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Srinivasa Ramanujam
Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erode
Erode (; īrōṭu), is a city in the Indian state of Tamil Nadu. It is located on the banks of the Kaveri river and is surrounded by the Western Ghats. Erode is the seventh largest urban agglomeration in Tamil Nadu. It is the administrative capital of Erode district and is administered by the Erode Municipal Corporation which was established in 2008. The region was ruled by the Cheras during the Sangam period between the 1st and the 4th centuries CE. The medieval Cholas conquered the region in the 10th century CE. The region was ruled by Vijayanagara Empire in the 15th century followed by the Nayaks who introduced the Palayakkarar system. In the later part of the 18th century, the it came under the Kingdom of Mysore and following the Anglo-Mysore Wars, the British East India Company annexed it to the Madras Presidency in 1799. The region played a prominent role in the second Poligar War (1801) when it was the area of operations of Dheeran Chinnamalai. Post Indian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan's Congruences
In mathematics, Ramanujan's congruences are the congruences for the partition function ''p''(''n'') discovered by Srinivasa Ramanujan: : \begin p(5k+4) & \equiv 0 \pmod 5, \\ p(7k+5) & \equiv 0 \pmod 7, \\ p(11k+6) & \equiv 0 \pmod . \end In plain words, the first congruence means that if a number is 4 more than a multiple of 5, i.e. it is in the sequence : 4, 9, 14, 19, 24, 29, . . . then the number of its partitions is a multiple of 5. Later, other congruences of this type were discovered, for numbers and for Tau-functions. Background In his 1919 paper, he proved the first two congruences using the following identities (using q-Pochhammer symbol notation): : \begin & \sum_^\infty p(5k+4)q^k=5\frac, \\ pt& \sum_^\infty p(7k+5)q^k=7\frac+49q\frac. \end He then stated that "It appears there are no equally simple properties for any moduli involving primes other than these". After Ramanujan died in 1920, G. H. Hardy extracted proofs of all three congruences from an unpub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fellow Of The Royal Society
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, including mathematics, engineering science, and medical science". Overview Fellowship of the Society, the oldest known scientific academy in continuous existence, is a significant honour. It has been awarded to :Fellows of the Royal Society, around 8,000 fellows, including eminent scientists Isaac Newton (1672), Benjamin Franklin (1756), Charles Babbage (1816), Michael Faraday (1824), Charles Darwin (1839), Ernest Rutherford (1903), Srinivasa Ramanujan (1918), Jagadish Chandra Bose (1920), Albert Einstein (1921), Paul Dirac (1930), Subrahmanyan Chandrasekhar (1944), Prasanta Chandra Mahalanobis (1945), Dorothy Hodgkin (1947), Alan Turing (1951), Lise Meitner (1955), Satyendra Nath Bose (1958), and Francis Crick (1959). More recently, fellow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan–Sato Series
In mathematics, a Ramanujan–Sato series generalizes Ramanujan's pi formulas such as, :\frac = \frac \sum_^\infty \frac \frac to the form :\frac = \sum_^\infty s(k) \frac by using other well-defined sequences of integers s(k) obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients \tbinom, and A,B,C employing modular forms of higher levels. Ramanujan made the enigmatic remark that there were "corresponding theories", but it was only in 2012 that H. H. Chan and S. Cooper found a general approach that used the underlying modular congruence subgroup \Gamma_0(n), while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators. Levels ''1–4A'' were given by Ramanujan (1914), level ''5'' by H. H. Chan and S. Cooper (2012), ''6A'' by Chan, Tanigawa, Yang, and Zudilin, ''6B'' by Sato (2002), ''6C'' by H. Chan, S. Chan, and Z. Liu (2004), ''6D'' by H. Chan and H. Verrill ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (number Theory)
In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and . No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of ends in the digit 4 or 9, the number of partitions of will be divisible by 5. Definition and examples For a positive integer , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan's Master Theorem
In mathematics, Ramanujan's master theorem, named after Srinivasa Ramanujan, is a technique that provides an analytic expression for the Mellin transform of an analytic function. The result is stated as follows: If a complex-valued function f(x) has an expansion of the form f(x)=\sum_^\infty \frac(-x)^k then the Mellin transform of f(x) is given by \int_0^\infty x^ f(x) \, dx = \Gamma(s)\,\varphi(-s) where \Gamma(s) is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics through Feynman diagrams. A similar result was also obtained by Glaisher. Alternative formalism An alternative formulation of Ramanujan's master theorem is as follows: \int_0^\infty x^\left(\,\lambda(0) - x\,\lambda(1) + x^2\,\lambda(2) -\,\cdots\,\right) dx = \frac\,\lambda(-s) which gets converted to the above form after substituting \lambda(n) \equiv \frac and us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rogers–Ramanujan Identities
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof . independently rediscovered and proved the identities. Definition The Rogers–Ramanujan identities are :G(q) = \sum_^\infty \frac = \frac =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots and :H(q) =\sum_^\infty \frac = \frac =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots . Here, (a;q)_n denotes the q-Pochhammer symbol. Combinatorial interpretation Consider the following: * \frac is the generating function for partitions with exactly n parts such that adjacent parts have difference at least 2. * \frac is the generating function for partitions such that each part is congruent to either 1 or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Theta Function
In mathematics, particularly q-analog, -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa Ramanujan. Definition The Ramanujan theta function is defined as :f(a,b) = \sum_^\infty a^\frac \; b^\frac for . The Jacobi triple product identity then takes the form :f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty. Here, the expression (a;q)_n denotes the q-Pochhammer symbol, -Pochhammer symbol. Identities that follow from this include :\varphi(q) = f(q,q) = \sum_^\infty q^ = and :\psi(q) = f\left(q,q^3\right) = \sum_^\infty q^\frac = and :f(-q) = f\left(-q,-q^2\right) = \sum_^\infty (-1)^n q^\frac = (q;q)_\infty This last being the Euler function, which is closely related to the Dedekind eta funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
