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History Of Combinatorics
The mathematical field of combinatorics was studied to varying degrees in numerous ancient societies. Its study in Europe dates to the work of Leonardo Fibonacci in the 13th century AD, which introduced Arabian and Indian ideas to the continent. It has continued to be studied in the modern era. Earliest records The earliest recorded use of combinatorial techniques comes from problem 79 of the Rhind papyrus, which dates to the 16th century BCE. The problem concerns a certain geometric series, and has similarities to Fibonacci's problem of counting the number of compositions of 1s and 2s that sum to a given total. In Greece, Plutarch wrote that Xenocrates of Chalcedon (396–314 BC) discovered the number of different syllables possible in the Greek language. This would have been the first attempt on record to solve a difficult problem in permutations and combinations. The claim, however, is implausible: this is one of the few mentions of combinatorics in Greece, and the numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pingala
Acharya Pingala ('; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody. The ' is a work of eight chapters in the late Sūtra style, not fully comprehensible without a commentary. It has been dated to the last few centuries BCE. In the 10th century CE, Halayudha wrote a commentary elaborating on the '. Pingala Maharshi was also said to be the brother of Pāṇini, the famous Sanskrit grammarian, considered the first descriptive linguist''. François & Ponsonnet (2013: 184).'' Combinatorics The ' presents the first known description of a binary numeral system in connection with the systematic enumeration of metres with fixed patterns of short and long syllables. Pingala's discussion of the combinatorics of metre corresponds to the binomial theorem. Halāyudha's 10th-century commentary on the ' includes a presentation of this theorem in what is now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and/or ''symmetry''. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids. Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lo Shu Square
The Luoshu (pinyin), Lo Shu ( Wade-Giles), or Nine Halls Diagram is an ancient Chinese diagram and named for the Luo River near Luoyang, Henan. The Luoshu appears in myths concerning the invention of writing by Cangjie and other culture heroes. It is a unique normal magic square of order three. It is usually paired with the River Map or Hetunamed in reference to the Yellow Riverand used with the River Map in various contexts involving Chinese geomancy, numerology, philosophy, and early natural science. Traditions The Lo Shu is part of the legacy of ancient Chinese mathematical and divinity (cf. the I Ching ) traditions, and is an important emblem in '' Feng Shui'' ()—the art of geomancy concerned with the placement of objects in relation to the flow of qi (), or "natural energy". History A Chinese legend concerning the pre-historic Emperor Yu () tells of the Lo Shu, often in connection with the ''Yellow River Map'' (Hetu) and the eight trigrams. In ancient China ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Go (board Game)
Go is an abstract strategy board game for two players in which the aim is to surround more territory than the opponent. The game was invented in China more than 2,500 years ago and is believed to be the oldest board game continuously played to the present day. A 2016 survey by the International Go Federation's 75 member nations found that there are over 46 million people worldwide who know how to play Go and over 20 million current players, the majority of whom live in East Asia. The playing pieces are called stones. One player uses the white stones and the other, black. The players take turns placing the stones on the vacant intersections (''points'') of a board. Once placed on the board, stones may not be moved, but stones are removed from the board if the stone (or group of stones) is surrounded by opposing stones on all orthogonally adjacent points, in which case the stone or group is ''captured''. The game proceeds until neither player wishes to make another move. When ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I Ching
The ''I Ching'' or ''Yi Jing'' (, ), usually translated ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination text that is among the oldest of the Chinese classics. Originally a divination manual in the Western Zhou period (1000750), the ''I Ching'' was transformed over the course of the Warring States and early imperial periods (500200) into a cosmological text with a series of philosophical commentaries known as the " Ten Wings". After becoming part of the Five Classics in the 2nd century BC, the ''I Ching'' was the subject of scholarly commentary and the basis for divination practice for centuries across the Far East, and eventually took on an influential role in Western understanding of East Asian philosophical thought. As a divination text, the ''I Ching'' is used for a traditional Chinese form of cleromancy known as ''I Ching'' divination, in which bundles of yarrow stalks are manipulated to produce sets of six apparently random numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Numbers
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :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, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. 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 Fibonacci numbers include ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the '' Khaṇḍakhādyaka'' ("edible bite", dated 665), a more practical text. Brahmagupta was the first to give rules for computing with '' zero''. The texts composed by Brahmagupta were in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived. In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. Life and career Brahmagupta was born in 598 CE according to his own statement. He lived in ''Bhillamāla'' in Gurjaradesa (modern Bhinmal in Rajasthan, India) during the reign of the Chavda dynasty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemacandra
Hemachandra was a 12th century () Indian Jain saint, scholar, poet, mathematician, philosopher, yogi, grammarian, law theorist, historian, lexicographer, rhetorician, logician, and prosodist. Noted as a prodigy by his contemporaries, he gained the title ''kalikālasarvajña'', "the knower of all knowledge in his times" and ''father of Gujarati language''. Born as Changadeva, he was ordained in the Śvētāmbara school of Jainism in 1110 and took the name Somachandra. In 1125 he became an adviser to King Kumarapala and wrote ''Arhanniti'', a work on politics from a Jain perspective. He also produced ''Trishashti-shalaka-purusha-charita'' (“Deeds of the 63 Illustrious Men”), a Sanskrit epic poem on the history of important figures of Jainism. Later in his life, he changed his name to Hemachandra. Early life Hemachandra was born in Dhandhuka, in present-day Gujarat, on Kartika Sud Purnima (the full moon day of Kartika month). His date of birth differs according to sourc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bhāskara II
Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiromani (सिद्धांतशिरोमणी), it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Sahyadri mountain ranges of Western Ghats, believed to be the town of Patan in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. He is the only ancient mathematician who has been immortalized on a monument. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him. Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as Maharashtrian Brahmins residing on the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prosody (poetry)
In poetry, metre ( Commonwealth spelling) or meter (American spelling; see spelling differences) is the basic rhythmic structure of a verse or lines in verse. Many traditional verse forms prescribe a specific verse metre, or a certain set of metres alternating in a particular order. The study and the actual use of metres and forms of versification are both known as prosody. (Within linguistics, " prosody" is used in a more general sense that includes not only poetic metre but also the rhythmic aspects of prose, whether formal or informal, that vary from language to language, and sometimes between poetic traditions.) Characteristics An assortment of features can be identified when classifying poetry and its metre. Qualitative versus quantitative metre The metre of most poetry of the Western world and elsewhere is based on patterns of syllables of particular types. The familiar type of metre in English-language poetry is called qualitative metre, with stressed syllables comin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
