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Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions and is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: : a_0 = a_1 = 1 and a_n = a_ + (n - 1)a_ for n > 1. The first few terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Involution
Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiation (archaic use of the term) Other uses * Involution (medicine), the shrinking of an organ (such as the uterus after pregnancy) * Neijuan, or involution, a Chinese social concept * ''Agricultural Involution'', a 1963 study of intensification of production through increased labour inputs * Involution (esoterism) The term involution has various meanings. In some instances it refers to a process prior to evolution which gives rise to the cosmos, in others it is an aspect of evolution, and in still others it is a process that follows the completion of evoluti ..., several notions of a counterpart to evolution * Involution (Meher Baba), the inner path of the human soul to the self as described by Meher Baba * Involution (Sri Aurobindo), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Reciprocal Cipher
Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both the encryption of plaintext and the decryption of ciphertext. The keys may be identical, or there may be a simple transformation to go between the two keys. The keys, in practice, represent a shared secret between two or more parties that can be used to maintain a private information link. The requirement that both parties have access to the secret key is one of the main drawbacks of symmetric-key encryption, in comparison to public-key encryption (also known as asymmetric-key encryption). However, symmetric-key encryption algorithms are usually better for bulk encryption. With exception of the one-time pad they have a smaller key size, which means less storage space and faster transmission. Due to this, asymmetric-key encryption is often used to exchange the secret key for symmetric-key encryption. Types Symmetric-key encryption can use either stream ciphers or block cip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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76 (number)
76 (seventy-six) is the natural number following 75 (number), 75 and preceding 77 (number), 77. In mathematics 76 is: * a composite number; a square-prime, of the form (''p''2, q) where q is a higher prime. It is the ninth of this general form and the seventh of the form (22.q). * a Lucas number. * a Telephone number (mathematics), telephone or involution number, the number of different ways of connecting 6 points with pairwise connections. * a nontotient. * a 14-gonal number. * a centered pentagonal number. * an Erdős–Woods number since it is possible to find sequences of 76 consecutive integers such that each inner member shares a factor with either the first or the last member. * with an aliquot sum of 64 (number), 64; within an aliquot sequence of two composite numbers (76,64 (number), 64,63 (number), 63,1 (number), 1,0) to the Prime in the 63-aliquot tree. * an automorphic number in base 10. It is one of two 2-digit numbers whose square, 5,776, and higher powers, end in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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26 (number)
26 (twenty-six) is the natural number following 25 and preceding 27. In mathematics *26 is the seventh discrete semiprime (2 \times 13) and the fifth with 2 as the lowest non-unitary factor thus of the form (2.q), where q is a higher prime. *26 is the smallest even number ''n'' such that both ''n'' + 1 and ''n'' − 1 are composite. *With an aliquot sum of 16, within an aliquot sequence of five composite numbers (26, 16, 15, 9, 4, 3, 1,0) to the Prime in the 3-aliquot tree. *26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1). *26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise connections. *There are 26 sporadic groups. *The 26-dimensional Lorentzian unimodular lattice II25,1 plays a significant role in sphere packing problems and the classification of finite simple groups. In particular, the Leech lattice is obtained in a simple way as a subquotient. *26 is the smallest numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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10 (number)
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language. Name The number "ten" originates from the Proto-Germanic root "*tehun", which in turn comes from the Proto-Indo-European root "*dekm-", meaning "ten". This root is the source of similar words for "ten" in many other Germanic languages, like Dutch, German, and Swedish. The use of "ten" in the decimal system is likely due to the fact that humans have ten fingers and ten toes, which people may have used to count by. Linguistics * A collection of ten items (most often ten years) is called a decade. * The ordinal adjective is ''decimal''; the distributive adjective is ''denary''. * Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten. * To reduce something by one tenth is to '' decimate''. (In ancient Rome, the killing o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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4 (number)
4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is a square number, the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures. Evolution of the Hindu-Arabic digit Brahmic numerals represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the digit, and the Kshatrapa and Pallava evolved the digit to a point where the speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, ending up with a digit very close to the original Brahmin cross. While the shape of the character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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2 (number)
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and the only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures. Mathematics The number 2 is the second natural number after 1. Each natural number, including 2, is constructed by succession, that is, by adding 1 to the previous natural number. 2 is the smallest and the only even prime number, and the first Ramanujan prime. It is also the first superior highly composite number, and the first colossally abundant number. An integer is determined to be even if it is divisible by two. When written in base 10, all multiples of 2 will end in 0, 2, 4, 6, or 8; more generally, in any even base, even numbers will end with an even digit. A digon is a polygon with two sides (or edges) and two vertices. Two distinct points in a plane are always sufficient to define a unique line in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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1 (number)
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. In mathematics The number 1 is the first natural number after 0. Each natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Heinrich August Rothe
Heinrich August Rothe (1773–1842) was a German mathematician, a professor of mathematics at Erlangen. He was a student of Carl Hindenburg and a member of Hindenburg's school of combinatorics. Biography Rothe was born in 1773 in Dresden, and in 1793 became a docent at the University of Leipzig. He became an extraordinary professor at Leipzig in 1796, and in 1804 he moved to Erlangen as a full professor, taking over the chair formerly held by Karl Christian von Langsdorf. He died in 1842, and his position at Erlangen was in turn taken by Johann Wilhelm Pfaff, the brother of the more famous mathematician Johann Friedrich Pfaff. Research The Rothe–Hagen identity, a summation formula for binomial coefficients, appeared in Rothe's 1793 thesis. It is named for him and for the later work of Johann Georg Hagen. The same thesis also included a formula for computing the Taylor series of an inverse function from the Taylor series for the function itself, related to the Lagrange inver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Commutative Property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures started to be studied. Definition A binary operation * on a set ''S'' is ''commutative'' if x * y = y * x for all x,y \in S. An operation that is not commutative is said to be ''noncommutative''. One says ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Function Composition
In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \circ f) is pronounced "the composition of and ". Reverse composition, sometimes denoted f \mapsto g , applies the operation in the opposite order, applying f first and g second. Intuitively, reverse composition is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composition of relations are true of composition of functions, such as #Properties, associativity. Examples * Composition of functions on a finite set (mathematics), set: If , and , then , as shown in the figure. * Composition of functions on an infinite set: If (where is the set of all real numbers) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |