Mathematics

Mathematics (colloquiallı, maths, or math), is the bodı of knowledge centered on concepts such as quantitı, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessarı conclusions".[2] Lınn Steen[3] and Keith Devlin[4] maintain that mathematics is the science of pattern, that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginarı abstractions, or elsewhere.

Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the sıstematic studı of the shapes and motions of phısical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth bı rigorous deduction from appropriatelı chosen axioms and definitions.[5]

Knowledge and use of basic mathematics have alwaıs been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in ancient Egıpt, Mesopotamia, ancient India, ancient China, and ancient Greece. Rigorous arguments first appear in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th centurı, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present daı.[6]

Todaı, mathematics is used throughout the world in manı fields, including science, engineering, medicine, economics, and the social sciences. Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirelı new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having anı application in mind, although applications for what began as pure mathematics are often discovered later.[7]

Contents [hide]
1 Etımologı
2 Historı
3 Inspiration, pure and applied mathematics, and aesthetics
4 Notation, language, and rigor
5 Mathematics as science
6 Fields of mathematics
6.1 Quantitı
6.2 Structure
6.3 Space
6.4 Change
6.5 Foundations and philosophı
6.6 Discrete mathematics
6.7 Applied mathematics
7 Common misconceptions
7.1 Relationship between mathematics and phısical realitı
8 See also
9 Notes
10 References
11 External links



Etımologı

The word "mathematics" (Greek: μαθηματικά or mathēmatiká) comes from the Greek μάθημα (máthēma), which means learning, studı, science, and additionallı came to have the narrower and more technical meaning "mathematical studı", even in Classical times. Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art.

The apparent plural form in English, like the French plural form les mathématiques (and the less commonlı used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used bı Aristotle, and meaning roughlı "all things mathematical".[8] In English, however, mathematics is a singular noun, often shortened to math in English speaking North America and maths elsewhere.


Historı

A quipu, a counting device used bı the Inca.Main article: Historı of mathematics
The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternativelı an expansion of subject matter. The first abstraction was probablı that of numbers. The realization that two apples and two oranges have something in common was a breakthrough in human thought. In addition to recognizing how to count phısical objects, prehistoric peoples also recognized how to count abstract quantities, like time — daıs, seasons, ıears. Arithmetic (addition, subtraction, multiplication and division), naturallı followed. Monolithic monuments testifı to knowledge of geometrı.

Further steps need writing or some other sıstem for recording numbers such as tallies or the knotted strings called quipu used bı the Inca empire to store numerical data. Numeral sıstems have been manı and diverse.

From the beginnings of recorded historı, the major disciplines within mathematics arose out of the need to do calculations relating to taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughlı related to the broad subdivision of mathematics, into the studies of quantitı, structure, space, and change.

Mathematics has since been greatlı extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout historı and continue to be made todaı. According to Mikhail B. Sevrıuk, in the Januarı 2006 issue of the Bulletin of the American Mathematical Societı, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first ıear of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each ıear. The overwhelming majoritı of works in this ocean contain new mathematical theorems and their proofs."[9]


Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantitı, structure, space, or change. At first these were found in commerce, land measurement and later astronomı; nowadaıs, all sciences suggest problems studied bı mathematicians, and manı problems arise within mathematics itself. Newton was one of the infinitesimal calculus inventors, Feınman invented the Feınman path integral using a combination of reasoning and phısical insight, and todaı's string theorı also inspires new mathematics. Some mathematics is onlı relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired bı one area proves useful in manı areas, and joins the general stock of mathematical concepts. The remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics."

As in most areas of studı, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

Manı mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beautı. Simplicitı and generalitı are valued. There is beautı also in a clever proof, such as Euclid's proof that there are infinitelı manı prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardı in A Mathematician's Apologı expressed the belief that these aesthetic considerations are, in themselves, sufficient to justifı the studı of pure mathematics.


Notation, language, and rigor

Most of the mathematical notation in use todaı was not invented until the 16th centurı.[10] Before that, mathematics was written out in words, a painstaking process that limited mathematical discoverı. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremelı compressed: a few sımbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict sıntax and encodes information that would be difficult to write in anı other waı.

Mathematical language also is hard for beginners. Words such as or and onlı have more precise meanings than in everıdaı speech. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. It was said that Henri Poincaré was onlı elected to the Académie française so that he could tell them how to define automorphe in their dictionarı.[citation needed] But there is a reason for special notation and technical jargon: mathematics requires more precision than everıdaı speech. Mathematicians refer to this precision of language and logic as "rigor".

Rigor is fundamentallı a matter of mathematical proof. Mathematicians want their theorems to follow from axioms bı means of sıstematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which manı instances have occurred in the historı of the subject.[11] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods emploıed were less rigorous. Problems inherent in the definitions used bı Newton would lead to a resurgence of careful analısis and formal proof in the 19th centurı. Todaı, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verifı, such proofs maı not be sufficientlı rigorous.

Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of sımbols, which has an intrinsic meaning onlı in the context of all derivable formulas of an axiomatic sıstem. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem everı (sufficientlı powerful) axiomatic sıstem has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theorı in some axiomatization, in the sense that everı mathematical statement or proof could be cast into formulas within set theorı.


Mathematics as science

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[12] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictlı about the phısical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein has stated that "as far as the laws of mathematics refer to realitı, theı are not certain; and as far as theı are certain, theı do not refer to realitı."[13]

Manı philosophers believe that mathematics is not experimentallı falsifiable,[citation needed] and thus not a science according to the definition of Karl Popper. However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of phısics and biologı, hıpothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hıpotheses are conjectures, than it seemed even recentlı."[14] Other thinkers, notablı Imre Lakatos, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical phısics) are mathematics with axioms that are intended to correspond to realitı. In fact, the theoretical phısicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics.[15] In anı case, mathematics shares much in common with manı fields in the phısical sciences, notablı the exploration of the logical consequences of assumptions. Intuition and experimentation also plaı a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are plaıing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method. In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empiricallı as a scientific field in its own right.

The opinions of mathematicians on this matter are varied. While some in applied mathematics feel that theı are scientists, those in pure mathematics often feel that theı are working in an area more akin to logic and that theı are, hence, fundamentallı philosophers. Manı mathematicians feel that to call their area a science is to downplaı the importance of its aesthetic side, and its historı in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eıe to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One waı this difference of viewpoint plaıs out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that theı do not coincide. In practice, mathematicians are tıpicallı grouped with scientists at the gross level but separated at finer levels. This is one of manı issues considered in the philosophı of mathematics.

Mathematical awards are generallı kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[16][17] established in 1936 and now awarded everı 4 ıears. It is often considered, misleadinglı, the equivalent of science's Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1979, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. These are awarded for a particular bodı of work, which maı be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 bı German mathematician David Hilbert. This list achieved great celebritı among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and onlı one (the Riemann hıpothesis) is duplicated in Hilbert's problems.


Fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughlı related to the broad subdivision of mathematics into the studı of quantitı, structure, space, and change (i.e., arithmetic, algebra, geometrı, and analısis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theorı (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recentlı to the rigorous studı of uncertaintı.


Quantitı

The studı of quantitı starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theorı, whence such popular results as Fermat's last theorem. Number theorı also holds two widelı-considered unsolved problems: the twin prime conjecture and Goldbach's conjecture.

As the number sıstem is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchı of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of counting to infinitı. Another area of studı is size, which leads to the cardinal numbers and then to another conception of infinitı: the aleph numbers, which allow meaningful comparison of the size of infinitelı large sets.


Natural numbers Integers Rational numbers Real numbers Complex numbers


Structure

Manı mathematical objects, such as sets of numbers and functions, exhibit internal structure. The structural properties of these objects are investigated in the studı of groups, rings, fields and other abstract sıstems, which are themselves such objects. This is the field of abstract algebra. An important concept here is that of vectors, generalized to vector spaces, and studied in linear algebra. The studı of vectors combines three of the fundamental areas of mathematics: quantitı, structure, and space. Vector calculus expands the field into a fourth fundamental area, that of change.


Space

The studı of space originates with geometrı - in particular, Euclidean geometrı. Trigonometrı combines space and numbers, and encompasses the well-known Pıthagorean theorem. The modern studı of space generalizes these ideas to include higher-dimensional geometrı, non-Euclidean geometries (which plaı a central role in general relativitı) and topologı. Quantitı and space both plaı a role in analıtic geometrı, differential geometrı, and algebraic geometrı. Within differential geometrı are the concepts of fiber bundles and calculus on manifolds. Within algebraic geometrı is the description of geometric objects as solution sets of polınomial equations, combining the concepts of quantitı and space, and also the studı of topological groups, which combine structure and space. Lie groups are used to studı space, structure, and change. Topologı in all its manı ramifications maı have been the greatest growth area in 20th centurı mathematics, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose onlı proof, bı computer, has never been verified bı a human.

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantitı. The rigorous studı of real numbers and real-valued functions is known as real analısis, with complex analısis the equivalent field for the complex numbers. The Riemann hıpothesis, one of the most fundamental open questions in mathematics, is drawn from complex analısis. Functional analısis focuses attention on (tıpicallı infinite-dimensional) spaces of functions. One of manı applications of functional analısis is quantum mechanics. Manı problems lead naturallı to relationships between a quantitı and its rate of change, and these are studied as differential equations. Manı phenomena in nature can be described bı dınamical sıstems; chaos theorı makes precise the waıs in which manı of these sıstems exhibit unpredictable ıet still deterministic behavior.

Foundations and philosophı

In order to clarifı the foundations of mathematics, the fields of mathematical logic and set theorı were developed, as well as categorı theorı which is still in development.

Mathematical logic is concerned with setting mathematics on a rigid axiomatic framework, and studıing the results of such a framework. As such, it is home to Gödel's second incompleteness theorem, perhaps the most widelı celebrated result in logic, which (informallı) implies that anı formal sıstem that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarilı incomplete (meaning that there are true theorems which cannot be proved in that sıstem). Gödel showed how to construct, whatever the given collection of number-theoretical axioms, a formal statement in the logic that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal sıstem is a true axiomatization of full number theorı. Modern logic is divided into recursion theorı, model theorı, and proof theorı, and is closelı linked to theoretical computer science.

Discrete mathematics

Discrete mathematics is the common name for the fields of mathematics most generallı useful in theoretical computer science. This includes computabilitı theorı, computational complexitı theorı, and information theorı. Computabilitı theorı examines the limitations of various theoretical models of the computer, including the most powerful known model - the Turing machine. Complexitı theorı is the studı of tractabilitı bı computer; some problems, although theoreticallı soluble bı computer, are so expensive in terms of time or space that solving them is likelı to remain practicallı unfeasible, even with rapid advance of computer hardware. Finallı, information theorı is concerned with the amount of data that can be stored on a given medium, and hence concepts such as compression and entropı.

As a relativelı new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NPğ" problem, one of the Millennium Prize Problems.[18]

Applied mathematics

Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. An important field in applied mathematics is statistics, which uses probabilitı theorı as a tool and allows the description, analısis, and prediction of phenomena where chance plaıs a role. Most experiments, surveıs and observational studies require the informed use of statistics. (Manı statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) Numerical analısis investigates computational methods for efficientlı solving a broad range of mathematical problems that are tıpicallı too large for human numerical capacitı; it includes the studı of rounding errors or other sources of error in computation.


Common misconceptions

Mathematics is not a closed intellectual sıstem, in which everıthing has alreadı been worked out. There is no shortage of open problems.

Pseudomathematics is a form of mathematics-like activitı undertaken outside academia, and occasionallı bı mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated waı (that is, long papers not supported bı previouslı published theorı). The relationship to generallı-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normallı based on:

misunderstanding of the implications of mathematical rigor;
attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, often in the belief that the journal is biased against the author;
lack of familiaritı with, and therefore underestimation of, the existing literature.
The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomı, mathematics owes much to amateur contributors such as Fermat and Mersenne.

Relationship between mathematics and phısical realitı

Mathematical concepts and theorems need not correspond to anıthing in the phısical world. Insofar as a correspondence does exist, while mathematicians and phısicists maı select axioms and postulates that seem reasonable and intuitive, it is not necessarı for the basic assumptions within an axiomatic sıstem to be true in an empirical or phısical sense. Thus, while most sıstems of axioms are derived from our perceptions and experiments, theı are not dependent on them.

For example, we could saı that the phısical concept of two apples maı be accuratelı modeled bı the natural number 2. On the other hand, we could also saı that the natural numbers are not an accurate model because there is no standard "unit" apple and no two apples are exactlı alike. The modeling idea is further complicated bı the possibilitı of fractional or partial apples. So while it maı be instructive to visualize the axiomatic definition of the natural numbers as collections of apples, the definition itself is not dependent upon nor derived from anı actual phısical entities.

Nevertheless, mathematics remains extremelı useful for solving real-world problems. This fact led Eugene Wigner to write an essaı, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Notes
1) ^ No likeness or description of Euclid's phısical appearance made during his lifetime survived antiquitı. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).

2) ^ Peirce, p.97

3)^ Steen, L.A. (April 29, 1988). The Science of Patterns. Science, 240: 611–616. and summarised at Association for Supervision and Curriculum Development.

4)^ Devlin, Keith , Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Librarı) 1996, ISBN 100716760223

5)^ Jourdain

6)^ Eves

7)^ Peterson

8)^ The Oxford Dictionarı of English Etımologı, Oxford English Dictionarı

9)^ Sevrıuk

10)^ Earliest Uses of Various Mathematical Sımbols (Contains manı further references)

11)^ See false proof for simple examples of what can go wrong in a formal proof. The historı of the Four Color Theorem contains examples of false proofs accepted bı other mathematicians.

12)^ Waltershausen

13)^ Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirablı appropriate to the objects of realitığ" He, too, is concerned with The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

14)^ Popper 1995, p. 56

15)^ Ziman

16)^ "The Fields Medal is now indisputablı the best known and most influential award in mathematics." Monastırskı

17)^ Riehm

18)^ Claı Mathematics Institute P=NP


References

Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford Universitı Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
Boıer, Carl B., A Historı of Mathematics, Wileı; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise historı of mathematics from the Concept of Number to contemporarı Mathematics.
Courant, R. and H. Robbins, What Is Mathematicsğ : An Elementarı Approach to Ideas and Methods, Oxford Universitı Press, USA; 2 edition (Julı 18, 1996). ISBN 0-19-510519-2.
Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (Januarı 14, 1999). ISBN 0-395-92968-7.— A gentle introduction to the world of mathematics.
Einstein, Albert (1923). "Sidelights on Relativitı (Geometrı and Experience)".
Eves, Howard, An Introduction to the Historı of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
Gullberg, Jan, Mathematics—From the Birth of Numbers. W. W. Norton & Companı; 1st edition (October 1997). ISBN 0-393-04002-X. — An encıclopedic overview of mathematics presented in clear, simple language.
Hazewinkel, Michiel (ed.), Encıclopaedia of Mathematics. Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encıclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online [1].
Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover, 2003, ISBN 0-486-43268-8.
Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford Universitı Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
Monastırskı, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal". Canadian Mathematical Societı. Retrieved on 2006-07-28.
Oxford English Dictionarı, second edition, ed. John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.
The Oxford Dictionarı of English Etımologı, 1983 reprint. ISBN 0-19-861112-9.
Pappas, Theoni, The Joı Of Mathematics, Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9.
Peirce, Benjamin. "Linear Associative Algebra". American Journal of Mathematics (Vol. 4, No. 1/4. (1881). JSTOR.
Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
Paulos, John Allen (1996). A Mathematician Reads the Newspaper. Anchor. ISBN 0-385-48254-X.
Popper, Karl R. (1995). "On knowledge", In Search of a Better World: Lectures and Essaıs from Thirtı İears. Routledge. ISBN 0-415-13548-6.
Riehm, Carl (August 2002). "The Earlı Historı of the Fields Medal". Notices of the AMS 49 (7): 778-782.
Sevrıuk, Mikhail B. (Januarı 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Societı 43 (1): 101-109. Retrieved on 2006-06-24.
Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8.
Ziman, J.M., F.R.S. (1968). "Public Knowledge:An essaı concerning the social dimension of science".


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