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Permutations
In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is &nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tuple
In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is only one 0-tuple, called the ''empty tuple''. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term ''"infinite tuple"'' is occasionally used for ''"infinite sequences"''. Tuples are usually written by listing the elements within parentheses "" and separated by commas; for example, denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning. An -tuple can be formally defined as the image of a function that has the set of the first natural numbers as its domain. Tuples may be also defined from ordered pairs by a recurrence starting from an ordered pair; indeed, an -tuple can be identified with the ordered pair of its first elements and its t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image (mathematics)
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each Element (mathematics), element of a given subset A of its Domain of a function, domain X produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain Y is the set of all elements of X that map to a member of B. The image of the function f is the set of all output values it may produce, that is, the image of X. The preimage of f is the preimage of the codomain Y. Because it always equals X (the domain of f), it is rarely used. Image and inverse image may also be defined for general Binary relation#Operations, binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a Function (mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics In Medieval Islam
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments of the period include extension of the place-value system to include decimal fractions, the systematised study of algebra and advances in geometry and trigonometry. The medieval Islamic world underwent significant developments in mathematics. Muhammad ibn Musa al-Khwārizmī played a key role in this transformation, introducing algebra as a distinct field in the 9th century. Al-Khwārizmī's approach, departing from earlier arithmetical traditions, laid the groundwork for the arithmetization of algebra, influencing mathematical thought for an extended period. Successors like Al-Karaji expanded on his work, contributing to advancements in various mathematical domains. The practicality and broad applicability of these mathematical metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Al-Khalil Ibn Ahmad Al-Farahidi
Abu ‘Abd ar-Raḥmān al-Khalīl ibn Aḥmad ibn ‘Amr ibn Tammām al-Farāhīdī al-Azdī al-Yaḥmadī (; 718 – 786 CE), known as al-Farāhīdī, or al-Khalīl, was an Arab philologist, lexicographer and leading grammarian of Basra in Iraq. He made the first dictionary of the Arabic language – and the oldest extant dictionary – '' Kitab al-'Ayn'' ( "The Source")Introduction to ''Early Medieval Arabic: Studies on Al-Khalīl Ibn Ahmad'', pg. 3. Ed. Karin C. Ryding. Washington, D.C.: Georgetown University Press, 1998. – introduced the now standard harakat (vowel marks in Arabic script) system, and was instrumental in the early development of ʿArūḍ (study of prosody),al-Khalīl ibn Aḥmad at the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Xenocrates
Xenocrates (; ; c. 396/5314/3 BC) of Chalcedon was a Greek philosopher, mathematician, and leader ( scholarch) of the Platonic Academy from 339/8 to 314/3 BC. His teachings followed those of Plato, which he attempted to define more closely, often with mathematical elements. He distinguished three forms of being: the sensible, the intelligible, and a third compounded of the two, to which correspond respectively, sense, intellect and opinion. He considered unity and duality to be gods which rule the universe, and the soul a self-moving number. God pervades all things, and there are daemonical powers, intermediate between the divine and the mortal, which consist in conditions of the soul. He held that mathematical objects and the Platonic Ideas are identical, unlike Plato who distinguished them. In ethics, he taught that virtue produces happiness, but external goods can minister to it and enable it to effect its purpose. Life Xenocrates was a native of Chalcedon. By the most ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plutarch
Plutarch (; , ''Ploútarchos'', ; – 120s) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo (Delphi), Temple of Apollo in Delphi. He is known primarily for his ''Parallel Lives'', a series of biographies of illustrious Greeks and Romans, and ''Moralia'', a collection of essays and speeches. Upon becoming a Roman citizen, he was possibly named Lucius Mestrius Plutarchus (). Family Plutarch was born to a prominent family in the small town of Chaeronea, about east of Delphi, in the Greek region of Boeotia. His family was long established in the town; his father was named Autobulus and his grandfather was named Lamprias. His brothers, Timon and Lamprias, are frequently mentioned in his essays and dialogues, which speak of Timon in particular in the most affectionate terms. Studies and life Plutarch studied mathematics and philosophy in Athens under Ammonius of Athens, Ammonius from AD 66 to 67. He attended th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pinyin
Hanyu Pinyin, or simply pinyin, officially the Chinese Phonetic Alphabet, is the most common romanization system for Standard Chinese. ''Hanyu'' () literally means 'Han Chinese, Han language'—that is, the Chinese language—while ''pinyin'' literally means 'spelled sounds'. Pinyin is the official romanization system used in China, Singapore, Taiwan, and by the United Nations. Its use has become common when transliterating Standard Chinese mostly regardless of region, though it is less ubiquitous in Taiwan. It is used to teach Standard Chinese, normally written with Chinese characters, to students in mainland China and Singapore. Pinyin is also used by various Chinese input method, input methods on computers and to lexicographic ordering, categorize entries in some Chinese dictionaries. In pinyin, each Chinese syllable is spelled in terms of an optional initial (linguistics), initial and a final (linguistics), final, each of which is represented by one or more letters. Initi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I Ching
The ''I Ching'' or ''Yijing'' ( ), usually translated ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination text that is among the oldest of the Chinese classics. The ''I Ching'' was originally a divination manual in the Western Zhou period (1000–750 BC). Over the course of the Warring States period, Warring States and early imperial periods (500–200 BC), it transformed into a Religious cosmology, cosmological text with a series of philosophical commentaries known as the Ten Wings. After becoming part of the Chinese Five Classics in the 2nd century BC, the ''I Ching'' was the basis for divination practice for centuries across the Far East and was the subject of scholarly commentary. Between the 18th and 20th centuries, it took on an influential role in Western understanding of East Asian philosophical thought. As a divination text, the ''I Ching'' is used for a Chinese form of cleromancy known as I Ching divination, ''I Ching'' div ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagram (I Ching)
The ''I Ching'' book consists of 64 hexagrams. A hexagram in this context is a figure composed of six stacked horizontal lines ( 爻 yáo), where each line is either Yang (an unbroken, or solid line), or Yin (broken, an open line with a gap in the center). The hexagram lines are traditionally counted from the bottom up, so the lowest line is considered line one while the top line is line six. Hexagrams are formed by combining the original eight trigrams in different combinations. Each hexagram is accompanied with a description, often cryptic, akin to parables. Each line in every hexagram is also given a similar description. The Chinese word for a hexagram is 卦 "guà", although that also means trigram. Types Classic and modern ''I Ching'' commentaries mention a number of different hexagram types: * Eight Trigrams * Original Hexagram * Future Hexagram * Contrasting (Reverse) Hexagram (is found by turning a hexagram upside down) * Complementary Hexagram (is found by chang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
