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Frénicle Standard Form
A magic square is in the Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions hold: # the element at position ,1(top left corner) is the smallest of the four corner elements; and # the element at position ,2(top edge, second from left) is smaller than the element in ,1 In 1693, Frénicle described all the 880 essentially different order-4 magic squares. Properties This standard form was devised since a magic square remains "essentially similar" if it is rotated or transposed, or flipped so that the order of rows is reversed. There exist 8 different magic squares sharing one standard form. For example, the following magic squares are all essentially similar, with only the final square being in the Frénicle standard form: 8 1 6 8 3 4 4 9 2 4 3 8 6 7 2 6 1 8 2 9 4 2 7 6 3 5 7 1 5 9 3 5 7 9 5 1 1 5 9 7 5 3 7 5 3 9 5 1 4 9 2 6 7 2 8 1 6 2 7 6 8 3 4 2 9 4 6 1 8 4 3 8 Generali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number of integers along one side (''n''), and the constant sum is called the ' magic constant'. If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be 'normal'. Some authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as 'trivial'. Some well-known examples, including the Sagrada Família magic square and the Parker square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant this gives a ''semimagic square (sometimes called orthomagic square). The mathematical study of magic squares typically deals with their construction, classification, and enumeration. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Frénicle De Bessy
Bernard Frénicle de Bessy (c. 1604 – 1674), was a French mathematician born in Paris, who wrote numerous mathematical papers, mainly in number theory and combinatorics. He is best remembered for , a treatise on magic squares published posthumously in 1693, in which he described all 880 essentially different normal magic squares of order 4. The Frénicle standard form, a standard representation of magic squares, is named after him. He solved many problems created by Fermat and also discovered the cube property of the number 1729 (Ramanujan number), later referred to as a taxicab number. He is also remembered for his treatise ''Traité des triangles rectangles en nombres'' published (posthumously) in 1676 and reprinted in 1729. Bessy was a member of many of the scientific circles of his day, including the French Academy of Sciences, and corresponded with many prominent mathematicians, such as Mersenne and Pascal. Bessy was also particularly close to Fermat, Descartes and Wal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT. Transpose of a matrix Definition The transpose of a matrix , denoted by , , , A^, , , or , may be constructed by any one of the following methods: # Reflect over its main diagonal (which runs from top-left to bottom-right) to obtain #Write the rows of as the columns of #Write the columns of as the rows of Formally, the -th row, -th column element of is the -th row, -th column element of : :\left mathbf^\operatorname\right = \left mathbf\right. If is an matrix, then is an matrix. In the case of square matric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2009
File:2009 Events Collage V2.png, From top left, clockwise: The vertical stabilizer of Air France Flight 447 is pulled out from the Atlantic Ocean; Barack Obama becomes the first African American to become President of the United States; 2009 Iranian presidential election protests, Protests erupt over the 2009 Iranian presidential election; US Airways Flight 1549 crash-lands in the Hudson River with no fatalities, with the event becoming known as the "Miracle on the Hudson"; The "King of Pop" Michael Jackson died in 2009; Bitcoin is initially launched by the pseudonymous name Satoshi Nakamoto; the 2009 L'Aquila earthquake strikes central Italy; The H1N1 virus was responsible for the 2009 swine flu pandemic., 300x300px, thumb rect 0 0 200 200 Air France Flight 447 rect 200 0 400 200 first inauguration of Barack Obama rect 400 0 600 200 2009 Iranian presidential election protests rect 0 200 300 400 2009 swine flu pandemic rect 300 200 600 400 US Airways Flight 1549 rect 0 400 200 600 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) The identity morphism ( identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms. The exact definition of an automorphism depends on the type of " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class (set Theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid Russell's paradox (see ). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. In Quine's set-theoretical writing, the phrase "ultimate class" is often used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated ( doubling ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Most-perfect Magic Square
A most-perfect magic square of order ''n'' is a magic square containing the numbers 1 to ''n''2 with two additional properties: # Each 2 × 2 subsquare sums to 2''s'', where ''s'' = ''n''2 + 1. # All pairs of integers distant ''n''/2 along a (major) diagonal sum to ''s''. __TOC__ Examples Two 12 × 12 most-perfect magic squares can be obtained adding 1 to each element of: 1 2 3 4 5 6 7 8 9 10 11 12'' ,'' 64 92 81 94 48 77 67 63 50 61 83 78 ,'' 31 99 14 97 47 114 28 128 45 130 12 113 ,'' 24 132 41 134 8 117 27 103 10 101 43 118 ,'' 23 107 6 105 39 122 20 136 37 138 4 121 ,'' 16 140 33 142 0 125 19 111 2 109 35 126 ,'' 75 55 58 53 91 70 72 84 89 86 56 69 ,'' 76 80 93 82 60 65 79 51 62 49 95 66 ,'' 115 15 98 13 131 30 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F'' "). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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