1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Sequence
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but 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 the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, 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 useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
3.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d
4.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
5.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0
6.
NIST
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The National Institute of Standards and Technology is a measurement standards laboratory, and a non-regulatory agency of the United States Department of Commerce. Its mission is to promote innovation and industrial competitiveness, in 1821, John Quincy Adams had declared Weights and measures may be ranked among the necessities of life to every individual of human society. From 1830 until 1901, the role of overseeing weights and measures was carried out by the Office of Standard Weights and Measures, president Theodore Roosevelt appointed Samuel W. Stratton as the first director. The budget for the first year of operation was $40,000, a laboratory site was constructed in Washington, DC, and instruments were acquired from the national physical laboratories of Europe. In addition to weights and measures, the Bureau developed instruments for electrical units, in 1905 a meeting was called that would be the first National Conference on Weights and Measures. Quality standards were developed for products including some types of clothing, automobile brake systems and headlamps, antifreeze, during World War I, the Bureau worked on multiple problems related to war production, even operating its own facility to produce optical glass when European supplies were cut off. Between the wars, Harry Diamond of the Bureau developed a blind approach radio aircraft landing system, in 1948, financed by the Air Force, the Bureau began design and construction of SEAC, the Standards Eastern Automatic Computer. The computer went into operation in May 1950 using a combination of vacuum tubes, about the same time the Standards Western Automatic Computer, was built at the Los Angeles office of the NBS and used for research there. A mobile version, DYSEAC, was built for the Signal Corps in 1954, due to a changing mission, the National Bureau of Standards became the National Institute of Standards and Technology in 1988. Following 9/11, NIST conducted the investigation into the collapse of the World Trade Center buildings. NIST had a budget for fiscal year 2007 of about $843.3 million. NISTs 2009 budget was $992 million, and it also received $610 million as part of the American Recovery, NIST employs about 2,900 scientists, engineers, technicians, and support and administrative personnel. About 1,800 NIST associates complement the staff, in addition, NIST partners with 1,400 manufacturing specialists and staff at nearly 350 affiliated centers around the country. NIST publishes the Handbook 44 that provides the Specifications, tolerances, the Congress of 1866 made use of the metric system in commerce a legally protected activity through the passage of Metric Act of 1866. NIST is headquartered in Gaithersburg, Maryland, and operates a facility in Boulder, nISTs activities are organized into laboratory programs and extramural programs. Effective October 1,2010, NIST was realigned by reducing the number of NIST laboratory units from ten to six, nISTs Boulder laboratories are best known for NIST‑F1, which houses an atomic clock. NIST‑F1 serves as the source of the official time. NIST also operates a neutron science user facility, the NIST Center for Neutron Research, the NCNR provides scientists access to a variety of neutron scattering instruments, which they use in many research fields
7.
Bernoulli polynomial
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In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in part because they are an Appell sequence. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled and this article also discusses the Bernoulli polynomials and the related Euler polynomials, and the Bernoulli and Euler numbers. The Bernoulli polynomials Bn admit a variety of different representations, which among them should be taken to be the definition may depend on ones purposes. B n = ∑ k =0 n b n − k x k, for n ≥0, the generating function for the Bernoulli polynomials is t e x t e t −1 = ∑ n =0 ∞ B n t n n. The generating function for the Euler polynomials is 2 e x t e t +1 = ∑ n =0 ∞ E n t n n. The Bernoulli polynomials are given by B n = D e D −1 x n where D = d/dx is differentiation with respect to x. It follows that ∫ a x B n d u = B n +1 − B n +1 n +1, the Bernoulli polynomials are the unique polynomials determined by ∫ x x +1 B n d u = x n. The integral transform = ∫ x x +1 f d u on polynomials f, F = f + f ′2 + f ″6 + f ‴24 + ⋯. This can be used to produce the inversion formulae below, an explicit formula for the Bernoulli polynomials is given by B m = ∑ n =0 m 1 n +1 ∑ k =0 n k m. Note the remarkable similarity to the convergent series expression for the Hurwitz zeta function. Indeed, one has B n = − n ζ where ζ is the Hurwitz zeta, thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n. The inner sum may be understood to be the nth forward difference of xm, thus, one may write B m = ∑ n =0 m n n +1 Δ n x m. This formula may be derived from an identity appearing above as follows, as long as this operates on an mth-degree polynomial such as xm, one may let n go from 0 only up to m. An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, an explicit formula for the Euler polynomials is given by E m = ∑ n =0 m 12 n ∑ k =0 n k m. This may also be written in terms of the Euler numbers Ek as E m = ∑ k =0 m E k 2 k m − k and we have ∑ k =0 x k p = B p +1 − B p +1 p +1. See Faulhabers formula for more on this, the Bernoulli numbers are given by B n = B n
8.
Taylor series
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In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the functions derivatives at a single point. The concept of a Taylor series was formulated by the Scottish mathematician James Gregory, a function can be approximated by using a finite number of terms of its Taylor series. Taylors theorem gives quantitative estimates on the error introduced by the use of such an approximation, the polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that functions Taylor polynomials as the degree increases, a function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an interval is known as an analytic function in that interval. The Taylor series of a real or complex-valued function f that is differentiable at a real or complex number a is the power series f + f ′1. Which can be written in the more compact sigma notation as ∑ n =0 ∞ f n, N where n. denotes the factorial of n and f denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and 0 and 0. are both defined to be 1, when a =0, the series is also called a Maclaurin series. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for 1/1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ so the Taylor series for 1/x at a =1 is 1 − +2 −3 + ⋯. The Taylor series for the exponential function ex at a =0 is x 00, + ⋯ =1 + x + x 22 + x 36 + x 424 + x 5120 + ⋯ = ∑ n =0 ∞ x n n. The above expansion holds because the derivative of ex with respect to x is also ex and this leaves the terms n in the numerator and n. in the denominator for each term in the infinite sum. The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a result, but rejected it as an impossibility. It was through Archimedess method of exhaustion that a number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later, in the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. The Kerala school of astronomy and mathematics further expanded his works with various series expansions, in the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a method for constructing these series for all functions for which they exist was finally provided by Brook Taylor. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, if f is given by a convergent power series in an open disc centered at b in the complex plane, it is said to be analytic in this disc
9.
Tangent function
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
10.
Hyperbolic function
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In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The inverse hyperbolic functions are the hyperbolic sine arsinh and so on. Just as the form a circle with a unit radius. The hyperbolic functions take a real argument called a hyperbolic angle, the size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a triangle covering this sector. Laplaces equations are important in areas of physics, including electromagnetic theory, heat transfer, fluid dynamics. In complex analysis, the hyperbolic functions arise as the parts of sine and cosine. When considered defined by a variable, the hyperbolic functions are rational functions of exponentials. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati, Riccati used Sc. and Cc. to refer to circular functions and Sh. and Ch. to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today, the abbreviations sh and ch are still used in some other languages, like French and Russian. The hyperbolic functions are, Hyperbolic sine, sinh x = e x − e − x 2 = e 2 x −12 e x =1 − e −2 x 2 e − x. Hyperbolic cosine, cosh x = e x + e − x 2 = e 2 x +12 e x =1 + e −2 x 2 e − x, the complex forms in the definitions above derive from Eulers formula. One also has sech 2 x =1 − tanh 2 x csch 2 x = coth 2 x −1 for the other functions, sinh = sinh 2 = sgn cosh −12 where sgn is the sign function. All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x and e − x, and it is possible to express the above functions as Taylor series, sinh x = x + x 33. + ⋯ = ∑ n =0 ∞ x 2 n +1, the function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh, the function cosh x has a Taylor series expression with only even exponents for x. Thus it is a function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the series expression of the exponential function
11.
Riemann zeta function
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More general representations of ζ for all s are given below. The Riemann zeta function plays a role in analytic number theory and has applications in physics, probability theory. As a function of a variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis. The values of the Riemann zeta function at even positive integers were computed by Euler, the first of them, ζ, provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of ζ, the values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, the Riemann zeta function ζ is a function of a complex variable s = σ + it. It can also be defined by the integral ζ =1 Γ ∫0 ∞ x s −1 e x −1 d x where Γ is the gamma function. The Riemann zeta function is defined as the continuation of the function defined for σ >1 by the sum of the preceding series. Leonhard Euler considered the series in 1740 for positive integer values of s. The above series is a prototypical Dirichlet series that converges absolutely to a function for s such that σ >1. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠1, for s =1 the series is the harmonic series which diverges to +∞, and lim s →1 ζ =1. Thus the Riemann zeta function is a function on the whole complex s-plane. For any positive even integer 2n, ζ = n +1 B2 n 2 n 2, where B2n is the 2nth Bernoulli number. For odd positive integers, no simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers. For nonpositive integers, one has ζ = B n +1 n +1 for n ≥0 In particular, ζ = −12, Similarly to the above, this assigns a finite result to the series 1 +1 +1 +1 + ⋯. ζ ≈ −1.4603545 This is employed in calculating of kinetic boundary layer problems of linear kinetic equations, ζ =1 +12 +13 + ⋯ = ∞, if we approach from numbers larger than 1. Then this is the harmonic series, but its Cauchy principal value lim ε →0 ζ + ζ2 exists which is the Euler–Mascheroni constant γ =0. 5772…. ζ ≈2.612, This is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems
12.
Jacob Bernoulli
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Jacob Bernoulli was one of the many prominent mathematicians in the Bernoulli family. He was a proponent of Leibnizian calculus and had sided with Leibniz during the Leibniz–Newton calculus controversy. He is known for his numerous contributions to calculus, and along with his brother Johann, was one of the founders of the calculus of variations and he also discovered the fundamental mathematical constant e. However, his most important contribution was in the field of probability, Jacob Bernoulli was born in Basel, Switzerland. Following his fathers wish, he studied theology and entered the ministry, but contrary to the desires of his parents, he also studied mathematics and astronomy. He traveled throughout Europe from 1676 to 1682, learning about the latest discoveries in mathematics and this included the work of Johannes Hudde, Robert Boyle, and Robert Hooke. During this time he produced an incorrect theory of comets. Bernoulli returned to Switzerland and began teaching mechanics at the University in Basel from 1683, in 1684 he married Judith Stupanus, and they had two children. During this decade, he began a fertile research career. His travels allowed him to establish correspondence with many leading mathematicians and scientists of his era, during this time, he studied the new discoveries in mathematics, including Christiaan Huygenss De ratiociniis in aleae ludo, Descartes Geometrie and Frans van Schootens supplements of it. He also studied Isaac Barrow and John Wallis, leading to his interest in infinitesimal geometry, apart from these, it was between 1684 and 1689 that many of the results that were to make up Ars Conjectandi were discovered. He was appointed professor of mathematics at the University of Basel in 1687, by that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the calculus in Nova Methodus pro Maximis et Minimis published in Acta Eruditorum. They also studied the publications of von Tschirnhaus and it must be understood that Leibnizs publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were the first to try to understand and apply Leibnizs theories. Jacob collaborated with his brother on various applications of calculus, by 1697, the relationship had completely broken down. His grave is in Basel Munster or Cathedral where the gravestone shown below is located, the lunar crater Bernoulli is also named after him jointly with his brother Johann. Jacob Bernoullis first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and his geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines. By 1689 he had published important work on series and published his law of large numbers in probability theory
13.
Takakazu Seki
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Seki Takakazu, also known as Seki Kōwa, was a Japanese mathematician in the Edo period. Seki laid foundations for the subsequent development of Japanese mathematics known as wasan and he created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. A contemporary of Gottfried Leibniz and Isaac Newton, Sekis work was independent and his successors later developed a school dominant in Japanese mathematics until the end of the Edo period. While it is not clear how much of the achievements of wasan are Sekis, since many of them only in writings of his pupils. For example, he is credited with the discovery of Bernoulli numbers, the resultant and determinant are attributed to him. This work was an advance on, for example, the comprehensive introduction of 13th-century Chinese algebra made as late as 1671. Not much is known about Kōwas personal life and his birthplace has been indicated as either Fujioka in Gunma prefecture, or Edo. His birth date ranges from 1635 to 1643 and he was born to the Uchiyama clan, a subject of Ko-shu han, and adopted into the Seki family, a subject of the Shogun. While in Ko-shu han, he was involved in a project to produce a reliable map of his employers land. He spent many years in studying 13th-century Chinese calendars to replace the accurate one used in Japan at that time. His mathematics was based on knowledge from the 13th to 15th centuries. This consisted of algebra with numerical methods, polynomial interpolation and its applications, Sekis work is more or less based on and related to these known methods. Chinese algebra discovered numerical evaluation of arbitrary degree algebraic equation with real coefficients, by using the Pythagorean theorem, they reduced geometric problems to algebra systematically. The number of unknowns in an equation was, however, quite limited and they used notations of an array of numbers to represent a formula, for example, for a x 2 + b x + c. Later, they developed a method that uses two-dimensional arrays, representing four variables at most, hence, a target of Seki and his contemporary Japanese mathematicians was the development of general multi-variable algebraic equations and elimination theory. In the Chinese approach to interpolation, the motivation was to predict the motion of celestial bodies from observed data. The method was applied to find various mathematical formulas. Seki learned this technique, most likely, through his examination of Chinese calendars
14.
Ars Conjectandi
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Ars Conjectandi is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The importance of early work had a large impact on both contemporary and later mathematicians, for example, Abraham de Moivre. Bernoulli wrote the text between 1684 and 1689, including the work of such as Christiaan Huygens, Gerolamo Cardano, Pierre de Fermat. Core topics from probability, such as expected value, were also a significant portion of important work. However, his influence on mathematical scene was not great, he wrote only one light tome on the subject in 1525 titled Liber de ludo aleae. In 1665 Pascal posthumously published his results on the eponymous Pascals triangle and he referred to the triangle in his work Traité du triangle arithmétique as the arithmetic triangle. In 1662, the book La Logique ou l’Art de Penser was published anonymously in Paris, the authors presumably were Antoine Arnauld and Pierre Nicole, two leading Jansenists, who worked together with Blaise Pascal. The Latin title of book is Ars cogitandi, which was a successful book on logic of the time. In the field of statistics and applied probability, John Graunt published Natural and Political Observations Made upon the Bills of Mortality also in 1662, De Witts work was not widely distributed beyond the Dutch Republic, perhaps due to his fall from power and execution by mob in 1672. Thus probability could be more than mere combinatorics, in the wake of all these pioneers, Bernoulli produced much of the results contained in Ars Conjectandi between 1684 and 1689, which he recorded in his diary Meditationes. The latter, however, did manage to provide Pascals and Huygens work, bernoulli’s progress over time can be pursued by means of the Meditationes. Three working periods with respect to his discovery can be distinguished by aims, finally, in the last period, the problem of measuring the probabilities is solved. Before the publication of his Ars Conjectandi, Bernoulli had produced a number of related to probability, Parallelismus ratiocinii logici et algebraici. In the Journal des Sçavans 1685, p.314 there appear two problems concerning the probability each of two players may have of winning in a game of dice. Solutions were published in the Acta Eruditorum 1690, pp. 219–223 in the article Quaestiones nonnullae de usuris, in addition, Leibniz himself published a solution in the same journal on pages 387-390. Theses logicae de conversione et oppositione enunciationum, a lecture delivered at Basel,12 February 1686. Theses XXXI to XL are related to the theory of probability, De Arte Combinatoria Oratio Inauguralis,1692. The Letter à un amy sur les parties du jeu de paume, that is, between 1703 and 1705, Leibniz corresponded with Jakob after learning about his discoveries in probability from his brother Johann
15.
Ada Lovelace
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Augusta Ada King-Noel, Countess of Lovelace was an English mathematician and writer, chiefly known for her work on Charles Babbages proposed mechanical general-purpose computer, the Analytical Engine. She was the first to recognise that the machine had applications beyond pure calculation, as a result, she is often regarded as the first to recognise the full potential of a computing machine and the first computer programmer. Ada Lovelace was the legitimate child of the poet George, Lord Byron. All Byrons other children were out of wedlock to other women. Byron separated from his wife a month after Ada was born and left England forever four months later, often ill, she spent most of her childhood sick. Ada married William King in 1835, King was made Earl of Lovelace in 1838, and she became Countess of Lovelace. Ada described her approach as poetical science and herself as an Analyst, Lovelace first met him in June 1833, through their mutual friend, and her private tutor, Mary Somerville. Between 1842 and 1843, Ada translated an article by Italian military engineer Luigi Menabrea on the engine and these notes contain what many consider to be the first computer program—that is, an algorithm designed to be carried out by a machine. Lovelaces notes are important in the history of computers. She also developed a vision of the capability of computers to go beyond mere calculating or number-crunching, while others, including Babbage himself. Her mindset of poetical science led her to ask questions about the Analytical Engine examining how individuals and she died of uterine cancer in 1852 at the age of 36. Byron expected his baby to be a boy and was disappointed when his wife gave birth to a girl. Augusta was named after Byrons half-sister, Augusta Leigh, and was called Ada by Byron himself, on 16 January 1816 Adas mother, Annabella, at Byrons behest, left for her parents home at Kirkby Mallory taking one-month-old Ada with her. On 21 April Byron signed the Deed of Separation, although very reluctantly, aside from an acrimonious separation, Annabella continually made allegations about Byrons immoral behaviour throughout her life. This set of events made Ada famous in Victorian society, Byron did not have a relationship with his daughter, and never saw her again. He died in 1824 when she was eight years old and her mother was the only significant parental figure in her life. Ada was not shown the family portrait of her father until her twentieth birthday and her mother became Baroness Wentworth in her own right in 1856. Annabella did not have a relationship with the young Ada and often left her in the care of her own mother Judith
16.
Ada Byron's notes on the analytical engine
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Augusta Ada King-Noel, Countess of Lovelace was an English mathematician and writer, chiefly known for her work on Charles Babbages proposed mechanical general-purpose computer, the Analytical Engine. She was the first to recognise that the machine had applications beyond pure calculation, as a result, she is often regarded as the first to recognise the full potential of a computing machine and the first computer programmer. Ada Lovelace was the legitimate child of the poet George, Lord Byron. All Byrons other children were out of wedlock to other women. Byron separated from his wife a month after Ada was born and left England forever four months later, often ill, she spent most of her childhood sick. Ada married William King in 1835, King was made Earl of Lovelace in 1838, and she became Countess of Lovelace. Ada described her approach as poetical science and herself as an Analyst, Lovelace first met him in June 1833, through their mutual friend, and her private tutor, Mary Somerville. Between 1842 and 1843, Ada translated an article by Italian military engineer Luigi Menabrea on the engine and these notes contain what many consider to be the first computer program—that is, an algorithm designed to be carried out by a machine. Lovelaces notes are important in the history of computers. She also developed a vision of the capability of computers to go beyond mere calculating or number-crunching, while others, including Babbage himself. Her mindset of poetical science led her to ask questions about the Analytical Engine examining how individuals and she died of uterine cancer in 1852 at the age of 36. Byron expected his baby to be a boy and was disappointed when his wife gave birth to a girl. Augusta was named after Byrons half-sister, Augusta Leigh, and was called Ada by Byron himself, on 16 January 1816 Adas mother, Annabella, at Byrons behest, left for her parents home at Kirkby Mallory taking one-month-old Ada with her. On 21 April Byron signed the Deed of Separation, although very reluctantly, aside from an acrimonious separation, Annabella continually made allegations about Byrons immoral behaviour throughout her life. This set of events made Ada famous in Victorian society, Byron did not have a relationship with his daughter, and never saw her again. He died in 1824 when she was eight years old and her mother was the only significant parental figure in her life. Ada was not shown the family portrait of her father until her twentieth birthday and her mother became Baroness Wentworth in her own right in 1856. Annabella did not have a relationship with the young Ada and often left her in the care of her own mother Judith
17.
Analytical Engine
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The Analytical Engine was a proposed mechanical general-purpose computer designed by English mathematician and computer pioneer Charles Babbage. It was first described in 1837 as the successor to Babbages difference engine, in other words, the logical structure of the Analytical Engine was essentially the same as that which has dominated computer design in the electronic era. Babbage was never able to complete construction of any of his machines due to conflicts with his chief engineer and it was not until the 1940s that the first general-purpose computers were actually built, more than a century after Babbage had proposed the pioneering Analytical Engine in 1837. Construction of this machine was never completed, Babbage had conflicts with his engineer, Joseph Clement. During this project, he realized that a more general design. The work on the design of the Analytical Engine started in 1835, the input, consisting of programs and data was to be provided to the machine via punched cards, a method being used at the time to direct mechanical looms such as the Jacquard loom. For output, the machine would have a printer, a curve plotter, the machine would also be able to punch numbers onto cards to be read in later. It employed ordinary base-10 fixed-point arithmetic, there was to be a store capable of holding 1,000 numbers of 40 decimal digits each. An arithmetical unit would be able to all four arithmetic operations, plus comparisons. Initially it was conceived as a difference engine curved back upon itself, in a circular layout. Later drawings depict a regularized grid layout, the programming language to be employed by users was akin to modern day assembly languages. Loops and conditional branching were possible, and so the language as conceived would have been Turing-complete as later defined by Alan Turing, there were three separate readers for the three types of cards. Babbage developed some two dozen programs for the Analytical Engine between 1837 and 1840, and one program later and these programs treat polynomials, iterative formulas, Gaussian elimination, and Bernoulli numbers. In 1842, the Italian mathematician Luigi Federico Menabrea published a description of the based on a lecture by Babbage in French. In 1843, the description was translated into English and extensively annotated by Ada Lovelace, in recognition of her additions to Menabreas paper, which included a way to calculate Bernoulli numbers using the machine, she has been described as the first computer programmer. Late in his life, Babbage sought ways to build a version of the machine. In 1878, a committee of the British Association for the Advancement of Science described the Analytical Engine as a marvel of mechanical ingenuity, but recommended against constructing it. The committee acknowledged the usefulness and value of the machine, but could not estimate the cost of building it, and were unsure whether the machine would function correctly after being built
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Algorithm
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In mathematics and computer science, an algorithm is a self-contained sequence of actions to be performed. Algorithms can perform calculation, data processing and automated reasoning tasks, an algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391, English adopted the French term, but it wasnt until the late 19th century that algorithm took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu and it begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as, Algorism is the art by which at present we use those Indian figures, the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals. An informal definition could be a set of rules that precisely defines a sequence of operations, which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually, but humans can do something equally useful, in the case of certain enumerably infinite sets, They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. An enumerably infinite set is one whose elements can be put into one-to-one correspondence with the integers, the concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a set of axioms. In logic, the time that an algorithm requires to complete cannot be measured, from such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete and abstract usage of the term. Algorithms are essential to the way computers process data, thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Although this may seem extreme, the arguments, in its favor are hard to refute. Gurevich. Turings informal argument in favor of his thesis justifies a stronger thesis, according to Savage, an algorithm is a computational process defined by a Turing machine. Typically, when an algorithm is associated with processing information, data can be read from a source, written to an output device. Stored data are regarded as part of the state of the entity performing the algorithm. In practice, the state is stored in one or more data structures, for some such computational process, the algorithm must be rigorously defined, specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be dealt with, case-by-case
19.
Charles Babbage
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Charles Babbage KH FRS was an English polymath. A mathematician, philosopher, inventor and mechanical engineer, Babbage is best remembered for originating the concept of a programmable computer. His varied work in other fields has led him to be described as pre-eminent among the many polymaths of his century, parts of Babbages uncompleted mechanisms are on display in the Science Museum in London. In 1991, a perfectly functioning difference engine was constructed from Babbages original plans, built to tolerances achievable in the 19th century, the success of the finished engine indicated that Babbages machine would have worked. Babbages birthplace is disputed, but according to the Oxford Dictionary of National Biography he was most likely born at 44 Crosby Row, Walworth Road, London, a blue plaque on the junction of Larcom Street and Walworth Road commemorates the event. His date of birth was given in his obituary in The Times as 26 December 1792, the parish register of St. Marys Newington, London, shows that Babbage was baptised on 6 January 1792, supporting a birth year of 1791. Babbage was one of four children of Benjamin Babbage and Betsy Plumleigh Teape and his father was a banking partner of William Praed in founding Praeds & Co. of Fleet Street, London, in 1801. In 1808, the Babbage family moved into the old Rowdens house in East Teignmouth, around the age of eight, Babbage was sent to a country school in Alphington near Exeter to recover from a life-threatening fever. For a short time he attended King Edward VI Grammar School in Totnes, South Devon, Babbage then joined the 30-student Holmwood academy, in Baker Street, Enfield, Middlesex, under the Reverend Stephen Freeman. The academy had a library that prompted Babbages love of mathematics and he studied with two more private tutors after leaving the academy. The first was a clergyman near Cambridge, through him Babbage encountered Charles Simeon and his evangelical followers and he was brought home, to study at the Totnes school, this was at age 16 or 17. The second was an Oxford tutor, under whom Babbage reached a level in Classics sufficient to be accepted by Cambridge, Babbage arrived at Trinity College, Cambridge, in October 1810. He was already self-taught in some parts of mathematics, he had read in Robert Woodhouse, Joseph Louis Lagrange. As a result, he was disappointed in the standard mathematical instruction available at the university, Babbage, John Herschel, George Peacock, and several other friends formed the Analytical Society in 1812, they were also close to Edward Ryan. In 1812 Babbage transferred to Peterhouse, Cambridge and he was the top mathematician there, but did not graduate with honours. He instead received a degree without examination in 1814 and he had defended a thesis that was considered blasphemous in the preliminary public disputation, but it is not known whether this fact is related to his not sitting the examination. Considering his reputation, Babbage quickly made progress and he lectured to the Royal Institution on astronomy in 1815, and was elected a Fellow of the Royal Society in 1816. After graduation, on the hand, he applied for positions unsuccessfully
20.
Computer program
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A computer program is a collection of instructions that performs a specific task when executed by a computer. A computer requires programs to function, and typically executes the programs instructions in a processing unit. A computer program is written by a computer programmer in a programming language. From the program in its form of source code, a compiler can derive machine code—a form consisting of instructions that the computer can directly execute. Alternatively, a program may be executed with the aid of an interpreter. A part of a program that performs a well-defined task is known as an algorithm. A collection of programs, libraries and related data are referred to as software. Computer programs may be categorized along functional lines, such as software or system software. The earliest programmable machines preceded the invention of the digital computer, in 1801, Joseph-Marie Jacquard devised a loom that would weave a pattern by following a series of perforated cards. Patterns could be weaved and repeated by arranging the cards, in 1837, Charles Babbage was inspired by Jacquards loom to attempt to build the Analytical Engine. The names of the components of the device were borrowed from the textile industry. In the textile industry, yarn was brought from the store to be milled, the device would have had a store—memory to hold 1,000 numbers of 40 decimal digits each. Numbers from the store would then have then transferred to the mill. It was programmed using two sets of perforated cards—one to direct the operation and the other for the input variables, however, after more than 17,000 pounds of the British governments money, the thousands of cogged wheels and gears never fully worked together. During a nine-month period in 1842–43, Ada Lovelace translated the memoir of Italian mathematician Luigi Menabrea, the memoir covered the Analytical Engine. The translation contained Note G which completely detailed a method for calculating Bernoulli numbers using the Analytical Engine and this note is recognized by some historians as the worlds first written computer program. In 1936, Alan Turing introduced the Universal Turing machine—a theoretical device that can model every computation that can be performed on a Turing complete computing machine and it is a finite-state machine that has an infinitely long read/write tape. The machine can move the back and forth, changing its contents as it performs an algorithm
21.
Pythagoras
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Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
22.
Archimedes
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Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
23.
Aryabhata
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Aryabhata or Aryabhata I was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya and the Arya-siddhanta, furthermore, in most instances Aryabhatta would not fit the metre either. Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga and this corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra, Bhāskara I describes Aryabhata as āśmakīya, one belonging to the Aśmaka country. During the Buddhas time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India. It has been claimed that the aśmaka where Aryabhata originated may be the present day Kodungallur which was the capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr, however, K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence. Aryabhata mentions Lanka on several occasions in the Aryabhatiya, but his Lanka is an abstraction and it is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I, identify Kusumapura as Pāṭaliputra, Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. Aryabhata is the author of treatises on mathematics and astronomy. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and it also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, a third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, direct details of Aryabhatas work are known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra and it is also occasionally referred to as Arya-shatas-aShTa, because there are 108 verses in the text. It is written in the terse style typical of sutra literature
24.
Al-Haytham
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Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, and philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and he was the first to explain that vision occurs when light bounces on an object and then is directed to ones eyes. He spent most of his close to the court of the Fatimid Caliphate in Cairo and earned his living authoring various treatises. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called The Physicist and he is also sometimes called al-Baṣrī after his birthplace Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born c.965 in Basra, which was part of the Buyid emirate. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. Legend has it that after deciding the scheme was impractical and fearing the caliphs anger, during this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. Among his students were Sorkhab, a Persian from Semnan who was his student for three years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen. Alhazen made significant contributions to optics, number theory, geometry, astronomy, Alhazens work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text. A Latin translation of the Kitab al-Manazir was made probably in the twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration and he made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the known as Alhazens problem. Alhazen wrote as many as 200 books, although only 55 have survived, some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew, the crater Alhazen on the Moon is named in his honour, as was the asteroid 59239 Alhazen. In honour of Alhazen, the Aga Khan University named its Ophthalmology endowed chair as The Ibn-e-Haitham Associate Professor, Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10, 000-dinar banknote issued in 2003, and on 10-dinar notes from 1982. The 2015 International Year of Light celebrated the 1000th anniversary of the works on optics by Ibn Al-Haytham, Alhazens most famous work is his seven-volume treatise on optics Kitab al-Manazir, written from 1011 to 1021. Optics was translated into Latin by a scholar at the end of the 12th century or the beginning of the 13th century
25.
Thomas Harriot
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Thomas Harriot — or spelled Harriott, Hariot, or Heriot — was an English astronomer, mathematician, ethnographer, and translator. He is sometimes credited with the introduction of the potato to the British Isles, Harriot was the first person to make a drawing of the Moon through a telescope, on 26 July 1609, over four months before Galileo. After graduating from St Mary Hall, Oxford, Harriot travelled to the Americas, accompanying the 1585 expedition to Roanoke island funded by Sir Walter Raleigh and led by Sir Ralph Lane. Harriot was a member of the venture, having translated and learned the Carolina Algonquian language from two Native Americans, Wanchese and Manteo. On his return to England he worked for the 9th Earl of Northumberland, at the Earls house, he became a prolific mathematician and astronomer to whom the theory of refraction is attributed. Born in 1560 in Oxford, England, Thomas Harriot attended St Mary Hall and his name appears in the halls registry dating from 1577. Prior to his expedition with Raleigh, Harriot wrote a treatise on navigation, in addition, he made efforts to communicate with Manteo and Wanchese, two Native Americans who had been brought to England. Harriot devised an alphabet to transcribe their Carolina Algonquian language. Harriot and Manteo spent many days in one company, Harriot interrogated Manteo closely about life in the New World. In addition, he recorded the sense of awe with which the Native Americans viewed European technology, Many things they sawe with us. as mathematical instruments, as the only Englishman who had learned Algonkin prior to the voyage, Harriot was vital to the success of the expedition. His account of the voyage, named A Briefe and True Report of the New Found Land of Virginia, was published in 1588. The True Report contains an account of the Native American population encountered by the expedition, it proved very influential upon later English explorers. He wrote, Whereby it may be hoped, if means of government be used, that they may in short time be brought to civility. At the same time, his views of Native Americans industry and capacity to learn were later largely ignored in favour of the parts of the True Report about extractable minerals and resources. As a scientific adviser during the voyage, Harriot was asked by Raleigh to find the most efficient way to stack cannonballs on the deck of the ship. His ensuing theory about the close-packing of spheres shows a resemblance to atomism and modern atomic theory. His correspondence about optics with Johannes Kepler, in which he described some of his ideas, Harriott was employed for many years by Henry Percy, 9th Earl of Northumberland, with whom he resided at Syon House, which was run by Henry Percys cousin Thomas Percy. Harriot himself was interrogated and briefly imprisoned but was soon released, Walter Warner, Robert Hues, William Lower, and other scientists were present around the Earl of Northumberlands mansion as they worked for him and assisted in the teaching of the familys children
26.
Johann Faulhaber
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Johann Faulhaber was a German mathematician. Born in Ulm, Faulhaber was a weaver who later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen, besides his work on the fortifications of cities, Faulhaber built water wheels in his home town and geometrical instruments for the military. Faulhaber made the first publication of Henry Briggss Logarithm in Germany, faulhabers major contribution was in calculating the sums of powers of integers. Jacob Bernoulli makes references to Faulhaber in his Ars Conjectandi, academia Algebrae, darinnen die miraculosische Inventiones, zu den höchsten Cossen weiters continuirt und profitiert werden
27.
Pierre de Fermat
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He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermats principle for light propagation and his Fermats Last Theorem in number theory, Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a leather merchant. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth, there is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban. He attended the University of Orléans from 1623 and received a bachelor in law in 1626. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apolloniuss De Locis Planis to one of the mathematicians there, there he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France and he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat, fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often little or no proof of his theorems. In some of these letters to his friends he explored many of the ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time and this naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, The basis of Fermats mathematics was the classical Greek treatises combined with Vietas new algebraic methods, Fermats pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, in these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of power functions. With his method, he was able to reduce this evaluation to the sum of geometric series, the resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pells equation, perfect numbers, amicable numbers and it was while researching perfect numbers that he discovered Fermats little theorem. Fermat developed the two-square theorem, and the polygonal number theorem, although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived
28.
Blaise Pascal
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Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a prodigy who was educated by his father. Pascal also wrote in defence of the scientific method, in 1642, while still a teenager, he started some pioneering work on calculating machines. After three years of effort and 50 prototypes, he built 20 finished machines over the following 10 years, following Galileo Galilei and Torricelli, in 1647, he rebutted Aristotles followers who insisted that nature abhors a vacuum. Pascals results caused many disputes before being accepted, in 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. Following a religious experience in late 1654, he began writing works on philosophy. His two most famous works date from this period, the Lettres provinciales and the Pensées, the set in the conflict between Jansenists and Jesuits. In that year, he wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids, Pascal had poor health, especially after the age of 18, and he died just two months after his 39th birthday. Pascal was born in Clermont-Ferrand, which is in Frances Auvergne region and he lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal, who also had an interest in science and mathematics, was a local judge, Pascal had two sisters, the younger Jacqueline and the elder Gilberte. In 1631, five years after the death of his wife, the newly arrived family soon hired Louise Delfault, a maid who eventually became an instrumental member of the family. Étienne, who never remarried, decided that he alone would educate his children, for they all showed extraordinary intellectual ability, the young Pascal showed an amazing aptitude for mathematics and science. Particularly of interest to Pascal was a work of Desargues on conic sections and it states that if a hexagon is inscribed in a circle then the three intersection points of opposite sides lie on a line. Pascals work was so precocious that Descartes was convinced that Pascals father had written it, in France at that time offices and positions could be—and were—bought and sold. In 1631 Étienne sold his position as president of the Cour des Aides for 65,665 livres. The money was invested in a government bond which provided, if not a lavish, then certainly a comfortable income which allowed the Pascal family to move to, but in 1638 Richelieu, desperate for money to carry on the Thirty Years War, defaulted on the governments bonds. Suddenly Étienne Pascals worth had dropped from nearly 66,000 livres to less than 7,300 and it was only when Jacqueline performed well in a childrens play with Richelieu in attendance that Étienne was pardoned
29.
Abraham de Moivre
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Abraham de Moivre was a French mathematician known for de Moivres formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmond Halley, even though he faced religious persecution he remained a steadfast Christian throughout his life. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux, De Moivre wrote a book on probability theory, The Doctrine of Chances, said to have been prized by gamblers. De Moivre first discovered Binets formula, the expression for Fibonacci numbers linking the nth power of the golden ratio φ to the nth Fibonacci number. He also was the first to postulate the central limit theorem, Abraham de Moivre was born in Vitry-le-François in Champagne on May 26,1667. His father, Daniel de Moivre, was a surgeon who believed in the value of education, though Abraham de Moivres parents were Protestant, he first attended Christian Brothers Catholic school in Vitry, which was unusually tolerant given religious tensions in France at the time. When he was eleven, his parents sent him to the Protestant Academy at Sedan, the Protestant Academy of Sedan had been founded in 1579 at the initiative of Françoise de Bourbon, the widow of Henri-Robert de la Marck. In 1682 the Protestant Academy at Sedan was suppressed, and de Moivre enrolled to study logic at Saumur for two years, in 1684, de Moivre moved to Paris to study physics, and for the first time had formal mathematics training with private lessons from Jacques Ozanam. It forbade Protestant worship and required all children be baptized by Catholic priests. De Moivre was sent to the Prieure de Saint-Martin, a school that the authorities sent Protestant children to for indoctrination into Catholicism, by the time he arrived in London, de Moivre was a competent mathematician with a good knowledge of many of the standard texts. To make a living, de Moivre became a tutor of mathematics. De Moivre continued his studies of mathematics after visiting the Earl of Devonshire and seeing Newtons recent book, looking through the book, he realized that it was far deeper than the books that he had studied previously, and he became determined to read and understand it. By 1692, de Moivre became friends with Edmond Halley and soon after with Isaac Newton himself, in 1695, Halley communicated de Moivres first mathematics paper, which arose from his study of fluxions in the Principia Mathematica, to the Royal Society. This paper was published in the Philosophical Transactions that same year, shortly after publishing this paper, de Moivre also generalized Newtons noteworthy binomial theorem into the multinomial theorem. The Royal Society became apprised of this method in 1697, after de Moivre had been accepted, Halley encouraged him to turn his attention to astronomy. The mathematician Johann Bernoulli proved this formula in 1710, at least a part of the reason was a bias against his French origins. In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a set up by the society. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, the full details of the controversy can be found in the Leibniz and Newton calculus controversy article
30.
Carl Gustav Jacob Jacobi
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Carl Gustav Jacob Jacobi was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory. His name is written as Carolus Gustavus Iacobus Iacobi in his Latin books. Jacobi was the first Jewish mathematician to be appointed professor at a German university, Jacobi was born of Ashkenazi Jewish parentage in Potsdam on 10 December 1804. He was the second of four children of banker Simon Jacobi and his elder brother Moritz von Jacobi would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages, in 1816, the twelve-year-old Jacobi went to the Potsdam Gymnasium, where students were being taught classical languages, German history as well as mathematics. As a result of the education received from his uncle, as well as his own remarkable abilities. However, as the University was not accepting students younger than 16 years old and he used this time to advance his knowledge, showing interest in all subjects, including Latin and Greek, philology, history and mathematics. During this period he made the first attempts at research trying to solve the quintic equation by radicals. In 1821 Jacobi went to study at the Berlin University, where initially he divided his attention between his passions for philology and mathematics, in philology he participated in the seminars of Böckh, drawing the professors attention with his talent. Jacobi did not follow a lot of mathematics classes at the University, however, he continued with his private study of the more advanced works of Euler, Lagrange and Laplace. By 1823 he understood that he needed to make a decision between his interests and he chose to devote all his attention to mathematics. In the same year he qualified to teach secondary school and was offered a position at the Joachimsthal Gymnasium in Berlin. Jacobi decided instead to continue to work towards a University position, in 1825 he obtained the degree of Doctor of Philosophy with a dissertation on the partial fraction decomposition of rational fractions defended before a commission led by Enno Dirksen. He followed immediately with his Habilitation and at the time converted to Christianity. Now qualifying for teaching University classes, the 21-year-old Jacobi lectured in 1825/26 on the theory of curves and surfaces at the University of Berlin, in 1827 he became a professor and in 1829, a tenured professor of mathematics at Königsberg University, and held the chair until 1842. Jacobi suffered a breakdown from overwork in 1843 and he then visited Italy for a few months to regain his health. On his return he moved to Berlin, where he lived as a royal pensioner until his death, during the Revolution of 1848 Jacobi was politically involved and unsuccessfully presented his parliamentary candidature on behalf of a Liberal club. This led, after the suppression of the revolution, to his royal grant being cut off – but his fame, in 1836, he had been elected a foreign member of the Royal Swedish Academy of Sciences
31.
Gottfried Wilhelm Leibniz
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Leibnizs notation has been widely used ever since it was published. It was only in the 20th century that his Law of Continuity and he became one of the most prolific inventors in the field of mechanical calculators. He also refined the number system, which is the foundation of virtually all digital computers. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism and he wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibnizs contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and he wrote in several languages, but primarily in Latin, French, and German. There is no complete gathering of the writings of Leibniz in English, Gottfried Leibniz was born on July 1,1646, toward the end of the Thirty Years War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his journal,21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, in English, On Sunday 21 June 1646, my son Gottfried Wilhelm is born into the world a quarter after six in the evening, in Aquarius. Leibniz was baptized on July 3 of that year at St. Nicholas Church, Leipzig and his father died when he was six and a half years old, and from that point on he was raised by his mother. Her teachings influenced Leibnizs philosophical thoughts in his later life, Leibnizs father had been a Professor of Moral Philosophy at the University of Leipzig, and the boy later inherited his fathers personal library. He was given access to it from the age of seven. Access to his fathers library, largely written in Latin, also led to his proficiency in the Latin language and he also composed 300 hexameters of Latin verse, in a single morning, for a special event at school at the age of 13. In April 1661 he enrolled in his fathers former university at age 15 and he defended his Disputatio Metaphysica de Principio Individui, which addressed the principle of individuation, on June 9,1663. Leibniz earned his masters degree in Philosophy on February 7,1664, after one year of legal studies, he was awarded his bachelors degree in Law on September 28,1665. His dissertation was titled De conditionibus, in early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria, the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666. His next goal was to earn his license and Doctorate in Law, in 1666, the University of Leipzig turned down Leibnizs doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth. Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, the title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure. Leibniz earned his license to practice law and his Doctorate in Law in November 1666 and he next declined the offer of an academic appointment at Altdorf, saying that my thoughts were turned in an entirely different direction
32.
Summation
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In mathematics, summation is the addition of a sequence of numbers, the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a sum, prefix sum. The numbers to be summed may be integers, rational numbers, real numbers, besides numbers, other types of values can be added as well, vectors, matrices, polynomials and, in general, elements of any additive group. For finite sequences of elements, summation always produces a well-defined sum. The summation of a sequence of values is called a series. A value of such a series may often be defined by means of a limit, another notion involving limits of finite sums is integration. The summation of the sequence is an expression whose value is the sum of each of the members of the sequence, in the example,1 +2 +4 +2 =9. Addition is also commutative, so permuting the terms of a sequence does not change its sum. There is no notation for the summation of such explicit sequences. If, however, the terms of the sequence are given by a pattern, possibly of variable length. For the summation of the sequence of integers from 1 to 100. In this case, the reader can guess the pattern. However, for more complicated patterns, one needs to be precise about the used to find successive terms. Using this sigma notation the above summation is written as, ∑ i =1100 i, the value of this summation is 5050. It can be found without performing 99 additions, since it can be shown that ∑ i =1 n i = n 2 for all natural numbers n, more generally, formulae exist for many summations of terms following a regular pattern. By contrast, summation as discussed in this article is called definite summation, when it is necessary to clarify that numbers are added with their signs, the term algebraic sum is used. Mathematical notation uses a symbol that compactly represents summation of many terms, the summation symbol, ∑. The i = m under the symbol means that the index i starts out equal to m
33.
Trigamma function
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In mathematics, the trigamma function, denoted ψ1, is the second of the polygamma functions, and is defined by ψ1 = d 2 d z 2 ln Γ. It follows from definition that ψ1 = d d z ψ where ψ is the digamma function. It may also be defined as the sum of the series ψ1 = ∑ n =0 ∞12, note that the last two formulae are valid when 1 − z is not a natural number. e. The Bernoulli numbers of the second kind, there are no roots on the real axis of ψ1, but there exist infinitely many pairs of roots zn, zn for Re z <0. Each such pair of roots approaches Re zn = −n + 1/2 quickly, +0.5978119. i and z2 = −1.4455692. +0.6992608. i are the first two roots with Im >0, gamma function Digamma function Polygamma function Catalans constant Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York. See section §6.4 Eric W. Weisstein, trigamma Function -- from MathWorld--A Wolfram Web Resource
34.
Gamma function
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In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if n is an integer, Γ =. The gamma function is defined for all numbers except the non-positive integers. The gamma function can be seen as a solution to the interpolation problem. The simple formula for the factorial, x. =1 ×2 × … × x, a good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers, the gamma function is the most useful solution in practice, being analytic, and it can be characterized in several ways. The Bohr–Mollerup theorem proves that these properties, together with the assumption that f be logarithmically convex, uniquely determine f for positive, from there, the gamma function can be extended to all real and complex values by using the unique analytic continuation of f. Also see Eulers infinite product definition below where the properties f =1 and f = x f together with the requirement that limn→+∞. nx / f =1 uniquely define the same function. The notation Γ is due to Legendre, if the real part of the complex number z is positive, then the integral Γ = ∫0 ∞ x z −1 e − x d x converges absolutely, and is known as the Euler integral of the second kind. The identity Γ = Γ z can be used to extend the integral formulation for Γ to a meromorphic function defined for all complex numbers z. It is this version that is commonly referred to as the gamma function. When seeking to approximate z. for a number z, it turns out that it is effective to first compute n. for some large integer n. And then use the relation m. = m. backwards n times. Furthermore, this approximation is exact in the limit as n goes to infinity, specifically, for a fixed integer m, it is the case that lim n → + ∞ n. m. =1, and we can ask that the formula is obeyed when the arbitrary integer m is replaced by an arbitrary complex number z lim n → + ∞ n. z. =1. Multiplying both sides by z. gives z. = lim n → + ∞ n. z, Z = lim n → + ∞1 ⋯ n ⋯ z = ∏ n =1 + ∞. Similarly for the function, the definition as an infinite product due to Euler is valid for all complex numbers z except the non-positive integers. By this construction, the function is the unique function that simultaneously satisfies Γ =1, Γ = z Γ for all complex numbers z except the non-positive integers
35.
Stirling formula
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In mathematics, Stirlings approximation is an approximation for factorials. It is a good-quality approximation, leading to accurate results even for small values of n and it is named after James Stirling, though it was first stated by Abraham de Moivre. The formula as used in applications is ln n. = n ln n − n + O or, for instance in the lower bound for comparison sorting. N n +12 e − n is always between √2π =2.5066. and e =2.71828, the formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n. one considers its natural logarithm as this is a slowly varying function, take limits to find that lim n → ∞ =1 − ∑ k =2 m k B k k + lim n → ∞ R m, n. = n ln +12 ln n + y + ∑ k =2 m k B k k n k −1 + O. Taking the exponential of both sides, and choosing any positive m, we get a formula involving an unknown quantity ey. For m =1, the formula is n, the quantity ey can be found by taking the limit on both sides as n tends to infinity and using Wallis product, which shows that ey = √2π. Therefore, we get Stirlings formula, n. =2 π n n, the formula may also be obtained by repeated integration by parts, and the leading term can be found through Laplaces method. Stirlings formula, without the factor √2πn that is irrelevant in applications. = ∑ j =1 n ln j with an integral, an alternative formula for n. using the gamma function is n. = ∫0 ∞ x n e − x d x. Rewriting and changing variables x = ny one gets n. = ∫0 ∞ e n ln x − x d x = e n ln n n ∫0 ∞ e n d y. Applying Laplaces method we have, ∫0 ∞ e n d y ∼2 π n e − n which recovers the Stirlings formula, ∼ e n ln n n 2 π n e − n =2 π n n. In fact further corrections can also be obtained using Laplaces method, for example, computing two-order expansion using Laplaces method yields ∫0 ∞ e n d y ∼2 π n e − n and gives Stirlings formula to two orders, n. ∼ e n ln n n 2 π n e − n =2 π n n, Stirlings formula is in fact the first approximation to the following series, n. ∼2 π n n. An explicit formula for the coefficients in this series was given by G. Nemes, the first graph in this section shows the relative error vs. n, for 1 through all 5 terms listed above. As n → ∞, the error in the series is asymptotically equal to the first omitted term
36.
Big-O notation
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Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, in computer science, big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows. Big O notation characterizes functions according to their rates, different functions with the same growth rate may be represented using the same O notation. The letter O is used because the rate of a function is also referred to as order of the function. A description of a function in terms of big O notation usually only provides a bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, Big O notation is also used in many other fields to provide similar estimates. Let f and g be two functions defined on some subset of the real numbers. That is, f = O if and only if there exists a real number M. In many contexts, the assumption that we are interested in the rate as the variable x goes to infinity is left unstated. If f is a product of several factors, any constants can be omitted, for example, let f = 6x4 − 2x3 +5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms, 6x4, −2x3, and 5, of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the rule, 6x4 is a product of 6. Omitting this factor results in the simplified form x4, thus, we say that f is a big-oh of. Mathematically, we can write f = O, one may confirm this calculation using the formal definition, let f = 6x4 − 2x3 +5 and g = x4. Applying the formal definition from above, the statement that f = O is equivalent to its expansion, | f | ≤ M | x 4 | for some choice of x0 and M. To prove this, let x0 =1 and M =13, Big O notation has two main areas of application. In mathematics, it is used to describe how closely a finite series approximates a given function. In computer science, it is useful in the analysis of algorithms, in both applications, the function g appearing within the O is typically chosen to be as simple as possible, omitting constant factors and lower order terms
37.
Chinese remainder theorem
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This theorem has this name because it is a theorem about remainders and was first discovered in the 3rd century AD by the Chinese mathematician Sunzi in Sunzi Suanjing. The Chinese remainder theorem is true over every principal ideal domain and it has been generalized to any commutative ring, with a formulation involving ideals. What amounts to an algorithm for solving this problem was described by Aryabhata, special cases of the Chinese remainder theorem were also known to Brahmagupta, and appear in Fibonaccis Liber Abaci. The result was later generalized with a solution called Dayanshu in Qin Jiushaos 1247 Mathematical Treatise in Nine Sections. The notion of congruences was first introduced and used by Gauss in his Disquisitiones Arithmeticae of 1801, Gauss introduces a procedure for solving the problem that had already been used by Euler but was in fact an ancient method that had appeared several times. Nk be integers greater than 1, which are often called moduli or divisors, Let us denote by N the product of the ni. The Chinese remainder theorem asserts that if the ni are pairwise coprime and this may be restated as follows in term of congruences, If the ni are pairwise coprime, and if a1. Ak are any integers, then there exists an x such that x ≡ a 1 ⋮ x ≡ a k. This means that for doing a sequence of operations in Z / N Z, one may do the same computation independently in each Z / n i Z. This may be faster than the direct computation if N. This is widely used, under the name multi-modular computation, for linear algebra over the integers or the rational numbers, the theorem can also be restated in the language of combinatorics as the fact that the infinite arithmetic progressions of integers form a Helly family. The existence and the uniqueness of the solution may be proven independently, however, the first proof of existence, given below, uses this uniqueness. Suppose that x and y are both solutions to all the congruences, as x and y give the same remainder, when divided by ni, their difference x − y is a multiple of each ni. As the ni are pairwise coprime, their product N divides also x − y, If x and y are supposed to be non negative and less than N, then their difference may be a multiple of N only if x = y. The map x ↦ maps congruence classes modulo N to sequences of congruence classes modulo ni, the proof of uniqueness shows that this map is injective. As the domain and the codomain of this map have the number of elements, the map is also surjective. This proof is simple but does not provide any direct way for computing a solution. Moreover, it cannot be generalized to situations where the following proof can
38.
Computational complexity theory
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A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity measures are used, such as the amount of communication, the number of gates in a circuit. One of the roles of computational complexity theory is to determine the limits on what computers can. Closely related fields in computer science are analysis of algorithms. More precisely, computational complexity theory tries to classify problems that can or cannot be solved with appropriately restricted resources, a computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a problem is referred to as a problem instance. In computational complexity theory, a problem refers to the question to be solved. In contrast, an instance of this problem is a rather concrete utterance, for example, consider the problem of primality testing. The instance is a number and the solution is yes if the number is prime, stated another way, the instance is a particular input to the problem, and the solution is the output corresponding to the given input. For this reason, complexity theory addresses computational problems and not particular problem instances, when considering computational problems, a problem instance is a string over an alphabet. Usually, the alphabet is taken to be the binary alphabet, as in a real-world computer, mathematical objects other than bitstrings must be suitably encoded. For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices and this can be achieved by ensuring that different representations can be transformed into each other efficiently. Decision problems are one of the objects of study in computational complexity theory. A decision problem is a type of computational problem whose answer is either yes or no. A decision problem can be viewed as a language, where the members of the language are instances whose output is yes. The objective is to decide, with the aid of an algorithm, if the algorithm deciding this problem returns the answer yes, the algorithm is said to accept the input string, otherwise it is said to reject the input. An example of a problem is the following
39.
SageMath
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SageMath is mathematical software with features covering many aspects of mathematics, including algebra, combinatorics, numerical mathematics, number theory, and calculus. The originator and leader of the SageMath project, William Stein, is a mathematician at the University of Washington, SageMath uses a Python-like syntax, supporting procedural, functional and object-oriented constructs. Features of SageMath include, A browser-based notebook for review and re-use of previous inputs and outputs, including graphics, compatible with Firefox, Opera, Konqueror, Google Chrome and Safari. Notebooks can be accessed locally or remotely and the connection can be secured with HTTPS, interfacing this way is documented officially to Sage. William Stein realized when designing Sage that there were many open-source mathematics software packages written in different languages, namely C, C++, Common Lisp, Fortran. Rather than reinventing the wheel, Sage integrates many specialized mathematics software packages into a common interface, however, Sage contains hundreds of thousands of unique lines of code adding new functions and creating the interface between its components. SageMath uses both students and professionals for development, the development of SageMath is supported by both volunteer work and grants. However, it was not until 2016 that the first full-time Sage developer was hired, only the major releases are listed below. SageMath practices the release early, release often concept, with releases every few weeks or months, in total, there have been over 300 releases, although their frequency has decreased. 2007, first prize in the software division of Les Trophées du Libre. 2012, one of the selected for the Google Summer of Code. SageMath has been cited in a variety of publications, both binaries and source code are available for SageMath from the download page. Cython can increase the speed of SageMath programs, as the Python code is converted into C, SageMath is free software, distributed under the terms of the GNU General Public License version 3. SageMath is available in many ways, The source code can be downloaded from the downloads page, although not recommended for end users, development releases of SageMath are also available. Many Linux distributions also include SageMath in their repositories, binaries can be downloaded for Linux, macOS and Solaris. A live CD containing a bootable Linux operating system is also available and this allows usage of SageMath without Linux installation. Users could have used a version of SageMath at sagenb. org. Users can use a single cell version of SageMath at sagecell. sagemath. org or embed a single SageMath cell into any web page
40.
Mathematica
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Wolfram Mathematica is a mathematical symbolic computation program, sometimes termed a computer algebra system or program, used in many scientific, engineering, mathematical, and computing fields. It was conceived by Stephen Wolfram and is developed by Wolfram Research of Champaign, the Wolfram Language is the programming language used in Mathematica. The kernel interprets expressions and returns result expressions, all content and formatting can be generated algorithmically or edited interactively. Standard word processing capabilities are supported, including real-time multi-lingual spell-checking, documents can be structured using a hierarchy of cells, which allow for outlining and sectioning of a document and support automatic numbering index creation. Documents can be presented in an environment for presentations. Notebooks and their contents are represented as Mathematica expressions that can be created, modified or analyzed by Mathematica programs or converted to other formats, the front end includes development tools such as a debugger, input completion, and automatic syntax highlighting. Among the alternative front ends is the Wolfram Workbench, an Eclipse based integrated development environment and it provides project-based code development tools for Mathematica, including revision management, debugging, profiling, and testing. There is a plugin for IntelliJ IDEA based IDEs to work with Wolfram Language code which in addition to syntax highlighting can analyse and auto-complete local variables, the Mathematica Kernel also includes a command line front end. Other interfaces include JMath, based on GNU readline and MASH which runs self-contained Mathematica programs from the UNIX command line, version 5.2 added automatic multi-threading when computations are performed on multi-core computers. This release included CPU specific optimized libraries, in addition Mathematica is supported by third party specialist acceleration hardware such as ClearSpeed. Support for CUDA and OpenCL GPU hardware was added in 2010, also, since version 8 it can generate C code, which is automatically compiled by a system C compiler, such as GCC or Microsoft Visual Studio. A free-of-charge version, Wolfram CDF Player, is provided for running Mathematica programs that have saved in the Computable Document Format. It can also view standard Mathematica files, but not run them and it includes plugins for common web browsers on Windows and Macintosh. WebMathematica allows a web browser to act as a front end to a remote Mathematica server and it is designed to allow a user written application to be remotely accessed via a browser on any platform. It may not be used to full access to Mathematica. Due to bandwidth limitations interactive 3D graphics is not fully supported within a web browser, Wolfram Language code can be converted to C code or to an automatically generated DLL. Wolfram Language code can be run on a Wolfram cloud service as a web-app or as an API either on Wolfram-hosted servers or in an installation of the Wolfram Enterprise Private Cloud. Communication with other applications occurs through a protocol called Wolfram Symbolic Transfer Protocol and it allows communication between the Wolfram Mathematica kernel and front-end, and also provides a general interface between the kernel and other applications
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Scientific notation
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Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations, on scientific calculators it is known as SCI display mode. In scientific notation all numbers are written in the form m × 10n, where the exponent n is an integer, however, the term mantissa may cause confusion because it is the name of the fractional part of the common logarithm. If the number is then a minus sign precedes m. In normalized notation, the exponent is chosen so that the value of the coefficient is at least one. Decimal floating point is an arithmetic system closely related to scientific notation. Any given integer can be written in the form m×10^n in many ways, in normalized scientific notation, the exponent n is chosen so that the absolute value of m remains at least one but less than ten. Thus 350 is written as 3. 5×102 and this form allows easy comparison of numbers, as the exponent n gives the numbers order of magnitude. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1, the 10 and exponent are often omitted when the exponent is 0. Normalized scientific form is the form of expression of large numbers in many fields, unless an unnormalized form. Normalized scientific notation is often called exponential notation—although the latter term is general and also applies when m is not restricted to the range 1 to 10. Engineering notation differs from normalized scientific notation in that the exponent n is restricted to multiples of 3, consequently, the absolute value of m is in the range 1 ≤ |m| <1000, rather than 1 ≤ |m| <10. Though similar in concept, engineering notation is rarely called scientific notation, engineering notation allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. A significant figure is a digit in a number that adds to its precision and this includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore,1,230,400 usually has five significant figures,1,2,3,0, and 4, when a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the place holding zeroes are no longer required, thus 1,230,400 would become 1.2304 ×106. However, there is also the possibility that the number may be known to six or more significant figures, thus, an additional advantage of scientific notation is that the number of significant figures is clearer
42.
Antiderivative
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In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F ′ = f, the process of solving for antiderivatives is called antidifferentiation and its opposite operation is called differentiation, which is the process of finding a derivative. The discrete equivalent of the notion of antiderivative is antidifference, the function F = x3/3 is an antiderivative of f = x2, as the derivative of x3/3 is x2. As the derivative of a constant is zero, x2 will have a number of antiderivatives, such as x3/3, x3/3 +1, x3/3 -2. Thus, all the antiderivatives of x2 can be obtained by changing the value of C in F = x3/3 + C, essentially, the graphs of antiderivatives of a given function are vertical translations of each other, each graphs vertical location depending upon the value of C. In physics, the integration of acceleration yields velocity plus a constant, the constant is the initial velocity term that would be lost upon taking the derivative of velocity because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion, C is called the arbitrary constant of integration. If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals. For instance F = { −1 x + C1 x <0 −1 x + C2 x >0 is the most general antiderivative of f =1 / x 2 on its natural domain ∪. Every continuous function f has an antiderivative, and one antiderivative F is given by the integral of f with variable upper boundary. Varying the lower boundary produces other antiderivatives and this is another formulation of the fundamental theorem of calculus. There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Examples of these are ∫ e − x 2 d x, ∫ sin x 2 d x, ∫ sin x x d x, ∫1 ln x d x, ∫ x x d x. From left to right, the first four are the function, the Fresnel function, the trigonometric integral. See also Differential Galois theory for a detailed discussion. Finding antiderivatives of elementary functions is often harder than finding their derivatives. For some elementary functions, it is impossible to find an antiderivative in terms of elementary functions. See the articles on elementary functions and nonelementary integral for further information, integrals which have already been derived can be looked up in a table of integrals
43.
Fundamental theorem of calculus
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The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the functions integral. This part of the guarantees the existence of antiderivatives for continuous functions. This part of the theorem has key practical applications because it simplifies the computation of definite integrals. The fundamental theorem of calculus relates differentiation and integration, showing that two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration, the first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory. Isaac Barrow proved a more generalized version of the theorem, while his student Isaac Newton completed the development of the mathematical theory. Gottfried Leibniz systematized the knowledge into a calculus for infinitesimal quantities, for a continuous function y = f whose graph is plotted as a curve, each value of x has a corresponding area function A, representing the area beneath the curve between 0 and x. The function A may not be known, but it is given that it represents the area under the curve. The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x, in other words, the area of this “sliver” would be A − A. There is another way to estimate the area of this same sliver, as shown in the accompanying figure, h is multiplied by f to find the area of a rectangle that is approximately the same size as this sliver. So, A − A ≈ f h In fact, this becomes a perfect equality if we add the red portion of the excess area shown in the diagram. So, A − A = f h + Rearranging terms, as h approaches 0 in the limit, the last fraction can be shown to go to zero. This is true because the area of the red portion of region is less than or equal to the area of the tiny black-bordered rectangle. More precisely, | f − A − A h | = | Red Excess | h ≤ h | f − f | h = | f − f |, by the continuity of f, the latter expression tends to zero as h does. Therefore, the left-hand side tends to zero as h does and that is, the derivative of the area function A exists and is the original function f, so, the area function is simply an antiderivative of the original function. Computing the derivative of a function and “finding the area” under its curve are opposite operations and this is the crux of the Fundamental Theorem of Calculus. Intuitively, the theorem states that the sum of infinitesimal changes in a quantity over time adds up to the net change in the quantity
44.
Digamma function
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In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function, ψ = d d x ln = Γ ′ Γ. It is the first of the polygamma functions, the digamma function is often denoted as ψ0, ψ0 or Ϝ. The gamma function obeys the equation Γ = z Γ and this may be written as ψ = − γ + ∫01 d x which follows from Leonhard Eulers integral formula for the harmonic numbers. e. ∑ n =0 ∞ u n = ∑ n =0 ∞ p q, where p and q are polynomials of n. Performing partial fraction on un in the field, in the case when all roots of q are simple roots. For the series to converge, lim n → ∞ n u n =0, otherwise the series will be greater than the harmonic series, the digamma has a rational zeta series, given by the Taylor series at z =1. This is ψ = − γ − ∑ k =1 ∞ ζ k, here, ζ is the Riemann zeta function. This series is derived from the corresponding Taylors series for the Hurwitz zeta function. The Newton series for the digamma follows from Eulers integral formula, the digamma function satisfies a reflection formula similar to that of the gamma function, ψ − ψ = π cot π x The digamma function satisfies the recurrence relation ψ = ψ +1 x. Thus, it can be said to telescope 1 / x and this satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula ψ = H n −1 − γ where γ is the Euler–Mascheroni constant. More generally, one has ψ = − γ + ∑ k =1 ∞, actually, ψ is the only solution of the functional equation F = F +1 x that is monotone on ℝ+ and satisfies F = −γ. This fact follows immediately from the uniqueness of the Γ function given its recurrence equation and this implies the useful difference equation, ψ − ψ = ∑ k =0 N −11 x + k There are numerous finite summation formulas for the digamma function. Although the infinite sum converges for no x, this becomes more accurate for larger values of x. To compute ψ for small x, the recurrence relation ψ =1 x + ψ can be used to shift the value of x to a higher value. Beal suggests using the recurrence to shift x to a value greater than 6 and then applying the above expansion with terms above x14 cut off. As x goes to infinity, ψ gets arbitrarily close to both ln and ln x. Going down from x +1 to x, ψ decreases by 1 / x, ln decreases by ln /, which is more than 1 / x, and ln x decreases by ln, which is less than 1 / x. From this we see that for any positive x greater than 1/2, ψ ∈ or, for any positive x, the exponential exp ψ is approximately x − 1/2 for large x, but gets closer to x at small x, approaching 0 at x =0
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Polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
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Binomial coefficient
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In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a coefficient is indexed by a pair of integers n ≥ k ≥0 and is written. It is the coefficient of the xk term in the expansion of the binomial power n. The value of the coefficient is given by the expression n. k, arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascals triangle. The properties of binomial coefficients have led to extending the definition to beyond the case of integers n ≥ k ≥0. Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier, the earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingalas Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī, alternative notations include C, nCk, nCk, Ckn, Cnk, and Cn, k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display, in this form the binomial coefficients are easily compared to k-permutations of n, written as P, etc. For natural numbers n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of n, the same coefficient also occurs in the binomial formula, which explains the name binomial coefficient. This shows in particular that is a number for any natural numbers n and k. Most of these interpretations are easily seen to be equivalent to counting k-combinations, several methods exist to compute the value of without actually expanding a binomial power or counting k-combinations. It also follows from tracing the contributions to Xk in n−1, as there is zero Xn+1 or X−1 in n, one might extend the definition beyond the above boundaries to include =0 when either k > n or k <0. This recursive formula then allows the construction of Pascals triangle, surrounded by white spaces where the zeros, or the trivial coefficients, a more efficient method to compute individual binomial coefficients is given by the formula = n k _ k. = n ⋯ k ⋯1 = ∏ i =1 k n +1 − i i and this formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, the denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. Due to the symmetry of the binomial coefficient with regard to k and n−k, calculation may be optimised by setting the limit of the product above to the smaller of k. This formula follows from the formula above by multiplying numerator and denominator by. As a consequence it involves many factors common to numerator and denominator and it is less practical for explicit computation unless common factors are first cancelled
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Triangular number
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime
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Square pyramidal number
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In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid. The first few square pyramidal numbers are,1,5,14,30,55,91,140,204,285,385,506,650,819 and this is a special case of Faulhabers formula, and may be proved by a mathematical induction. An equivalent formula is given in Fibonaccis Liber Abaci, in modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial L of a polyhedron P is a polynomial that counts the number of points in a copy of P that is expanded by multiplying all its coordinates by the number t. The Ehrhart polynomial of a pyramid base is a unit square with integer coordinates. The square pyramidal numbers can also be expressed as sums of binomial coefficients, the smaller tetrahedral number represents 1 +3 +6 + ⋯ + T and the larger 1 +3 +6 + ⋯ + T. Offsetting the larger and adding, we arrive at 1,1 +3,3 +6 …, Square pyramidal numbers are also related to tetrahedral numbers in a different way, P n =14. The sum of two square pyramidal numbers is an octahedral number. Augmenting a pyramid whose base edge has n balls by adding to one of its faces a tetrahedron whose base edge has n −1 balls produces a triangular prism. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism and this geometric dissection leads to another relation, P n = n −. Besides 1, there is one other number that is both a square and a pyramid number,4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918, in the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes. Also, P n = − which is the difference of two pentatope numbers and this can be seen by expanding, n − n = n = n and dividing through by 24. A common mathematical puzzle involves finding the number of squares in a n by n square grid. This number can be derived as follows, The number of 1 ×1 boxes found in the grid is n2, the number of 2 ×2 boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of 2 ×2 boxes, the number of k × k boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of k × k boxes and it follows that the number of squares in an n × n square grid is, n 2 +2 +2 +2 + … +12 = n 6
49.
Trigonometric functions
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
50.
Cotangent
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
51.
Hyperbolic tangent
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In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The inverse hyperbolic functions are the hyperbolic sine arsinh and so on. Just as the form a circle with a unit radius. The hyperbolic functions take a real argument called a hyperbolic angle, the size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a triangle covering this sector. Laplaces equations are important in areas of physics, including electromagnetic theory, heat transfer, fluid dynamics. In complex analysis, the hyperbolic functions arise as the parts of sine and cosine. When considered defined by a variable, the hyperbolic functions are rational functions of exponentials. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati, Riccati used Sc. and Cc. to refer to circular functions and Sh. and Ch. to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today, the abbreviations sh and ch are still used in some other languages, like French and Russian. The hyperbolic functions are, Hyperbolic sine, sinh x = e x − e − x 2 = e 2 x −12 e x =1 − e −2 x 2 e − x. Hyperbolic cosine, cosh x = e x + e − x 2 = e 2 x +12 e x =1 + e −2 x 2 e − x, the complex forms in the definitions above derive from Eulers formula. One also has sech 2 x =1 − tanh 2 x csch 2 x = coth 2 x −1 for the other functions, sinh = sinh 2 = sgn cosh −12 where sgn is the sign function. All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x and e − x, and it is possible to express the above functions as Taylor series, sinh x = x + x 33. + ⋯ = ∑ n =0 ∞ x 2 n +1, the function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh, the function cosh x has a Taylor series expression with only even exponents for x. Thus it is a function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the series expression of the exponential function
52.
Hyperbolic cotangent
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In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The inverse hyperbolic functions are the hyperbolic sine arsinh and so on. Just as the form a circle with a unit radius. The hyperbolic functions take a real argument called a hyperbolic angle, the size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a triangle covering this sector. Laplaces equations are important in areas of physics, including electromagnetic theory, heat transfer, fluid dynamics. In complex analysis, the hyperbolic functions arise as the parts of sine and cosine. When considered defined by a variable, the hyperbolic functions are rational functions of exponentials. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati, Riccati used Sc. and Cc. to refer to circular functions and Sh. and Ch. to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today, the abbreviations sh and ch are still used in some other languages, like French and Russian. The hyperbolic functions are, Hyperbolic sine, sinh x = e x − e − x 2 = e 2 x −12 e x =1 − e −2 x 2 e − x. Hyperbolic cosine, cosh x = e x + e − x 2 = e 2 x +12 e x =1 + e −2 x 2 e − x, the complex forms in the definitions above derive from Eulers formula. One also has sech 2 x =1 − tanh 2 x csch 2 x = coth 2 x −1 for the other functions, sinh = sinh 2 = sgn cosh −12 where sgn is the sign function. All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x and e − x, and it is possible to express the above functions as Taylor series, sinh x = x + x 33. + ⋯ = ∑ n =0 ∞ x 2 n +1, the function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh, the function cosh x has a Taylor series expression with only even exponents for x. Thus it is a function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the series expression of the exponential function
53.
Laurent series
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In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843, karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death. The path of integration γ is counterclockwise around a closed, rectifiable path containing no self-intersections, enclosing c, the expansion for f will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled γ. If we take γ to be a circle | z − c | = ϱ, where r < ϱ < R, the fact that these integrals are unchanged by a deformation of the contour γ is an immediate consequence of Greens theorem. Laurent series with complex coefficients are an important tool in complex analysis, consider for instance the function f = e −1 / x 2 with f =0. As a real function, it is differentiable everywhere, as a complex function however it is not differentiable at x =0. By replacing x by −1/x2 in the series for the exponential function. The graph opposite shows e−1/x2 in black and its Laurent approximations ∑ n =0 N n x −2 n n. for N =1,2,3,4,5,6,7 and 50. As N → ∞, the approximation becomes exact for all x except at the singularity x =0. More generally, Laurent series can be used to express holomorphic functions defined on an annulus, suppose ∑ n = − ∞ ∞ a n n is a given Laurent series with complex coefficients an and a complex center c. Then there exists an inner radius r and outer radius R such that. To say that the Laurent series converges, we mean that both the positive degree power series and the negative degree power series converge, furthermore, this convergence will be uniform on compact sets. Finally, the convergent series defines a holomorphic function ƒ on the open annulus, outside the annulus, the Laurent series diverges. That is, at point of the exterior of A. It is possible that r may be zero or R may be infinite, at the other extreme, its not necessarily true that r is less than R. These radii can be computed as follows, r = lim sup n → ∞ | a − n |1 n 1 R = lim sup n → ∞ | a n |1 n and we take R to be infinite when this latter lim sup is zero
54.
Smooth manifold
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In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas, one may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart, in formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. In other words, where the domains of overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the charts to one another are called transition maps. Differentiability means different things in different contexts including, continuously differentiable, k times differentiable, smooth, furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for theories such as classical mechanics, general relativity. It is possible to develop a calculus for differentiable manifolds and this leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry, the emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen and these ideas found a key application in Einsteins theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces, the widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space. This formalizes the notion of patching together pieces of a space to make a manifold – the manifold produced also contains the data of how it has been patched together, However, different atlases may produce the same manifold, a manifold does not come with a preferred atlas. And, thus, one defines a manifold to be a space as above with an equivalence class of atlases. There are a number of different types of manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following, a differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable
55.
Orientability
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In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the rule to define a clockwise direction of loops in the surface. More generally, orientability of a surface, or manifold. Equivalently, a surface is orientable if a figure such as in the space cannot be moved around the space. The notion of orientability can be generalised to higher-dimensional manifolds as well, a manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. A surface S in the Euclidean space R3 is orientable if a two-dimensional figure cannot be moved around the surface, an abstract surface is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous manner. That is to say that a loop going around one way on the surface can never be continuously deformed to a loop going around the opposite way and this turns out to be equivalent to the question of whether the surface contains no subset that is homeomorphic to the Möbius strip. Thus, for surfaces, the Möbius strip may be considered the source of all non-orientability, for an orientable surface, a consistent choice of clockwise is called an orientation, and the surface is called oriented. For surfaces embedded in Euclidean space, an orientation is specified by the choice of a continuously varying surface normal n at every point, If such a normal exists at all, then there are always two ways to select it, n or −n. More generally, an orientable surface admits exactly two orientations, and the distinction between a surface and an orientable surface is subtle and frequently blurred. Examples Most surfaces we encounter in the world are orientable. Spheres, planes, and tori are orientable, for example, but Möbius strips, real projective planes, and Klein bottles are non-orientable. They, as visualized in 3-dimensions, all have just one side, the real projective plane and Klein bottle cannot be embedded in R3, only immersed with nice intersections. Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a surface would think there is an other side. The essence of one-sidedness is that the ant can crawl from one side of the surface to the other going through the surface or flipping over an edge. In general, the property of being orientable is not equivalent to being two-sided, however, this holds when the ambient space is orientable. For example, a torus embedded in K2 × S1 can be one-sided, Orientation by triangulation Any surface has a triangulation, a decomposition into triangles such that each edge on a triangle is glued to at most one other edge
56.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
57.
Stirling numbers of the second kind
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In mathematics, particularly in combinatorics, a Stirling number of the second kind is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S or. Stirling numbers of the second kind occur in the field of mathematics called combinatorics, Stirling numbers of the second kind are one of two kinds of Stirling numbers, the other kind being called Stirling numbers of the first kind. Mutually inverse triangular matrices can be formed from the Stirling numbers of each according to the parameters n, k. The Stirling numbers of the kind, written S or or with other notations. Equivalently, they count the number of different equivalence relations with precisely k equivalence classes that can be defined on an n element set, in fact, there is a bijection between the set of partitions and the set of equivalence relations on a given set. They can be calculated using the explicit formula, =1 k. Various notations have been used for Stirling numbers of the second kind, the brace notation was used by Imanuel Marx and Antonio Salmeri in 1962 for variants of these numbers. This led Knuth to use it, as here, in the first volume of The Art of Computer Programming. However, according to the edition of The Art of Computer Programming. The notation S was used by Richard Stanley in his book Enumerative Combinatorics, if we let n = x ⋯ be the falling factorial, we can characterize the Stirling numbers of the second kind by ∑ k =0 n k = x n. Analogously, the ordered Bell numbers can be computed from the Stirling numbers of the kind as a n = ∑ k =0 n k. Below is an array of values for the Stirling numbers of the second kind, As with the binomial coefficients, this table could be extended to k > n. Stirling numbers of the second kind obey the recurrence relation = k + for k >0 with initial conditions =1 and = =0 for n >0. For instance, the number 25 in column k=3 and row n=5 is given by 25=7+, to understand this recurrence, observe that a partition of the n+1 objects into k nonempty subsets either contains the n+1-th object as a singleton or it does not. The number of ways that the singleton is one of the subsets is given by since we must partition the n objects into the available k-1 subsets. In the other case the object belongs to a subset containing other objects. The number of ways is given by k since we partition all objects other than the n+1-th into k subsets, summing these two values gives the desired result. Some more recurrences are as follows, = ∑ j = k n, = ∑ j = k n n − j, = ∑ j =0 k j
58.
Bernoulli number
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In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are B0 =1, B±1 = ±1/2, B2 = 1/6, B3 =0, B4 = −1/30, B5 =0, B6 = 1/42, B7 =0, B8 = −1/30. The superscript ± is used by this article to designate the two conventions for Bernoulli numbers. They differ only in the sign of the n =1 term, B−n are the first Bernoulli numbers, in this convention, B−1 = −1/2. B+n are the second Bernoulli numbers, which are called the original Bernoulli numbers. In this convention, B+1 = +1/2, since Bn =0 for all odd n >1, and many formulas only involve even-index Bernoulli numbers, some authors write Bn to mean B2n. This article does not follow this notation, the Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Sekis discovery was published in 1712 in his work Katsuyo Sampo, Bernoullis, also posthumously. Ada Lovelaces note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbages machine, as a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program. Bernoulli numbers feature prominently in the form expression of the sum of the mth powers of the first n positive integers. For m, n ≥0 define S m = ∑ k =1 n k m =1 m +2 m + ⋯ + n m and this expression can always be rewritten as a polynomial in n of degree m +1. The coefficients of polynomials are related to the Bernoulli numbers by Bernoullis formula, S m =1 m +1 ∑ k =0 m B k + n m +1 − k. For example, taking m to be 1 gives the triangular numbers 0,1,3,6, … A000217,1 +2 + ⋯ + n =12 =12. Taking m to be 2 gives the square pyramidal numbers 0,1,5,14,12 +22 + ⋯ + n 2 =13 =13. Some authors use the convention for Bernoulli numbers and state Bernoullis formula in this way. Bernoullis formula is sometimes called Faulhabers formula after Johann Faulhaber who also found ways to calculate sums of powers. Faulhabers formula was generalized by V. Guo and J. Zeng to a q-analog, many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of the most useful ones are mentioned, an equation, an explicit formula, a generating function
59.
Bernoulli polynomials
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In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in part because they are an Appell sequence. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled and this article also discusses the Bernoulli polynomials and the related Euler polynomials, and the Bernoulli and Euler numbers. The Bernoulli polynomials Bn admit a variety of different representations, which among them should be taken to be the definition may depend on ones purposes. B n = ∑ k =0 n b n − k x k, for n ≥0, the generating function for the Bernoulli polynomials is t e x t e t −1 = ∑ n =0 ∞ B n t n n. The generating function for the Euler polynomials is 2 e x t e t +1 = ∑ n =0 ∞ E n t n n. The Bernoulli polynomials are given by B n = D e D −1 x n where D = d/dx is differentiation with respect to x. It follows that ∫ a x B n d u = B n +1 − B n +1 n +1, the Bernoulli polynomials are the unique polynomials determined by ∫ x x +1 B n d u = x n. The integral transform = ∫ x x +1 f d u on polynomials f, F = f + f ′2 + f ″6 + f ‴24 + ⋯. This can be used to produce the inversion formulae below, an explicit formula for the Bernoulli polynomials is given by B m = ∑ n =0 m 1 n +1 ∑ k =0 n k m. Note the remarkable similarity to the convergent series expression for the Hurwitz zeta function. Indeed, one has B n = − n ζ where ζ is the Hurwitz zeta, thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n. The inner sum may be understood to be the nth forward difference of xm, thus, one may write B m = ∑ n =0 m n n +1 Δ n x m. This formula may be derived from an identity appearing above as follows, as long as this operates on an mth-degree polynomial such as xm, one may let n go from 0 only up to m. An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, an explicit formula for the Euler polynomials is given by E m = ∑ n =0 m 12 n ∑ k =0 n k m. This may also be written in terms of the Euler numbers Ek as E m = ∑ k =0 m E k 2 k m − k and we have ∑ k =0 x k p = B p +1 − B p +1 p +1. See Faulhabers formula for more on this, the Bernoulli numbers are given by B n = B n
60.
Stirling numbers of the first kind
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In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles, the original definition of Stirling numbers of the first kind was algebraic. These numbers, usually written s, are signed integers whose sign, positive or negative and that is the approach taken on this page. Most identities on this page are stated for unsigned Stirling numbers, note that the notations on this page are not universal. The unsigned Stirling numbers of the first kind are denoted in various ways by different authors, common notations are c, | s | and. They count the number of permutations of n elements with k disjoint cycles, =6 permutations of three elements, there is one permutation with three cycles, three permutations with two cycles and two permutations with one cycle. The unsigned Stirling numbers also arise as coefficients of the rising factorial, thus, for example, = x =1 ⋅ x 3 +3 ⋅ x 2 +2 ⋅ x, which matches the computations in the preceding paragraph. Stirling numbers of the first kind are given by s = n − k and they are the coefficients in the expansion n = ∑ k =0 n s x k, where n is the falling factorial n = x ⋯. The unsigned Stirling numbers of the first kind can be calculated by the recurrence relation = n + for k >0 and it follows immediately that the Stirling numbers of the first kind satisfy the recurrence s = − n s + s. We prove the recurrence relation using the definition of Stirling numbers in terms of rising factorials, distributing the last term of the product, we have = x ⋯ = n + x. The coefficient of xk on the side of this equation is. The coefficient of xk in n is n ⋅, while the coefficient of xk in x is, since the two sides are equal as polynomials, the coefficients of xk on both sides must be equal, and the result follows. We prove the recurrence relation using the definition of Stirling numbers in terms of permutations with a number of cycles. Consider forming a permutation of n +1 objects from a permutation of n objects by adding a distinguished object, there are exactly two ways in which this can be accomplished. We could do this by forming a cycle, i. e. leaving the extra object alone. This increases the number of cycles by 1 and so accounts for the term in the recurrence formula and we could also insert the new object into one of the existing cycles. Consider an arbitrary permutation of n objects with k cycles, an, so that the permutation is represented by … ⏟ k c y c l e s. To form a new permutation of n +1 objects and k cycles one must insert the new object into this array, there are n ways to perform this insertion, inserting the new object immediately following any of the n already present
61.
Eulerian number
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In combinatorics, the Eulerian number A, is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element. They are the coefficients of the Eulerian polynomials, A n = ∑ m =0 n A t m, the Eulerian polynomials are defined by the exponential generating function ∑ n =0 ∞ A n x n n. The Eulerian polynomials can be computed by the recurrence A0 =1, A n = t A n −1 ′ + A n −1, n ≥1. An equivalent way to write this definition is to set the Eulerian polynomials inductively by A0 =1, A n = ∑ k =0 n −1 A k n −1 − k, n ≥1. Other notations for A are E and ⟨ n m ⟩, in 1755 Leonhard Euler investigated in his book Institutiones calculi differentialis polynomials α1 =1, α2 = x +1, α3 = x2 + 4x +1, etc. These polynomials are a form of what are now called the Eulerian polynomials An. For a given value of n >0, the m in A can take values from 0 to n −1. For fixed n there is a permutation which has 0 ascents. There is also a permutation which has n −1 ascents. Therefore A and A are 1 for all values of n, reversing a permutation with m ascents creates another permutation in which there are n − m −1 ascents. Values of A can be calculated by hand for small values of n and m, for example For larger values of n, A can be calculated using the recursive formula A = A + A. For example A = A + A =3 ×1 +2 ×4 =11, values of A for 0 ≤ n ≤9 are, The above triangular array is called the Euler triangle or Eulers triangle, and it shares some common characteristics with Pascals triangle. The sum of row n is the factorial n, an explicit formula for A is A = ∑ k =0 m +1 k n. This assumes that 00 =0 and A =1, Worpitzkys identity expresses xn as the linear combination of Eulerian numbers with binomial coefficients, x n = ∑ m =0 n −1 A. It follows from Worpitzkys identity that ∑ k =1 N k n = ∑ m =0 n −1 A, another interesting identity is e 1 − e x = ∑ n =0 ∞ A n n. The numerator on the side is the Eulerian polynomial. For a fixed function f, R → C which is integrable on we have the integral formula ∫01 ⋯ ∫01 f d x 1 ⋯ d x n = ∑ k A f n. The Eulerian number of the kind, denoted ⟨ ⟨ n m ⟩ ⟩
62.
Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
63.
Imaginary unit
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The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 +1 =0. The term imaginary is used there is no real number having a negative square. There are two square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used, in the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. For the history of the unit, see Complex number § History. The imaginary number i is defined solely by the property that its square is −1, with i defined this way, it follows directly from algebra that i and −i are both square roots of −1. In polar form, i is represented as 1eiπ/2, having a value of 1. In the complex plane, i is the point located one unit from the origin along the imaginary axis, more precisely, once a solution i of the equation has been fixed, the value −i, which is distinct from i, is also a solution. Since the equation is the definition of i, it appears that the definition is ambiguous. However, no ambiguity results as long as one or other of the solutions is chosen and labelled as i and this is because, although −i and i are not quantitatively equivalent, there is no algebraic difference between i and −i. Both imaginary numbers have equal claim to being the number whose square is −1, the issue can be a subtle one. See also Complex conjugate and Galois group, a more precise explanation is to say that the automorphism group of the special orthogonal group SO has exactly two elements — the identity and the automorphism which exchanges CW and CCW rotations. All these ambiguities can be solved by adopting a rigorous definition of complex number. For example, the pair, in the usual construction of the complex numbers with two-dimensional vectors. The imaginary unit is sometimes written √−1 in advanced mathematics contexts, however, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the square root function. Similarly,1 i =1 −1 =1 −1 = −11 = −1 = i, the calculation rules a ⋅ b = a ⋅ b and a b = a b are only valid for real, non-negative values of a and b. These problems are avoided by writing and manipulating expressions like i√7, for a more thorough discussion, see Square root and Branch point
64.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler