value "a" is zero, for any number "a"
While for a discrete random variable one could say that an event with probability zero is impossible, this can not be said in the case of a continuous random variable, because then no value would be possible
This paradox is solved by réalizing that the probability that "X" attains a value in an uncountable set (for example an interval) can not be found by adding the probabilities for individual values
By another convention, the term "continuous random variable" is reserved for random variables that have probability density functions
A random variable with the Cantor distribution is continuous according to the first convention, and according to the second, is neither continuous nor discrete nor a weighted average of continuous and discrete random variables
In practical applications random variables are often either discrete or continuous
Fungsi Héaviside step, ngaran nu dipaké keur ngahargaan ka Oliver Heaviside, nyaéta a fungsi diskontinyu nu mana nileyna nyaéta nol keur asupan negatip sarta hiji keur nu séjénna: formula_1 The function is used in the mathematics of signal processing to represent a signal that switches on at a specified time and stays switched on indefinitely
Its derivative is formally the Dirac delta function
It is the cumulative distribution function of a variabel acak which is almost surely 0
(See constant random variable.)
The Héaviside function is the integral of the Dirac delta function
The value of H(0) is of very little importance, since the function is often used within an integral
Some writers give H(0) = 0, some H(0) = 1
H(0) = 0.5 is often used, since it maximizes the symmetry of the function
This makés the definition: formula_2 This is sometimes notated by a subscript, as in H0.5(x), which méans that H(0) = 0.5
This notation is also used to represent a completely different concept, however; an x offset: formula_3 where "c" is a positive offset in the "x"-dimension of the transition from 0 to 1
In other words, "H"3("x") = "H"("x" − 3) would be zero until "x" = 3, and would transition to 1 for "x" &gt; 3
The méaning of the subscript should be given in context
The question of the Fourier transform of H is an interesting example for the théory of distributions
It is often stated that it is 1/x, up to a normalizing constant
But néar x=0 that cannot be justified: the definition must be given in terms of "principal value" limit, and the transform isn't to be tréated simply as a function
The corresponding convolution operator is the "Hilbert transform"
Often an integral representation of the step function is useful, formula_4 in the limit formula_5
Basa program nyaéta basa jieunan nu bisa dipaké keur ngadalikeun paripolah hiji mesin, utamana komputer
Basa program, saperti basa alami, dihartikeun ku aturan "sintaksis" jeung "semantik" keur ngajelaskeun struktur jeung hartina
Loba basa program nu mibanda sababaraha wanda dina nuliskeun "sintaks" jeung "semantik"; sababaraha basa program ngan dihartikeun ku sawanda dina cara maké sacara resmi
Basa program dipaké keur mantuan "komunikasi" dina raraga ngokolakeun pangaweruh tur nembongkeun "algoritma" kalawan présisi
Sababaraha pangarang méré watesan nu leuwih heureut yén "basa program" nyaéta basa nu bisa dipaké keur ngagambarkeun "sakabeh" "algoritma" nu mungkin; kadangkala watesan "basa komputer" dipaké keur basa jieunan nu leuwih heureut
Rébuan basa program nu béda-beda geus dijieun, tur méh unggal taun ngahasilkeun basa program nu anyar
Sifat penting keur nangtukeun pilihan basa program nyaéta: Basa non-komputasi, saperti basa markup siga HTML atawa formal grammar siga BNF, ilaharna teu disebut basa program
Ilaharna basa program ditambahkeun kana basa non-itungan ieu
Kaperluan utama basa program nyaéta keur méré paréntah kana komputer
Saperti, basa program béda jeung éksprési jalma nu mana merlukeun présisi nu luhur tur lengkep
Waktu ngagunakan hiji basa alam keur ngobrol jeung jalma séjén, pangarang jeung nu ngobrol bisa bingung tur nyieun saeutik kasalahan, sarta duana masih bisa ngarti
béda jeung komputer nu sarua jeung naon anu diparéntahkeun ka manéhna, teu bisa ngarti kana kode nu nyieun program "nu dimaksud" keur nulis
Kombinasi harti basa, program, jeung asupan program kudu sacara husus ngartikeun paripolah luar nu bakal kajadian lamun éta program dijalankeun
Loba basa geus dirancang ti anggalna, dirobah pikeun minuhan kabutuhan anyar, digabungkeun jeung basa séjén, sarta pamustunganana ditinggalkeun
Sanaos aya usaha pikeun ngararancang hiji basa komputer "universal" anu tiasa dianggo pikeun sadaya tujuan, aranjeunna sadayana gagal ditampi dina peran ieu
Kabutuhan pikeun rupa-rupa basa komputer bijil tina rupa-rupa basa kontéks anu digunakeunana : One common trend in the development of programming languages has been to add more ability to solve problems using a higher level of abstraction
The éarliest programming languages were tied very closely to the underlying hardware of the computer
As new programming languages have developed, féatures have been added that let programmers express idéas that are more removed from simple translation into underlying hardware instructions
Because programmers are less tied to the needs of the computer, their programs can do more computing with less effort from the programmer
This lets them write more programs in the same amount of time
Natural language processors have been proposed as a way to eliminate the need for a specialized language for programming
However, this goal remains distant and its benefits are open to debate
Edsger Dijkstra took the position that the use of a formal language is essential to prevent the introduction of méaningless constructs, and dismissed natural language programming as "foolish."
Alan Perlis was similarly dismissive of the idéa
A programming language's surface form is known as its syntax
Most programming languages are purely textual; they use sequences of text including words, numbers, and punctuation, much like written natural languages
On the other hand, there are some programming languages which are more graphical in nature, using spatial relationships between symbols to specify a program
The syntax of a language describes the possible combinations of symbols that form a syntactically correct program
The méaning given to a combination of symbols is handled by semantics
Since most languages are textual, this article discusses textual syntax
Programming language syntax is usually defined using a combination of regular expressions (for lexical structure) and Backus-Naur Form (for grammatical structure)
Below is a simple grammar, based on Lisp: codice_1 This grammar specifies the following: The following are examples of well-formed token sequences in this grammar: 'codice_2', 'codice_3', 'codice_4' Not all syntactically correct programs are semantically correct
Many syntactically correct programs are nonetheless ill-formed, per the language's rules; and may (depending on the language specification and the soundness of the implementation) result in an error on translation or execution
In some cases, such programs may exhibit undefined behavior
Even when a program is well-defined within a language, it may still have a méaning that is not intended by the person who wrote it
Using natural language as an example, it may not be possible to assign a méaning to a grammatically correct sentence or the sentence may be false: The following C language fragment is syntactically correct, but performs an operation that is not semantically defined (because p is a null pointer, the operations p-&gt;réal and p-&gt;im have no méaning): complex *p = NULL; complex abs_p =
*p = NULL; complex abs_p = sqrt (p-&gt;réal * p-&gt;réal + p-&gt;im * p-&gt;im); The grammar needed to specify a programming language can be classified by its position in the Chomsky hierarchy
The syntax of most programming languages can be specified using a Type-2 grammar, i.e., they are context-free grammars
A type system defines how a programming language classifies values and expressions into "types", how it can manipulate those types and how they interact
This generally includes a description of the data structures that can be constructed in the language
The design and study of type systems using formal mathematics is known as "type theory"
Internally, all data in modérn digital computers are stored simply as zeros or ones (binary)
Typed versus untyped languages
A language is "typed" if operations defined for one data type cannot be performed on values of another data type
For example, "codice_5" is a string
In most programming languages, dividing a number by a string has no méaning
Most modérn programming languages will therefore reject any program attempting to perform such an operation
In some languages, the méaningless operation will be detected when the program is compiled ("static" type checking), and rejected by the compiler, while in others, it will be detected when the program is run ("dynamic" type checking), resulting in a runtime exception
A special case of typed languages are the "single-type" languages
These are often scripting or markup languages, such as Rexx or SGML, and have only one data type — most commonly character strings which are used for both symbolic and numeric data
In contrast, an "untyped language", such as most assembly languages, allows any operation to be performed on any data, which are generally considered to be sequences of bits of various lengths
High-level languages which are untyped include BCPL and some varieties of Forth
In practice, while few languages are considered typed from the point of view of type theory (verifying or rejecting "all" operations), most modérn languages offer a degree of typing
Many production languages provide méans to bypass or subvert the type system
Static versus dynamic typing
In "static typing" all expressions have their types determined prior to the program being run (typically at compile-time)
For example, 1 and (2+2) are integer expressions; they cannot be passed to a function that expects a string, or stored in a variable that is defined to hold dates
Statically-typed languages can be "manifestly typed" or "type-inferred"
In the first case, the programmer must explicitly write types at certain textual positions (for example, at variable declarations)
In the second case, the compiler "infers" the types of expressions and declarations based on context
Most mainstréam statically-typed languages, such as C++ and Java, are manifestly typed
Complete type inference has traditionally been associated with less mainstréam languages, such as Haskell and ML
However, many manifestly typed languages support partial type inference; for example, Java and C# both infer types in certain limited cases
"Dynamic typing", also called "latent typing", determines the type-safety of operations at runtime; in other words, types are associated with "runtime values" rather than "textual expressions"
As with type-inferred languages, dynamically typed languages do not require the programmer to write explicit type annotations on expressions
Among other things, this may permit a single variable to refer to values of different types at different points in the program execution
However, type errors cannot be automatically detected until a piece of code is actually executed, making debugging more difficult
Ruby, Lisp, JavaScript, and Python are dynamically typed
Weak and strong typing
"Weak typing" allows a value of one type to be tréated as another, for example tréating a string as a number
This can occasionally be useful, but it can also allow some kinds of program faults to go undetected at compile time
"Strong typing" prevents the above
Attempting to mix types raises an error
Strongly-typed languages are often termed "type-safe" or "safe"
Type safety can prevent particular kinds of program faults occurring (because constructs containing them are flagged at compile time)
An alternative definition for "weakly typed" refers to languages, such as Perl, JavaScript, and C++ which permit a large number of implicit type conversions; Perl in particular can be characterized as a dynamically typed programming language in which type checking can take place at runtime
See type system
This capability is often useful, but occasionally dangerous; as it would permit operations whose objects can change type on demand
"Strong" and "static" are generally considered orthogonal concepts, but usage in the literature differs
Some use the term "strongly typed" to méan "strongly, statically typed", or, even more confusingly, to méan simply "statically typed"
Thus C has been called both strongly typed and wéakly, statically typed
Once data has been specified, the machine must be instructed to perform operations on the data
The "execution semantics" of a language defines how and when the various constructs of a language should produce a program behavior
For example, the semantics may define the strategy by which expressions are evaluated to values, or the manner in which control structures conditionally execute statements
Most programming languages have an associated core library (sometimes known as the 'Standard library', especially if it is included as part of the published language standard), which is conventionally made available by all implementations of the language
Core libraries typically include definitions for commonly used algorithms, data structures, and mechanisms for input and output
A language's core library is often tréated as part of the language by its users, although the designers may have tréated it as a separate entity
Many language specifications define a core that must be made available in all implementations, and in the case of standardized languages this core library may be required
The line between a language and its core library therefore differs from language to language
Indeed, some languages are designed so that the méanings of certain syntactic constructs cannot even be described without referring to the core library
For example, in Java, a string literal is defined as an instance of the java.lang.String class; similarly, in Smalltalk, an anonymous function expression (a "block") constructs an instance of the library's BlockContext class
Conversely, Scheme contains multiple coherent subsets that suffice to construct the rest of the language as library macros, and so the language designers do not even bother to say which portions of the language must be implemented as language constructs, and which must be implemented as parts of a library
A language's designers and users must construct a number of artifacts that govern and enable the practice of programming
The most important of these artifacts are the language "specification" and "implementation"
The specification of a programming language is intended to provide a definition that language users and implementors can use to determine the behavior of a program, given its source code
A programming language specification can take several forms, including the following: Implementation
An implementation of a programming language provides a way to execute that program on one or more configurations of hardware and software
There are, broadly, two approaches to programming language implementation: "compilation" and "interpretation"
It is generally possible to implement a language using either technique
The output of a compiler may be executed by hardware or a program called an interpreter
In some implementations that maké use of the interpreter approach there is no distinct boundary between compiling and interpreting
For instance, some implementations of the BASIC programming language compile and then execute the source a line at a time
Programs that are executed directly on the hardware usually run several orders of magnitude faster than those that are interpreted in software
One technique for improving the performance of interpreted programs is just-in-time compilation
Here the virtual machine monitors which sequences of bytecode are frequently executed and translates them to machine code for direct execution on the hardware
The first programming languages predate the modérn computer
The 19th century had "programmable" looms and player piano scrolls which implemented what are today recognized as examples of domain-specific programming languages
By the beginning of the twentieth century, punch cards encoded data and directed mechanical processing
In the 1930s and 1940s, the formalisms of Alonzo Church's lambda calculus and Alan Turing's Turing machines provided mathematical abstractions for expressing algorithms; the lambda calculus remains influential in language design
In the 1940s, the first electrically powered digital computers were créated
The computers of the éarly 1950s, notably the UNIVAC I and the IBM 701 used machine language programs
First generation machine language programming was quickly superseded by a second generation of programming languages known as Assembly languages
Later in the 1950s, assembly language programming, which had evolved to include the use of macro instructions, was followed by the development of three higher-level programming languages: FORTRAN, LISP, and COBOL
Updated versions of all of these are still in general use, and importantly, éach has strongly influenced the development of later languages
At the end of the 1950s, the language formalized as Algol 60 was introduced, and most later programming languages are, in many respects, descendants of Algol
The format and use of the éarly programming languages was héavily influenced by the constraints of the interface
The period from the 1960s to the late 1970s brought the development of the major language paradigms now in use, though many aspects were refinements of idéas in the very first Third-generation programming languages: éach of these languages spawned an entire family of descendants, and most modérn languages count at léast one of them in their ancestry
The 1960s and 1970s also saw considerable debate over the merits of "structured programming", and whether programming languages should be designed to support it
Edsger Dijkstra, in a famous 1968 letter published in the Communications of the ACM, argued that GOTO statements should be eliminated from all "higher level" programming languages
The 1960s and 1970s also saw expansion of techniques that reduced the footprint of a program as well as improved productivity of the programmer and user
The card deck for an éarly 4GL was a lot smaller for the same functionality expressed in a 3GL deck
Consolidation and growth
The 1980s were yéars of relative consolidation
C++ combined object-oriented and systems programming
The United States government standardized Ada, a systems programming language intended for use by defense contractors
In Japan and elsewhere, vast sums were spent investigating so-called "fifth generation" languages that incorporated logic programming constructs
The functional languages community moved to standardize ML and Lisp
Rather than inventing new paradigms, all of these movements elaborated upon the idéas invented in the previous decade
One important trend in language design during the 1980s was an incréased focus on programming for large-scale systems through the use of "modules", or large-scale organizational units of code
[[Modula-2]], Ada, and ML all developed notable module systems in the 1980s, although other languages, such as [[PL/I]], alréady had extensive support for modular programming
Module systems were often wedded to [[generic programming]] constructs
The rapid growth of the [[Internet]] in the mid-1990's créated opportunities for new languages
[[Perl]], originally a Unix scripting tool first reléased in 1987, became common in dynamic [[Web site]]s. [[Java (programming language)|Java]] came to be used for server-side programming
These developments were not fundamentally novel, rather they were refinements to existing languages and paradigms, and largely based on the C family of programming languages
Programming language evolution continues, in both industry and reséarch
Current directions include security and reliability verification, new kinds of modularity ([[mixin]]s, [[Delegation (programming)|delegates]], [[aspect-oriented programming|aspects]]), and database integration
The [[4GL]]s are examples of languages which are domain-specific, such as [[SQL]], which manipulates and returns [[set]]s of data rather than the scalar values which are canonical to most programming languages
[[Perl]], for example, with its '[[here document]]' can hold multiple 4GL programs, as well as multiple JavaScript programs, in part of its own perl code and use variable interpolation in the 'here document' to support multi-language programming
Measuring language usage
It is difficult to determine which programming languages are most widely used, and what usage méans varies by context
One language may occupy the gréater number of programmer hours, a different one have more lines of code, and a third utilize the most CPU time
Some languages are very popular for particular kinds of applications
For example, [[COBOL]] is still strong in the corporate data center, often on large [[Mainframe computer|mainframes]]; [[Fortran (programming language)|FORTRAN]] in engineering applications; [[C (programming language)|C]] in embedded applications and operating systems; and other languages are regularly used to write many different kinds of applications
Various methods of méasuring language popularity, éach subject to a different bias over what is méasured, have been proposed: Taxonomies
There is no overarching classification scheme for programming languages
A given programming language does not usually have a single ancestor language
Languages commonly arise by combining the elements of several predecessor languages with new idéas in circulation at the time
Idéas that originate in one language will diffuse throughout a family of related languages, and then léap suddenly across familial gaps to appéar in an entirely different family
The task is further complicated by the fact that languages can be classified along multiple axes
For example, Java is both an object-oriented language (because it encourages object-oriented organization) and a concurrent language (because it contains built-in constructs for running multiple [[Thread (computer science)|threads]] in parallel)
[[Python (programming language)|Python]] is an object-oriented [[scripting language]]
In broad strokes, programming languages divide into "[[programming paradigm]]s" and a classification by "intended domain of use"
Paradigms include [[procedural programming]], [[object-oriented programming]], [[functional programming]], and [[logic programming]]; some languages are hybrids of paradigms or multi-paradigmatic
An [[assembly language]] is not so much a paradigm as a direct modél of an underlying machine architecture
By purpose, programming languages might be considered general purpose, system programming languages, scripting languages, domain-specific languages, or concurrent/distributed languages (or a combination of these)
Some general purpose languages were designed largely with educational goals
A programming language may also be classified by factors unrelated to programming paradigm
For instance, most programming languages use [[English language]] keywords, while a [[Categorical list of programming languages#Non-English-based languages|minority do not]]
Other languages may be classified as being [[Esoteric programming language|esoteric]] or not
[[Kategori:Jejer basa pamrograman| ]] [[Kategori:Basa komputer]]
Dina statistik, nilai-p tina variabel random T nyaéta probabilitas Pr(T ≤ tobserved) nu mana T bakal dianggap leuwih gedé atawa sarua jeung nilai observasi tobserved, dina kayaan null hypothesis dianggap bener
Dina basa séjén, anggapan yén null hypothesis sederhana ditolak lamun tes statistic "T" leuwih gedé tinimbang nilai kritis "c"
Kira-kira dina sabagéan kasus T nu di-observasi sarua jeung tobserved
Mangka nilai-p tina T dina éta kasus probabiliti yén T bakal sarua atawa leuwih ti tobserved
The inverse transform sampling method is a method of sampling a number at random from any probability distribution, given its cumulative distribution function (cdf)
The problem that the inverse transform sampling method solves is as follows: Many programming languages have the ability to generate pseudo-randomnumbers which are effectively distributed according to the standard sebaran seragam
If a random variable has that distribution, then the probability of its falling within any subinterval ("a", "b") of the interval from 0 to 1 is just the length "b" - "a" of that subinterval
The inverse transform sampling method works as follows: The following diagram may help the réader to visualise how the method works: Sampling using the inverse transform method Tempo ogé
The rejection sampling method
Dina aljabar dasar, interval (atawa heuleut) hartina hiji kumpulan nu ngandung sakabéh wilangan nyata antara dua wilangan nu disebutkeun, katut, bisa jadi, dua wilangan éta
Notasi interval nyaéta éksprési variabel nu kawengku dina hiji interval; misalna "5 &lt; x &lt; 9"
Dumasar kasapukan konvénsi, interval "(10,20)" nunjukkeun sakabéh angka nyata antara 10 jeung 20, teu kaasup 10 atawa 20
Di sisi séjén, interval "[10,20]" ngawengku sakabéh awilangan nyata antara 10 jeung 20 "katut" angka 10 jeung 20
Kamungkinan lianna dibéréndélkeun di handap
Dina matematika tingkat luhur, harti formal interval téh nyaéta hiji subsét "S" tina hiji totally ordered set "T" nu ngandung sarat "x" jeung "y" kaasup "S", "x" &lt; "z" &lt; "y", lajeng "z" anggota "S"
Sakumaha disebutkeun di luhur, jadi penting nalika "T" = R, sét wilangan nyata
Interval R kabagi kana sababaraha tipe ("a" jeung "b" wilangan nyata, "a" &lt; "b"): In éach case where they appéar above, "a" and "b" are known as endpoints of the interval
Note that a square bracket [ or ] indicates that the endpoint is included in the interval, while a round bracket ( or ) indicates that it is not
For more information about the notation used above, see Naive set theory
Intervals of type (1), (5), (7), (9) and (11) are called open intervals (because they are open sets) and intervals (2), (6), (8), (9), (10) and (11) closed intervals (because they are closed sets)
Intervals (3) and (4) are sometimes called half-closed (or, not surprisingly, half-open) intervals
Notice that intervals (9) and (11) are both open "and" closed, which is not the same thing as being half-open and half-closed
Intervals (1), (2), (3), (4), (10) and (11) are called bounded intervals and intervals (5), (6), (7), (8) and (9) unbounded intervals
Interval (10) is also known as a singleton
The length of the bounded intervals (1), (2), (3), (4) is "b"-"a" in éach case
The total length of a sequence of intervals is the sum of the lengths of the intervals
No allowance is made for the intersection of the intervals
For instance, the total length of the sequence {(1,2),(1.5,2.5)} is 1+1=2, despite the fact that the union of the sequence is an interval of length 1.5
Intervals play an important role in the théory of integration, because they are the simplest sets whose "size" or "measure" or "length" is éasy to define (see above)
The concept of méasure can then be extended to more complicated sets, léading to the Borel measure and eventually to the Lebesgue measure
Intervals are precisely the connected subsets of R. They are also precisely the convex subsets of R. Since a continuous image of a connected set is connected, it follows that if "f": R→R is a continuous function and "I" is an interval, then its image "f"("I") is also an interval
This is one formulation of the intermediate value theorem
Intervals in partial orders
In order theory, one usually considers partially ordered sets
However, the above notations and definitions can immediately be applied to this general case as well
Of special interest in this general setting are intervals of the form ["a","b"]
For a partially ordered set ("P", ≤) and two elements "a" and "b" of "P", one defines the set ["a", "b"] = { "x" | "a" ≤ "x" ≤ "b" } One may choose to restrict this definition to pairs of elements with the property that "a" ≤ "b"
Alternatively, the intervals without this condition will just coincide with the empty set, which in the former case would not be considered as an interval
Interval arithmetic, also called interval mathematics, interval analysis, and interval computations, has been introduced in 1956 by M. Warmus
It defines a set of operations which can be applied on intervals : Division by an interval containing zero is not possible
The addition and multiplication operations are commutative, associative and sub-distributive: the set "X" ( "Y" + "Z" ) is a subset of "XY" + "XZ"
Cara séjén keur nuliskeun interval, ilaharna katempo di France sarta sababarha nagara Eropa séjénna, nyaéta:
"This article deals with the concept of an integral in mathematical calculus
For other meanings of "integral" see integration."
Dina kalkulus, fungsi integral mangrupa "generalisasi" area, massa, volume, sum, sarta total
Teu siga di prosés differentiation, aya sababaraha harti nu béda ngeunaan integral, gumantung kana béda téhnikna
Sanajan kitu, dua cara nu béda dina fungsi integrasi bakal méré hasil nu sarua lamun duanana digawekeun
Integral kontinyu, fungsi nilai-riil positip "f" tina variable riil "x" antara sisi kénca "a" sarta sisi katuhu "b" nembongkeun batas wewengkeun ku garis "x=a", "x=b", sumbu-"x", sarta kurva dihartikeun ku grapik "f"
Leuwih resmi, lamun anggap "S"={("x","y"):"a"≤"x"≤"b",0≤"y"≤"f(x)"}, mangka integral "f" antara "a" jeung "b" mangrupa measure "S"
Leibniz ngawanohkeun notasi baku long s keur integral
Integral dina paragrap samemegna bisa ditulis formula_1
Tanda ∫ ngalambakeun integral, "a" jeung "b" mangrupa titik tungtung interval, "f(x)" nyaéta fungsi nu di-integralkeun, sarta "dx" notasi keur variabel integrasi
Sajarahna, "dx" ngagambarkeun wilangan nu takhingga, sarta s panjang singkatan keur "jumlah"
Sanajan kitu, téori integral modérn diwangun ku dasar nu béda sarta simbol tradisional ngan sakadar notation
As an example, if "f" is the constant function "f(x)=3", then the integral of "f" between 0 and 10 is the aréa of the rectangle bounded by the lines "x="0, "x="10, "y="0, and "y=3"
The aréa is 10"c", so the value of the integral is 30
Integrals can be taken over regions other than intervals
In general, the integral over a set "E" of a function "f" is written ∫Ef(x)dx
Here "x" need not be a réal number, but, for instance, a vector in R3
Fubini's theorem shows that such integrals can be rewritten as an iterated integral
In other words, the integral can be calculated by integrating one coordinate at a time
If a function has an integral, it is said to be "integrable"
The function for which the integral is calculated is called the integrand
Integrals are sometimes called definite integrals to emphasize that they result in a number, not another function
This is to distinguish them from indefinite integrals, which are another name for an antiderivative
If the domain of the function is the real numbers, and if the region of integration is an interval, then the greatest lower bound of the interval is called the "lower limit of integration", and the least upper bound is called the "upper limit of integration"
The most basic technique for computing integrals of one réal variable is based on the Fundamental Theorem of Calculus
It proceeds like this: Note that the integral is not actually the antiderivative (it is a number), but the fundamental théorem allows us to use antiderivatives to evaluate integrals
The difficult step is finding an antiderivative of "f"
It is rarely possible to glance at a function and write down its antiderivative
More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals
Most of these techniques rewrite one integral as a different one which is hopefully more tractable
Techniques include: Even if these techniques fail, it may still be possible to evaluate the integral
The next most common technique is residue calculus
There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform the integral of a square into an infinite sum
Occasionally an integral can be evaluated by a trick; for an example of this, see Gaussian integral
Computation of volumes of solids of revolution can usually be done with disk integration or shell integration
Specific results which have been worked out by various techniques are collected in the list of integrals
Approximation of definite integrals
Definite integrals may be approximated using several methods
One popular method, called the rectangle method or the trapezoidal rule, relies on dividing the function into a series of rectangles and finding the sum
Another well-known method is Simpson's rule
Some integrals cannot be found exactly, and others are so complex that finding the exact answer would be extremely time-consuming or computationally-intensive
Approximation, however, is a process which relies only on variable substitution, multiplication, addition, and division
It can be done éasily and quickly by modérn graphing calculators and computers
Many réal-world applications of calculus rely on integral approximation because of the complexity of formulas and unnecessary nature of an exact answer
Integrals and computerized algebra systems
Many professionals, educators, and students now use computerized algebra systems to maké difficult (or simply tedious) algebra and calculus problems éasier
The design of such a computer algebra system is nontrivial as systematic methods of antidifferentiation are difficult to formulate
One difficulty is that it is not always possible to find "nice formulae" for antiderivatives
For instance, there is a (nontrivial) proof that there is no nice function (e.g., involving sin, cos, exp, polynomials, roots and so on) whose derivative is exp(-"x"2)
As such, computerized algebra systems have no hope of being able to find an antiderivative for this particular function
Unfortunately, functions that have nice antiderivatives are the exception
If one writes a large random expression involving exponentials and polynomials, the odds are almost nil that it will have an antiderivative
(This statement can be made formal, but it is difficult to do so.)
One of the difficulties is to decide what set of functions to use as building blocks for antiderivatives
Usually, we need a set of antiderivatives closed under, say, multiplication and composition
This set of antiderivatives should also include polynomials, perhaps quotients, exponentials, logarithms, sines and cosines
The Risch-Norman algorithm is able to compute any integral of such a shape; that is, if the antiderivative involves polynomials, sines, cosines, etc..., the Risch-Norman algorithm will be able to compute it
Extended versions of this algorithm are implemented in Mathematica and Maple computer algebra system
Some special integrands occur often enough to warrant special study
In particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the Hypergeometric function, fungsi gamma jeung saterusna.)
Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging
Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae
On the other hand, very complex formulae are unlikely to have closed-form antiderivatives, so this advantage is dubious
Not all integrals can be evaluated using a single limit process
An integral which can only be evaluated by considering it as the limit of integrals on successively larger and larger integrals is called an improper integral
Improper integrals usually turn up when the range of the function is infinite or, in the case of the Riemann integral, when the domain is infinite
One common example of an improper integral is the Cauchy principal value
Definitions of the integral
The most important integrals are the Riemann integral and the Lebesgue integral
The Riemann integral was créated by Bernhard Riemann and was the first rigorous definition of the integral
The Lebesgue integral was créated by Henri Lebesgue to integrate a wider class of functions and to prove very strong theorems about interchanging limits and integrals
Although the Riemann and Lebesgue integrals are the most important ones, a number of others exist, including but not limited to:
Dina matematik, "integral" mangrupa konsép umum ngeunaan daerah tina gambar nu teratur ka wewengkon nu diwatesan ku fungsi
Integrasi Lebesgue mangrupa rarangkay pagawéan keur ngalegaan integral ka fungsi kelas nu kacida gedéna
Integral Lebesgue kacida pentingna dina widang matematik nu disebut real analysis sarta sababaraha widang séjénna
Integral Lebesgue dumasar kana ngaran Henri Lebesgue (1875-1941)
Cara maca ngaranna nu gampang nyaéta "leh BEG"
Fungsi integral "f" bisa dihartikeun salaku wewengkon "S" sahandapeun grapik "f"
Ieu gampang dipikaharti keur fungsi kulawarga saperti polinomial, tapi naon hartina keur fungsi nu leuwih asing
Sacara umum, naha kelas fungsi nu mangrupa "wewengkon sahandapeun kurva" asup akal
Keur ngajawab ieu patarosan merlukan téori jeung prakték
As part of a general movement toward formalism in mathematics in the nineteenth century, attempts were made to put the integral calculus on a firm foundation
The Riemann integral, proposed by Bernhard Riemann (1826-1866), is a broadly successful attempt to provide such a foundation for the integral
Riemann's definition starts with the construction of a sequence of éasily-calculated integrals which converge to the integral of a given function
This definition is successful in the sense that it gives the expected answer for many alréady-solved problems, and gives useful results for many other problems
However, the behavior of the Riemann integral in limit processes is difficult to analyze
This is of prime importance, for instance, in the study of Fourier series, Fourier transforms and other topics
The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign
The Lebesgue definition considers a different class of éasily-calculated integrals than the Riemann definition, which is the main réason the Lebesgue integral is better behaved
The Lebesgue definition also makes it possible to calculate integrals for a broader class of functions
For example, the function which is 0 where its argument is irrational and 1 otherwise has a Lebesgue integral, but it does not have a Riemann integral
We now give a highly technical description
It is possible to skip directly to the discussion héading for further technical and historical justification of the Lebesgue integral if the réader is so inclined
Construction of the Lebesgue integral
Let μ be a (non-negative) measure on a sigma-algebra "X" over a set "E"
(In real analysis, "E" will typically be Euclidean "n"-space R"n" or some Lebesgue measurable subset of it, "X" will be the sigma-algebra of all Lebesgue méasurable subsets of "E", and μ will be the Lebesgue méasure
In probability and statistics, μ will be a probability méasure on a probability space "E".)
We build up an integral for réal-valued functions defined on "E" as follows
Fix a set "S" in "X" and let "f" be the function on "E" whose value is 0 outside of "S" and 1 inside of "S" (i.e., "f"("x") = 1 if "x" is in "S", otherwise "f"("x") = 0.)
This is called the "indicator function" or "characteristic function" of "S" and is denoted 1"S"
To assign a value to ∫1"S" consistent with the given méasure μ, the only réasonable choice is to set: formula_1 We extend by linéarity to the linear span of indicating functions: formula_2 where the sum is finite and the coefficients "a""k" are réal numbers
Such a finite linear combination of indicating functions is called a simple function
Note that a simple function can be written in many ways as a linéar combination of characteristic functions, but the integral will always be the same
Now the difficulties begin as we attempt to take limits so that we can integrate more general functions
It turns out that the following process works and is most fruitful
Let "f" be a non-negative function supported on the set "E" (we allow it to attain the value +∞, in other words, "f" takes values in the extended real number line.)
We define ∫"f" to be the supremum of ∫"s" where "s" varies over all simple functions which are under "f" (that is, "s"("x") ≤ "f"("x") for all "x".)
This is analogous to the lower sums of Riemann
However, we will not build an upper sum, and this fact is important in getting a more general class of integrable functions
One can be more explicit and mention the méasure and domain of integration: formula_3 There is the question of whether this definition makes sense (do simple function or indicating function keep the same integral?)
There is also the question of whether this corresponds in any way to a Riemann notion of integration
It is not so hard to prove that the answer to both questions is yes
We have defined ∫"f" for "any" non-negative function on "E"; however for some functions ∫"f" will be infinite
Furthermore, desirable additive and limit properties of the integral are not satisfied, unless we require that all our functions are "measurable", méaning that the pre-image of any interval is in "X"
We will maké this assumption from now on
To handle signed functions, we need a few more definitions
If "f" is a function of the méasurable set "E" to the réals (including ± ∞), then we can write "f" = "g" - "h" where "g"("x") = ("f"("x") if "f"("x")&gt;0, 0 otherwise) and "h"("x") = (-"f"("x") if "f"("x") &lt; 0, 0 otherwise)
Note that both "g" and "h" are non-negative functions
Also note that |"f"| = "g" + "h"
If ∫|"f"| is finite, then "f" is called "Lebesgue integrable"
In this case, both ∫"g" and ∫"h" are finite, and it makes sense to define ∫"f" by ∫"g" - ∫"h"
It turns out that this definition is the correct one
Complex valued functions can be similarly integrated, by considering the réal part and the imaginary part separately
Properties of the Lebesgue integral
Every réasonable notion of integral needs to be linear and monotone, and the Lebesgue integral is: if "f" and "g" are integrable functions and "a" and "b" are réal numbers, then "af" + "bg" is integrable and ∫("af" + "bg") = "a"∫"f" + "b"∫"g"; if "f" ≤ "g", then ∫"f" ≤ ∫"g"
Two functions which only differ on a set of μ-méasure zero have the same integral, or more precisely: if μ({"x" : "f"("x") ≠ "g"("x")}) = 0, then "f" is integrable if and only if "g" is, and in this case ∫ "f" = ∫ "g"
One of the most important advantages that the Lebesgue integral carries over the Riemann integral is the éase with which we can perform limit processes
Three théorems are key here
The monotone convergence theorem states that if "f""k" is a sequence of non-negative méasurable functions such that "f""k"("x") ≤ "f""k"+1("x") for all "k", and if "f" = lim "f""k", then ∫"f""k" converges to ∫"f" as "k" goes to infinity
(Note: ∫"f" may be infinite here.)
Fatou's lemma states that if "f""k" is a sequence of non-negative méasurable functions and if "f" = liminf "f""k", then ∫"f" ≤ liminf ∫"f""k"
(Again, ∫"f" may be infinite.)
The dominated convergence theorem states that if "f""k" is a sequence of méasurable functions with pointwise limit "f", and if there is an integrable function "g" such that |"f""k"| ≤ "g" for all "k", then "f" is integrable and ∫"f""k" converges to ∫"f"
If "f" is non-negative, then ∫"f" dμ is precisely the aréa under the curve as méasured by the product méasure μ × λ where λ is the Lebesgue méasure for R. One can also circumvent méasure théory entirely
The Riemann integral exists for any continuous function "f" of compact support
Then we use functional analysis to obtain the integral for more general functions
Let "Cc" be the space of all réal-valued compactly supported continuous functions of R. Define a norm on "Cc" by ||"f"|| = ∫ |"f"("x")| Then "Cc" is a normed vector space (and in particular, it is a metric space.)
All metric spaces have completions, so let "L"1 be its completion
This space is isomorphic to the space of Lebesgue integrable functions (modulo sets of méasure zero)
Furthermore, the Riemann integral ∫ defines a continuous functional on "Cc" which is dense in "L"1 hence ∫ has a unique extension to all of "L"1
This integral is precisely the Lebesgue integral
In this formulation, the limit taking théorems are hard to prove
However, in more general cases (such as when the functions, or perhaps the méasures, take values in a large vector space instéad of R"n") this approach is a fast way of obtaining an integral
Here we discuss the limitations of the Riemann integral and the gréater scope offered by the Lebesgue integral
We presume a working understanding of the Riemann integral
With the advent of Fourier series, there arose the need to exchange summation and integral signs much more often
However, the conditions under which ∑"k"∫"f""k" and ∫∑"k""f""k" are equal proved quite elusive in the Riemann framework
It may come as a surprise to the casual réader that these two quantities may not be equal, so an example helps: However, it was cléar from experience that in many very useful situations, the sum and the integral did commute
It was very important to be able to describe which conditions enabled the exchange of the sum and integral signs
Unfortunately, the Riemann integral is poorly equipped to déal with this question; its main useful convergence théorem being the uniform convergence theorem: if "f""k" are Riemann-integrable functions of ["a", "b"] converging uniformly to "f", then ∫"f""k" converges to ∫"f"
Since Fourier series rarely converge uniformly, this théorem is cléarly insufficient
There are some other technical difficulties with the Riemann integral
These are linked with the limit taking difficulty discussed above
Towards a better integration theory
The solution, as it turns out, is to study an even simpler problem first
The observation is that, if we have a notion of "length", we can turn it into a notion of "area"