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Classical Analysis
Mathematical analysis
Mathematical analysis
is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.[1][2] These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis
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Strange Attractor
In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system.[1] System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate. If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right)
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Madhava Of Sangamagrama
Madhava of Sangamagrama (c. 1340 – c. 1425), was a mathematician and astronomer from the town of Sangamagrama (believed to be present-day Aloor, Irinjalakuda
Irinjalakuda
in Thrissur
Thrissur
District), Kerala, India. He is considered the founder of the Kerala
Kerala
school of astronomy and mathematics
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Infinitesimals
In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small.[1] The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small"
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The Method Of Mechanical Theorems
The Method of Mechanical Theorems (Greek: Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as The Method, is considered one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes
Archimedes
to Eratosthenes,[1] the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles (sometimes referred to as infinitesimals).[2][3] The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes
Archimedes
Palimpsest
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Chinese Mathematics
Mathematics
Mathematics
in China emerged independently by the 11th century BC.[1] The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry. Ancient Chinese mathematicians made advances in algorithm development and algebra
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Liu Hui
Liu
Liu
Hui (fl. 3rd century CE) was a Chinese mathematician who lived in the state of Cao Wei
Cao Wei
during the Three Kingdoms
Three Kingdoms
period (220–280) of China. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as The Nine Chapters on the Mathematical Art, in which he was possibly the first mathematician to discover, understand and use negative numbers. He was a descendant of the Marquis of Zi District (菑鄉侯) of the Eastern Han dynasty, whose marquisate is in present-day Zichuan District, Zibo, Shandong. He completed his commentary to the Nine Chapters in the year 263
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Zu Chongzhi
Zu Chongzhi
Zu Chongzhi
(429–500 AD), courtesy name Wenyuan, was a Chinese mathematician, astronomer, writer and politician during the Liu Song and Southern Qi
Southern Qi
Dynasties.Contents1 Life and works 2 Astronomy 3 Mathematics 4 The South Pointing Chariot 5 Named after him 6 Notes 7 References 8 Further reading 9 External linksLife and works[edit] Chongzhi's ancestry was from modern Baoding, Hebei. To flee from the ravage of war, Zu's grandfather Zu Chang moved to the Yangtze, as part of the massive population movement during the Eastern Jin. Zu Chang (祖昌) at one point held the position of Chief Minister for the Palace Buildings (大匠卿) within the Liu Song
Liu Song
and was in charge of government construction projects. Zu's father, Zu Shuozhi (祖朔之) also served the court and was greatly respected for his erudition. Zu was born in Jiankang
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Cavalieri's Principle
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:[1]2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes.Today Cavalieri's principle
Cavalieri's principle
is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration
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Sphere
A sphere (from Greek σφαῖρα — sphaira, "globe, ball"[1]) is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to a circular object in two dimensions). Like a circle, which geometrically is an object in two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in three-dimensional space.[2] This distance r is the radius of the ball, and the given point is the center of the mathematical ball
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Indian Mathematics
Indian mathematics
Indian mathematics
emerged in the Indian subcontinent[1] from 1200 BCE[2] until the end of the 18th century, after which Indian mathematicians were directly part of the development of global mathematics. In the classical period of Indian mathematics
Indian mathematics
(400 CE to 1600 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji
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Bhāskara II
Bhāskara[1] (also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I) (1114–1185), was an Indian mathematician and astronomer. He was born in Bijapur in Karnataka.[2] Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.[3] His main work Siddhānta Shiromani, ( Sanskrit
Sanskrit
for "Crown of Treatises")[4] is divided into four parts called Lilāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya,[5] which are also sometimes considered four independent works.[6] These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively
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Rolle's Theorem
In calculus, Rolle's theorem
Rolle's theorem
essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero.Contents1 Standard version of the theorem 2 History 3 Examples3.1 First example 3.2 Second example4 Generalization4.1 Remarks5 Proof of the generalized version 6 Generalization to higher derivatives6.1 Proof7 Generalizatio
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Power Series
In mathematics, a power series (in one variable) is an infinite series of the form ∑ n = 0 ∞ a n ( x − c ) n = a 0 + a 1 ( x − c ) 1 + a 2 ( x − c ) 2 + ⋯ displaystyle sum _ n=0 ^ infty a_ n left(x-cright)^ n =a_ 0 +a_ 1 (x-c)^ 1 +a_ 2 (x-c)^ 2 +cdots where an represents the coefficient of the nth term and c is a constant. an is independent of x and may be expressed as a function of n (e.g
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Greek Mathematics
Greek mathematics
Greek mathematics
refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture and language. Greek mathematics of the period following Alexander the Great
Alexander the Great
is sometimes called Hellenistic mathematics
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Taylor Series
In mathematics, a Taylor series
Taylor series
is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The concept of a Taylor series
Taylor series
was formulated by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor
Brook Taylor
in 1715. If the Taylor series
Taylor series
is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series
Taylor series
in the 18th century. A function can be approximated by using a finite number of terms of its Taylor series. Taylor's theorem
Taylor's theorem
gives quantitative estimates on the error introduced by the use of such an approximation
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