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Fibonacci Numbers
In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the '' Fibonacci Quarterly''. Applications of Fibon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci number, Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description . The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, , even though we do not have a formula for the ''n''th perfect number. Computable and definable sequences An integer sequence is computable function, computable if th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pine Cone
A conifer cone, or in formal botanical usage a strobilus, : strobili, is a seed-bearing organ on gymnosperm plants, especially in conifers and cycads. They are usually woody and variously conic, cylindrical, ovoid, to globular, and have scales and bracts arranged around a central axis, but can be fleshy and berry-like. The cone of Pinophyta (conifer clade) contains the reproductive structures. The woody cone is the female cone, which produces seeds. The male cone, which produces pollen, is usually ephemeral and much less conspicuous even at full maturity. The name "cone" derives from Greek ''konos'' (pine cone), which also gave name to the geometric cone. The individual plates of a cone are known as ''scales''. In conifers where the cone develops over more than one year (such as pines), the first year's growth of a seed scale on the cone, showing up as a protuberance at the end of the two-year-old scale, is called an ''umbo'', while the second year's growth is called the '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemachandra
Hemacandra was a 12th century () Śvetāmbara Jaina acharya, ācārya, scholar, poet, mathematician, philosopher, yogi, wikt:grammarian, grammarian, Law, law theorist, historian, Lexicography, lexicographer, rhetorician, logician, and Prosody (linguistics), prosodist. Noted as a prodigy by his contemporaries, he gained the title ''kalikālasarvajña'', "the knower of all knowledge in his times" and is also regarded as father of the Gujarati language. Born as Caṅgadeva, he was ordained in the Śvētāmbara, Śvetāmbara school of Jainism in 1110 and took the name Somacandra. In 1125 he became an adviser to King Kumārapāla and wrote ''Arhannīti'', a work on politics from Jaina perspective. He also produced ''Triśaṣṭi-śalākā-puruṣacarita'' (“Deeds of the 63 Illustrious Men”), a Sanskrit epic poem on the history of important figures of Jainism. Later when he was consecrated as ācārya, his name was changed to Hemacandra. Early life Hemacandra was born in Dhand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mora (linguistics)
A mora (plural ''morae'' or ''moras''; often symbolized μ) is a smallest unit of timing, equal to or shorter than a syllable, that theoretically or perceptually exists in some spoken languages in which phonetic length (such as vowel length) matters significantly. For example, in the Japanese language, the name of the city '' Ōsaka'' () consists of three syllables (''O-sa-ka'') but four morae (), since the first syllable, ''Ō'', is pronounced with a long vowel (the others being short). Thus, a short vowel contains one mora and is called ''monomoraic'', while a long vowel contains two and is called ''bimoraic''. Extra-long syllables with three morae (''trimoraic'') are relatively rare. Such metrics based on syllables are also referred to as syllable weight. In Japanese, certain consonants also stand on their own as individual morae and thus are monomoraic. The term comes from the Latin word for 'linger, delay', which was also used to translate the Greek word : ('time') in it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virahanka
Virahanka (Devanagari: विरहाङ्क) was an Indian prosodist who is also known for his work on mathematics. He may have lived in the 6th century, but it is also possible that he worked as late as the 8th century. His work on prosody builds on the ''Chhanda-sutras'' of Pingala (4th century BCE), and was the basis for a 12th-century commentary by Gopala. He was the first to propose the so-called Fibonacci Sequence. See also *Indian mathematicians Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians in the modern era. One of such works is Hindu numeral system which is predominantly used today and is likely ... References External links ''The So-called Fibonacci Numbers in Ancient and Medieval India'' by Parmanand Singh 8th-century Indian mathematicians Fibonacci numbers Medieval Sanskrit grammarians Ancient Indian mathematical works {{asia-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natya Shastra
The ''Nāṭya Shāstra'' (, ''Nāṭyaśāstra'') is a Sanskrit treatise on the performing arts. The text is attributed to sage Bharata, and its first complete compilation is dated to between 200 BCE and 200 CE, but estimates vary between 500 BCE and 500 CE. The text consists of 36 chapters with a cumulative total of 6,000 poetic verses describing performance arts. The subjects covered by the treatise include dramatic composition, structure of a play and the construction of a stage to host it, genres of acting, body movements, make up and costumes, role and goals of an art director, the musical scales, musical instruments and the integration of music with art performance. The ''Nāṭya Śāstra'' is notable as an ancient encyclopedic treatise on the arts, one which has influenced dance, music and literary traditions in India. It is also notable for its aesthetic "Rasa" theory, which asserts that entertainment is a desired effect of performance arts but not t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bharata Muni
Bharata (Devanagari: भरत) was a '' muni'' (sage) of ancient India. He is traditionally attributed authorship of the influential performing arts treatise '' Natya Shastra'', which covers ancient Indian dance, poetics, dramaturgy, and music. Identity He is thought to have lived between 200 BCE and 200 CE, but estimates vary between 500 BCE and 500 CE. ''Nāṭya Śāstra'' Bharata is known only as being traditionally attributed authorship of the treatise '' Natya Shastra''. All other early Sanskrit treatises were similarly attributed to mythical sages. The text draws on his authority, as existing in the public imagination. The ''Nāṭya Śāstra'' is notable as an ancient encyclopedic treatise on the performing arts The performing arts are arts such as music, dance, and drama which are performed for an audience. They are different from the visual arts, which involve the use of paint, canvas or various materials to create physical or static art o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sanskrit Prosody
Sanskrit prosody or Chandas refers to one of the six Vedangas, or limbs of Vedic studies.James Lochtefeld (2002), "Chandas" in The Illustrated Encyclopedia of Hinduism, Vol. 1: A-M, Rosen Publishing, , page 140 It is the study of poetic metres and verse in Sanskrit. This field of study was central to the composition of the Vedas, the scriptural canons of Hinduism; in fact, so central that some later Hindu and Buddhist texts refer to the Vedas as ''Chandas''. The Chandas, as developed by the Vedic schools, were organized around seven major metres, each with its own rhythm, movements and aesthetics. Sanskrit metres include those based on a fixed number of syllables per verse, and those based on fixed number of morae per verse. Extant ancient manuals on Chandas include Pingala's ''Chandah Sutra'', while an example of a medieval Sanskrit prosody manual is Kedara Bhatta's ''Vrittaratnakara''. The most exhaustive compilations of Sanskrit prosody describe over 600 metres. This i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Sanskrit Prosody
Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci'', is first found in a modern source in a 1838 text by the Franco-Italian mathematician Guglielmo Libri and is short for ('son of Bonacci'). However, even as early as 1506, Perizolo, a notary of the Holy Roman Empire, mentions him as "Lionardo Fibonacci". Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of (''Book of Calculation'') and also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in . Biography Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official who directed a trading post in Bugia, modern-day Béjaïa, Algeria. Fibonacci travelled with him as a young boy, and it was in Bugia where he was educat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U_n(P, Q) and V_n(P, Q). More generally, Lucas sequences U_n(P, Q) and V_n(P, Q) represent sequences of polynomials in P and Q with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers (see below). Lucas sequences are named after the French mathematician Édouard Lucas. Recurrence relations Given two integer parameters P and Q, the Lucas sequences of the first kind U_n(P,Q) and of the second kind V_n(P,Q) are defined by the recurrence relations: :\begin U_0(P,Q)&=0, \\ U_1(P,Q)&=1, \\ U_n(P,Q)&=P\cdot U_(P,Q)-Q\cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
