For example, the Netflix data set could have further obscured user identity by adding noise, permuting ordering and removing some activity at random, without noticeably compromising the result.
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Mathews invented the "Mathews' Algorithm," a method for producing literary works by transposing or permuting elements according to a predetermined set of rules.
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A 1954 patent application refers to "the Bell Telephone Gray code". Other names include "cyclic binary code", "cyclic progression code", "cyclic permuting binary" or "cyclic permuted binary" (CPB).
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The group application focuses on finite permutation groups. Basic properties of a group can be calculated like characters and conjugacy classes. Combined with a polytope, this application can compute properties associated with a group acting on a polytope by permuting the polytope's vertices, facets, or coordinates.
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Each clue in a Jumble word puzzle is a word that has been “jumbled” by permuting the letters of each word to make an anagram. A dictionary of such anagrams may be used to solve puzzles or verify that a jumbled word is unique when creating puzzles.
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The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to an S3 group of outer automorphisms of Spin(8).
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By symmetry, each edge of the K6 belongs to three perfect matchings. Incidentally, this partitioning of vertices into edge-vertices and matching-vertices shows that the Tutte-Coxeter graph is bipartite. Based on this construction, Coxeter showed that the Tutte–Coxeter graph is a symmetric graph; it has a group of 1440 automorphisms, which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b). The inner automorphisms of this group correspond to permuting the six vertices of the K6 graph; these permutations act on the Tutte–Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set.
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Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation ∗ (i.e., ) we can form five new operations: (the opposite operation), / and \, and their opposites. That makes a total of six quasigroup operations, which are called the conjugates or parastrophes of ∗.
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Unlike the situation with Latin squares, when two isotopic quasigroups are represented by Cayley tables (bordered Latin squares), the permutations and operate only on the border headings and do not move columns and rows, while operates on the body of the table. Permuting the rows and columns of a Cayley table (including the headings) does not change the quasigroup it defines, however, the Latin square associated with this table will be permuted to an isotopic Latin square. Thus, normalizing a Cayley table (putting the border headings in some fixed predetermined order by permuting rows and columns including the headings) preserves the isotopy class of the associated Latin square. Furthermore, if two normalized Cayley tables represent isomorphic quasigroups then their associated Latin squares are also isomorphic.
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The German procedure that sent an encrypted doubled key was the mistake that gave Rejewski a way in. Rejewski viewed the Enigma as permuting the plaintext letters into ciphertext. For each character position in a message, the machine used a different permutation.The permutations would be determined by the plugboard, the rotor order, the rotor positions, and the reflector.
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A cable-knit piece of fabric Cable knitting is a style of knitting in which textures of crossing layers are achieved by permuting stitches. For example, given four stitches appearing on the needle in the order ', one might cross the first two (in front of or behind) the next two, so that in subsequent rows those stitches appear in the new order '.
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A nice feature of SYNTAX (compared to Lex/Yacc) is its built-in algorithmPierre Boullier and Martin Jourdan. A New Error Repair and Recovery Scheme for Lexical and Syntactic Analysis. Science of Computer Programming 9(3): 271-286 (1987). for automatically recovering from lexical and syntactic errors, by deleting extra characters or tokens, inserting missing characters or tokens, permuting characters or tokens, etc.
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The two evolutionary models mentioned above describe ways in which genes may be circularly permuted, resulting in a circularly permuted mRNA after transcription. Proteins can also be circularly permuted via post- translational modification, without permuting the underlying gene. Circular permutations can happen spontaneously through autocatalysis, as in the case of concanavalin A. Alternately, permutation may require restriction enzymes and ligases.
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In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when used to multiply another matrix, say , results in permuting the rows (when pre-multiplying, to form ) or columns (when post-multiplying, to form ) of the matrix .
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The method describes solving the cube in a layer-by-layer fashion. First a "cross" is made on the first layer, consisting of the center piece and four edges. The first layer corners and edges of the second layer are put into their correct positions simultaneously (four pairs). The last layer is solved by first orienting and then permuting the last layer of the cube using a few sets of algorithms.
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Everything else proceeds as described above: upon completion, one has a unital associative algebra; one can take a quotient in either of the two ways described above. The above is exactly how the universal enveloping algebra for Lie superalgebras is constructed. One need only to carefully keep track of the sign, when permuting elements. In this case, the (anti-)commutator of the superalgebra lifts to an (anti-)commuting Poisson bracket.
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According to Singmaster's report on the 1982 World Rubik's Cube Championship, Fridrich was then using a basic layer method, while Dutch competitor Guus Razoux Schultz had a primitive F2L system. The last layer steps OLL and PLL involve first orienting the last layer pieces, then permuting them into their correct positions. This step was proposed by Hans Dockhorn and Anneke Treep. Fridrich switched to F2L later in 1982.
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Fix a partition λ of n and a commutative ring k. The partition determines a Young diagram with n boxes. A Young tableau of shape λ is a way of labelling the boxes of this Young diagram by distinct numbers 1, \dots, n. A tabloid is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row.
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The defining property of the Vandermonde polynomial is that it is alternating in the entries, meaning that permuting the X_i by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the polynomial – in fact, it is the basic alternating polynomial, as will be made precise below. It thus depends on the order, and is zero if two entries are equal – this also follows from the formula, but is also consequence of being alternating: if two variables are equal, then switching them both does not change the value and inverts the value, yielding V_n = -V_n, and thus V_n = 0 (assuming the characteristic is not 2, otherwise being alternating is equivalent to being symmetric). Conversely, the Vandermonde polynomial is a factor of every alternating polynomial: as shown above, an alternating polynomial vanishes if any two variables are equal, and thus must have (X_i - X_j) as a factor for all i eq j.
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Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a (finite- dimensional) vector space is called a representation of the group. It allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set with elements by permuting the elements of the set.
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Repeat this procedure until reaching the end of the vertical line. Then the player is given the thing written at the bottom of the line. If the elements written above the Ghost Leg are treated as a sequence, and after the Ghost Leg is used, the same elements are written at the bottom, then the starting sequence has been transformed to another permutation. Hence, Ghost Leg can be treated as a kind of permuting operator.
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Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row.
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Heat maps originated in 2D displays of the values in a data matrix. Larger values were represented by small dark gray or black squares (pixels) and smaller values by lighter squares. Loua (1873) used a shading matrix to visualize social statistics across the districts of Paris. Sneath (1957) displayed the results of a cluster analysis by permuting the rows and the columns of a matrix to place similar values near each other according to the clustering.
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The infinite symmetric product SP(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xn) with (x1,...,xn,e) and identifying any point with any other point given by permuting its coordinates. In other words its underlying set is the free commutative monoid generated by X (with unit e), and is the abelianization of the James reduced product.
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The set T3 of rooted ternary trees consists of rooted trees where every node (or non-leaf vertex) has exactly three children (leaves or subtrees). Small ternary trees are shown at right. Note that rooted ternary trees with n nodes are equivalent to rooted trees with n vertices of degree at most 3 (by ignoring the leaves). In general, two rooted trees are isomorphic when one can be obtained from the other by permuting the children of its nodes.
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Let x_1, \dots, x_n be a sequence of positive real numbers and P a permutation operator, then the following properties hold: #\min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n). #:Each generalized mean always lies between the smallest and largest of the x values. #M_p(x_1, \dots, x_n) = M_p(P(x_1, \dots, x_n)). #:Each generalized mean is a symmetric function of its arguments; permuting the arguments of a generalized mean does not change its value.
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For a graph G on n vertices, let e_k denote the number of colorings using exactly k colors up to renaming colors (so colorings that can be obtained from one another by permuting colors are counted as one; colorings obtained by automorphisms of G are still counted separately). In other words, e_k counts the number of partitions of the vertex set into k (non-empty) independent sets. Then k! \cdot e_k counts the number of colorings using exactly k colors (with distinguishable colors).
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A solution commonly used by speedcubers was developed by Jessica Fridrich. This method is called CFOP standing for "cross, F2L, OLL, PLL". It is similar to the layer- by-layer method but employs the use of a large number of algorithms, especially for orienting and permuting the last layer. The cross is done first, followed by first layer corners and second layer edges simultaneously, with each corner paired up with a second-layer edge piece, thus completing the first two layers (F2L).
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The Gosset graph can be explicitly constructed as follows: the 56 vertices are the vectors in R8, obtained by permuting the coordinates and possibly taking the opposite of the vector (3, 3, −1, −1, −1, −1, −1, −1). Two such vectors are adjacent when their inner product is 8\. An alternative construction is based on the 8-vertex complete graph K8. The vertices of the Gosset graph can be identified with two copies of the set of edges of K8.
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The Hilbert space for n particles is given by the tensor product \otimes_n H . The permutation group of S_n acts on this space by permuting the entries. By definition the expectation values for an observable a of n indistinguishable particles should be invariant under these permutation. This means that for all \psi \in H and \sigma \in S_n : (\sigma \Psi )^t a (\sigma \Psi) = \Psi^t a \Psi, or equivalently for each \sigma \in S_n : \sigma^t a \sigma = a .
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The group held a number of meetings over the years, which included Brazilian finance minister Ciro Gomes, Chilean senator Carlos Ominami, Argentinian politicians Dante Caputo and Rodolfo Terragno, and Mexican politician and future president Vicente Fox.Castaneda, Jorge. "Mexico: Permuting Power", New Left Review 7 (January–February 2001); accessed 13 December 2015. The meetings resulted in a document entitled the "Buenos Aires Consensus" in 1997,"A Latin American Alternative" document signed at Buenos Aires in November 1997; accessed 11 December 2015.
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Painting of Pierrot, the object of Schoenberg's atonal suite Pierrot Lunaire, painted by Antoine Watteau A second direction in the search for a new tonality was twelve-tone serialism. Arnold Schoenberg developed the twelve-tone method of composition as an alternative to the structure provided by the diatonic system. His method entails building a piece using a series of the twelve notes of the chromatic scale, permuting it and superimposing it on itself to create the composition. Schoenberg did not arrive immediately at the serial method.
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The reasoning is that if the null hypothesis of there being no relation between the two matrices is true, then permuting the rows and columns of the matrix should be equally likely to produce a larger or a smaller coefficient. In addition to overcoming the problems arising from the statistical dependence of elements within each of the two matrices, use of the permutation test means that no reliance is being placed on assumptions about the statistical distributions of elements in the matrices. Many statistical packages include routines for carrying out the Mantel test.
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Robin Beanland, the game's third composer, only wrote the elevator music that can be heard in certain levels. All the sound effects were created by Norgate and a lot of effort was put into combining and permuting sounds in different ways to give the game a satisfying feel. According to Hollis, whenever the player shoots a gun, up to nine different sound effects will randomly trigger. When the game was reviewed by Nintendo shortly before it was released, the company was slightly concerned about the amount of violence and gunplay.
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This is then followed by orienting the last layer, then permuting the last layer (OLL and PLL respectively). Fridrich's solution requires learning roughly 120 algorithms but allows the Cube to be solved in only 55 moves on average. A now well-known method was developed by Lars Petrus. In this method, a 2×2×2 section is solved first, followed by a 2×2×3, and then the incorrect edges are solved using a three-move algorithm, which eliminates the need for a possible 32-move algorithm later.
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For example, for three variables , , and , the expressions :a \wedge b \wedge c :a \wedge (b \vee c) :(a \wedge b)\vee c, and :a\vee b\vee c are all read-once (as are the other functions obtained by permuting the variables in these expressions). However, the Boolean median operation, given by the expression :(a\vee b)\wedge (a\vee c)\wedge (b\wedge c) is not read-once: this formula has more than one copy of each variable, and there is no equivalent formula that uses each variable only once., p. 520.
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Beginning in 1970 interests in Asian classical music and a wish to be able to find the intervals he had been using in his work led Young to pursue studies with Pandit Pran Nath. Fellow students included Zazeela, composers Terry Riley, Michael Harrison, and Yoshi Wada, philosophers Henry Flynt and Catherine Christer Hennix and many others. Young considers The Well-Tuned Piano—a permuting composition of themes and improvisations for just-intuned solo piano—to be his masterpiece. Young gave the world premiere of The Well-Tuned Piano in Rome in 1974, ten years after the creation of the piece.
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Bit 4 selects between two possible key schedules: one using the key to seed a pseudorandom number generator, the other using BassOmatic itself. Making such variations key-dependent means that some keys must be weaker than others; the key space is not flat. The chosen key schedule produces a total of 8 permutation tables, each a permutation of the numbers 0 to 255. Each round consists of 4 operations: XORing the block with one of the permutation tables, shredding or permuting individual bits throughout the block, an unkeyed diffusion called raking, and a substitution step using the permutation tables as S-boxes.
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A map of the 24 permutations and the 23 swaps used in Heap's algorithm permuting the four letters A (amber), B (blue), C (cyan) and D (dark red) Wheel diagram of all permutations of length n=4 generated by Heap's algorithm, where each permutation is color-coded (1=blue, 2=green, 3=yellow, 4=red). Heap's algorithm generates all possible permutations of objects. It was first proposed by B. R. Heap in 1963. The algorithm minimizes movement: it generates each permutation from the previous one by interchanging a single pair of elements; the other elements are not disturbed.
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There is some freedom of choice regarding the technical details of how sequents and structural rules are formalized. As long as every derivation in LK can be effectively transformed to a derivation using the new rules and vice versa, the modified rules may still be called LK. First of all, as mentioned above, the sequents can be viewed to consist of sets or multisets. In this case, the rules for permuting and (when using sets) contracting formulae are obsolete. The rule of weakening will become admissible, when the axiom (I) is changed, such that any sequent of the form \Gamma , A \vdash A , \Delta can be concluded.
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Cable needles are typically very short (a few inches), and are used to hold stitches temporarily while others are being knitted. Cable patterns are made by permuting the order of stitches; although one or two stitches may be held by hand or knit out of order, cables of three or more generally require a cable needle. The third needle type consists of circular needles, which are long, flexible double-pointed needles. The two tapered ends (typically long) are rigid and straight, allowing for easy knitting; however, the two ends are connected by a flexible strand (usually nylon) that allows the two ends to be brought together.
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The CFOP method (Cross – F2L – OLL – PLL), sometimes known as the Fridrich method, is one of the most commonly used methods in speedsolving a 3×3×3 Rubik's Cube. This method was first developed in the early 1980s combining innovations by a number of speed cubers. Czech speedcuber and the namesake of the method Jessica Fridrich is generally credited for popularizing it by publishing it online in 1997. The method works on a layer-by-layer system, first solving a cross typically on the bottom, continuing to solve the first two layers (F2L), orienting the last layer (OLL), and finally permuting the last layer (PLL).
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For example, there are well-known algorithms for cycling three corners without changing the rest of the puzzle or flipping the orientation of a pair of edges while leaving the others intact. Some algorithms do have a certain desired effect on the cube (for example, swapping two corners) but may also have the side-effect of changing other parts of the cube (such as permuting some edges). Such algorithms are often simpler than the ones without side-effects and are employed early on in the solution when most of the puzzle has not yet been solved and the side-effects are not important. Most are long and difficult to memorise.
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In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product :C × C × ... × C or Cn by the group action of the symmetric group on n letters permuting the factors. It exists as a smooth algebraic variety ΣnC; if C is a compact Riemann surface it is therefore a complex manifold. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients. For C the projective line (say the Riemann sphere) ΣnC can be identified with projective space of dimension n.
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Now take S to be the set of sequences of k elements selected from our n-element set without repetition. On one hand, there is an easy bijection of S with the Cartesian product corresponding to the numerator n(n-1)\cdots(n-k+1), and on the other hand there is a bijection from the set C of pairs of a k-combination and a permutation σ of k to S, by taking the elements of C in increasing order, and then permuting this sequence by σ to obtain an element of S. The two ways of counting give the equation :n(n-1)\cdots(n-k+1)=\binom nk k!, and after division by k! this leads to the stated formula for \tbinom nk.
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However, one could also say "two different linear combinations can have the same value" in which case the reference is to the expression. The subtle difference between these uses is the essence of the notion of linear dependence: a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each vi; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations. In a given situation, K and V may be specified explicitly, or they may be obvious from context.
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221 polytope. This symmetric projection contains 2 rings of 12 vertices, and 3 vertices coinciding at the center. The intersection graph of the 27 lines on a cubic surface is a locally linear graph that is the complement of the Schläfli graph. That is, two vertices are adjacent in the Schläfli graph if and only if the corresponding pair of lines are skew.. The Schläfli graph may also be constructed from the system of eight-dimensional vectors :(1, 0, 0, 0, 0, 0, 1, 0), :(1, 0, 0, 0, 0, 0, 0, 1), and :(−1/2, −1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2), and the 24 other vectors obtained by permuting the first six coordinates of these three vectors.
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Within each directory, directory entries appear as directory items, whose right-hand key values are a CRC32C hash of their filename. Their data is a location key, or the key of the inode item it points to. Directory items together can thus act as an index for path-to-inode lookups, but are not used for iteration because they are sorted by their hash, effectively randomly permuting them. This means user applications iterating over and opening files in a large directory would thus generate many more disk seeks between non-adjacent files—a notable performance drain in other file systems with hash-ordered directories such as ReiserFS, ext3 (with Htree- indexes enabled) and ext4, all of which have TEA-hashed filenames.
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The vertices of a 600-cell centered at the origin of 4-space, with edges of length (where φ = is the golden ratio), can be given as follows: 16 vertices of the form: :(±, ±, ±, ±), and 8 vertices obtained from :(0, 0, 0, ±1) by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of :(±φ, ±1, ±, 0). Note that the first 16 vertices are the vertices of a tesseract, the second eight are the vertices of a 16-cell, and that all 24 vertices together are vertices of a 24-cell. The final 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.
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Since the order of multiplication does not matter, one can switch and and the values of and will not change: one can say that and are symmetric polynomials in and . In fact, they are the elementary symmetric polynomials – any symmetric polynomial in and can be expressed in terms of and The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one "break the symmetry" and recover the roots? Thus solving a polynomial of degree is related to the ways of rearranging ("permuting") terms, which is called the symmetric group on letters, and denoted . For the quadratic polynomial, the only way to rearrange two terms is to swap them ("transpose" them), and thus solving a quadratic polynomial is simple.
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Consider a finite group G permuting those indeterminates over K. By standard Galois theory, the set of fixed points of this group action is a subfield of L, typically denoted L^G. The rationality question for K \subset L^G is called Noether's problem and asks if this field of fixed points is or is not a purely transcendental extension of K. In the paper on Galois theory she studied the problem of parameterizing the equations with given Galois group, which she reduced to "Noether's problem". (She first mentioned this problem in where she attributed the problem to E. Fischer.) She showed this was true for n = 2, 3, or 4. found a counter-example to the Noether's problem, with n = 47 and G a cyclic group of order 47.
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Bayesian inference can be used to produce phylogenetic trees in a manner closely related to the maximum likelihood methods. Bayesian methods assume a prior probability distribution of the possible trees, which may simply be the probability of any one tree among all the possible trees that could be generated from the data, or may be a more sophisticated estimate derived from the assumption that divergence events such as speciation occur as stochastic processes. The choice of prior distribution is a point of contention among users of Bayesian-inference phylogenetics methods. Implementations of Bayesian methods generally use Markov chain Monte Carlo sampling algorithms, although the choice of move set varies; selections used in Bayesian phylogenetics include circularly permuting leaf nodes of a proposed tree at each step and swapping descendant subtrees of a random internal node between two related trees.
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Symbolically, Tn = (S1)n. The configuration space of unordered, not necessarily distinct points is accordingly the orbifold Tn/Sn, which is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates). For n = 2, the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 this quotient may be described as a solid torus with cross-section an equilateral triangle, with a twist; equivalently, as a triangular prism whose top and bottom faces are connected with a 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical.
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To some debaters, Significance derives from the word "substantially", which appears in most resolutions, and one can argue that Significance has been subsumed by the option for the Negative team to argue nontopicality on that word against the Affirmative team, then the Negative would lose on the stricture against permuting. In Push Debate, topicality does not need extraordinary defense nor flimpsy probing, and the traditional stock issue Significance is preserved if nothing could be done about Inherency that would be nontopical. The difference is between saying "our plan is significantly (or substantially) topical because it is a specific implementation of the resolution", which does not mean much other than it is minimal in terms of Grounding, and "our plan's solvency is significant (or substantial)", which is what judges are looking for about plans and the resolution in the "benevolent debate" that is not bogged down in wordiness.
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Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory, and algebraic geometry. A first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange, who observed that permuting the zeros of a cubic polynomial in the expression : (with being a third root of unity) only yields two values. This way, Lagrange conceptually explained the classical solution method of Scipione del Ferro and François Viète, which proceeds by reducing a cubic equation for an unknown to a quadratic equation for . Together with a similar observation for equations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups. Vandermonde, also in 1770, and to a fuller extent, Carl Friedrich Gauss, in his Disquisitiones Arithmeticae (1801), studied the equation : for a prime and, again using modern language, the resulting cyclic Galois group.
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A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime. A circular prime with at least two digits can only consist of combinations of the digits 1, 3, 7 or 9, because having 0, 2, 4, 6 or 8 as the last digit makes the number divisible by 2, and having 0 or 5 as the last digit makes it divisible by 5. The complete listing of the smallest representative prime from all known cycles of circular primes (The single- digit primes and repunits are the only members of their respective cycles) is 2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, R19, R23, R317, R1031, R49081, R86453, R109297, and R270343, where Rn is a repunit prime with n digits.
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In other words, it is the unique binary search tree of the rational numbers in which the parent of any vertex q has a smaller denominator than q (or if q and its parent are both integers, in which the parent is smaller than q). It follows from the theory of Cartesian trees that the lowest common ancestor of any two numbers q and r in the Stern–Brocot tree is the rational number in the closed interval [q, r] that has the smallest denominator among all numbers in this interval. Permuting the vertices on each level of the Stern–Brocot tree by a bit-reversal permutation produces a different tree, the Calkin–Wilf tree, in which the children of each number a/b are the two numbers a/(a + b) and (a + b)/b. Like the Stern–Brocot tree, the Calkin–Wilf tree contains each positive rational number exactly once, but it is not a binary search tree.
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The set of n×n generalized permutation matrices with entries in a field F forms a subgroup of the general linear group GL(n,F), in which the group of nonsingular diagonal matrices Δ(n, F) forms a normal subgroup. Indeed, the generalized permutation matrices are the normalizer of the diagonal matrices, meaning that the generalized permutation matrices are the largest subgroup of GL in which diagonal matrices are normal. The abstract group of generalized permutation matrices is the wreath product of F× and Sn. Concretely, this means that it is the semidirect product of Δ(n, F) by the symmetric group Sn: :Δ(n, F) ⋉ Sn, where Sn acts by permuting coordinates and the diagonal matrices Δ(n, F) are isomorphic to the n-fold product (F×)n. To be precise, the generalized permutation matrices are a (faithful) linear representation of this abstract wreath product: a realization of the abstract group as a subgroup of matrices.
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Seamlessly weaving together excerpts from the Polieri-Thibaudeau scores with new material, Barraqué used his technique of proliferating series to obtain new tone rows from the one originally used for the theatre music: C–F–A–F–B–E–E–G–A–B–D–C. In this case, the method involved comparing the inversion of the row: C–A–G–C–E–D–B– E–B–F–F–A Observing that this is equivalent to permuting the original row in the order 1–3–8–12–6–11–10–7–5–4–2–9, applying the same permutation to the inverted row produced C–G–E–A–D–F–F–B–E–C–G–B, a row that includes three conventional triads: C minor (notes 1–3), D major (notes 4–6), and C minor (notes 9–11). These chords, not chosen deliberately by the composer but engendered "automatically" by the method, gave Barraqué "serial permission to re-encounter elements of the tonal past, as figures in a dream" .
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Although Desargues' theorem chooses different roles for its ten lines and points, the Desargues configuration itself is more symmetric: any of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity. The Desargues configuration has a symmetry group S5 of order 120; that is, there are 120 different ways of permuting the points and lines of the configuration in a way that preserves its point-line incidences . The three-dimensional construction of the Desargues configuration makes these symmetries more readily apparent: if the configuration is generated from five planes in general position in three dimensions, then each of the 120 different permutations of these five planes corresponds to a symmetry of the configuration . The Desargues configuration is self-dual, meaning that it is possible to find a correspondence from points of one Desargues configuration to lines of a second configuration, and from lines of the first configuration to points of a second configuration, in such a way that all of the configuration's incidences are preserved .
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