High Quality Content by WIKIPEDIA articles! In Euclidean plane geometry, tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles. A tangent line t to a circle C intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In...

ISBN: 978-6-1303-4827-4
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High Quality Content by WIKIPEDIA articles! In mathematics, an tangle can mean one of two related concepts: * In John Conway's definition, a n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2n marked points on the ball's boundary. * In link theory, a tangle is an embedding of n arcs and m circles into mathbf{R}^2 times [0,1] – the difference from the previous definition is that it includes circles as well as arches, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance. Данное издание не является оригинальным....

ISBN: 978-6-1311-5645-8
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High Quality Content by WIKIPEDIA articles! Tanh-sinh quadrature is a method for numerical integration introduced by Hidetosi Takahasi and Masatake Mori in 1974. It uses the change of variables x = tanh(tfrac12 pi sinh t), to transform an integral on (?1, 1) to an integral on the entire real line. After this transformation, the integrand decays with a double exponential rate, and thus, this method is also known as the double exponential (DE) formula. For a given step size h, the integral is approximated by the sum int_{-1}^1 f(x) , dx approx sum_{k=-infty}^infty w_k f(x_k). with the abscissas x_k = tanh(tfrac12 pi sinh kh) and the weights w_k = frac{tfrac12 h pi cosh kh}{cosh^2(tfrac12 pi sinh kh)}. Данное издание не...

ISBN: 978-6-1311-5632-8
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High Quality Content by WIKIPEDIA articles! In mathematics, a tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory. The name is taken from Tannaka-Krein duality, a theory about compact groups G and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some...

ISBN: 978-6-1311-5617-5
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High Quality Content by WIKIPEDIA articles! In computer science, Tarjan's off-line least common ancestors algorithm (more precisely, least should actually be lowest) is an algorithm for computing lowest common ancestors for pairs of nodes in a tree, based on the union-find data structure. The lowest common ancestor of two nodes d and e in a rooted tree T is the node g that is an ancestor of both d and e and that has the greatest depth in T. It is named after Robert Tarjan, who discovered the technique in 1979. Tarjan's algorithm is offline; that is, unlike other lowest common ancestor algorithms, it requires that all pairs of nodes for which the lowest common ancestor is desired must be specified in advance. The simplest version of the...

ISBN: 978-6-1311-7097-3
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High Quality Content by WIKIPEDIA articles! The basic idea of the algorithm is this: a depth-first search begins from a start node. The strongly connected components form the subtrees of the search tree, the roots of which are the roots of the strongly connected components. The nodes are placed on a stack in the order in which they are visited. When the search returns from a subtree, the nodes are taken from the stack and it is determined whether each node is the root of a strongly connected component. If a node is the root of a strongly connected component, then it and all of the nodes taken off before it form that strongly connected component. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1311-7126-0
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High Quality Content by WIKIPEDIA articles! Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory. Other modern axiomizations of Euclidean geometry are those by Hilbert and George Birkhoff. Like other modern axiomatizations of Euclidean geometry, Tarski's employs a formal system consisting of symbol strings, called sentences, whose construction respects formal syntactical rules, and rules of proof that determine the allowed manipulations of the sentences. Unlike some other modern axiomatizations, such as Birkhoff's and Hilbert's, Tarski's axiomatization has no primitive objects...

ISBN: 978-6-1303-4848-9
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High Quality Content by WIKIPEDIA articles! Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklos Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich's decomposition uses about 1050 different pieces. In particular, it is impossible to dissect a circle and make a square using pieces that could be cut with scissors (that is, having Jordan curve boundary). The pieces used in Laczkovich's proof are non-measurable subsets. Laczkovich actually proved the reassembly can be done...

ISBN: 978-6-1311-7212-0
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High Quality Content by WIKIPEDIA articles! In 1936, Alfred Tarski set out an axiomatization of the real numbers and their arithmetic, consisting of only the 8 axioms shown below and a mere four primitive notions: the set of reals denoted R, a binary total order over R, denoted by infix <, a binary operation of addition over R, denoted by infix +, and the constant 1. The literature occasionally mentions this axiomatization but never goes into detail, notwithstanding its economy and elegant metamathematical properties. This axiomatization appears little known, possibly because of its second-order nature. Tarski's axiomatization can be seen as a version of the more usual definition of real numbers as the unique Dedekind-complete ordered...

ISBN: 978-6-1311-7176-5
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High Quality Content by WIKIPEDIA articles! In 1931, Kurt Godel published his famous incompleteness theorems, which he proved in part by showing how to represent syntax within (first-order) arithmetic. Each expression of the language of arithmetic is assigned a distinct number. This procedure is known variously as Godel-numbering, coding, and more generally, as arithmetization. In particular, various sets of expressions are coded as sets of numbers. It turns out that for various syntactic properties (such as being a formula, being a sentence, etc.), these sets are computable. Moreover, any computable set of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of arithmetic defining the set...

ISBN: 978-6-1311-7221-2
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High Quality Content by WIKIPEDIA articles! In mathematics, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by A. Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд...

ISBN: 978-6-1311-7145-1
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklos Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich's decomposition uses about 1050 different pieces. In particular, it is impossible to dissect a circle and make a square using pieces that could be cut with scissors (that is, having Jordan curve boundary). The pieces used in Laczkovich's proof are non-measurable subsets. Данное издание не является...

ISBN: 978-6-1311-7201-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from etale cohomology. It is unsolved in the general case, as of 2009[update], and, like the Hodge conjecture to which it is related at the level of some important analogies, it is generally taken to be one of the major problems in the field. Tate's original statement runs as follows. Let V be a smooth algebraic variety over a field k, which is finitely-generated over its prime field. Let G be the absolute Galois group of k. Fix a prime number l. Write H*(V) for the l-adic cohomology (coefficients in the l-adic...

ISBN: 978-6-1311-7239-7
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High Quality Content by WIKIPEDIA articles! In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules is a motivation, but the cohomology can be defined using various constructions. There is a dual theory, group homology, and a generalization to non-abelian coefficients. These algebraic ideas are closely related to topological ideas. A great deal is known about the cohomology of groups, including interpretations of low dimensional cohomology, functorality, and how to change groups. The history of group...

ISBN: 978-6-1311-7234-2
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High Quality Content by WIKIPEDIA articles! In number theory and algebraic geometry, the Tate twist, named after John Tate, is an operation on Galois modules. For example, if K is a field, GK is its absolute Galois group, and ? : GK ? AutK(V) is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers, then the Tate twist of V, denoted V(1), is the representation on the tensor product V?Qp(1), where Qp(1) is the p-adic cyclotomic character (i.e. the Tate module of the group of roots of unity in the separable closure Ks of K). More generally, if m is a positive integer, the mth Tate twist of V, denoted V(m), is the tensor product of V with the m-fold tensor product of Qp(1). Denoting by Qp(?1) the...

ISBN: 978-6-1311-7252-6
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High Quality Content by WIKIPEDIA articles! Because the eta function is easy to compute numerically from either power series, it is often helpful in computation to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms. The picture on this page shows the modulus of the Euler function: the additional factor of q1 / 24 between this and eta makes almost no visual difference whatsoever (it only introduces a tiny pinprick at the origin). Thus, this picture can be taken as a picture of eta as a function of q. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1311-7262-5
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold. (Multiple covers of 2-tori with self-intersection –1 are also counted.) Taubes proved the information contained in this invariant is equivalent to invariants derived from the Seiberg–Witten equations in a series of four long papers. Much of the analytical complexity connected to this invariant comes from properly counting multiply-covered pseudoholomorphic curves. The crux is a topologically defined index for pseudoholomorphic curves which controls embeddedness...

ISBN: 978-6-1311-7267-0
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High Quality Content by WIKIPEDIA articles! In mathematics, a taut foliation is a codimension 1 foliation of a 3-manifold with the property there is a single transverse circle intersecting every leaf. By transverse circle, it is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Taut foliations were brought to prominence by the work of William Thurston and David Gabai. It is closely related to the concept of Reebless foliation. A taut foliation cannot have a Reeb component, since the component would act like a "dead-end" from which a transverse curve...

ISBN: 978-6-1311-7275-5
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High Quality Content by WIKIPEDIA articles! In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T*Q of a manifold Q. The exterior derivative of this form defines a symplectic form giving T*Q the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1311-7326-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, tautological bundle is a term for a particularly natural vector bundle occurring over a Grassmannian, and more specially over projective space. Canonical bundle as a name dropped out of favour, on the grounds that 'canonical' is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the canonical class in algebraic geometry could scarcely be avoided. Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space W. If G is a Grassmannian, and Vg is the subspace of W corresponding to g in G, this is already almost the data required for a vector bundle: namely a vector space for each point g,...

ISBN: 978-6-1311-7314-1
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics a projective space is a set of elements similar to the set P(V) of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively. The idea of a projective space relates to perspective, more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points which lie on a projection line (i.e. a "line-of-sight"), intersecting with the focal point of the camera, are projected onto a common image point. In this case the vector space is R3 with the camera focal point at the origin and the projective space...

ISBN: 978-6-1311-7321-9
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High Quality Content by WIKIPEDIA articles! This solution was later used to attack the problem of the brachistochrone curve. Jakob Bernoulli solved the problem using calculus in a paper (Acta Eruditorum, 1690) that saw the first published use of the term integral. The tautochrone problem was studied more closely when it was realized that a pendulum, which follows a circular path, was not isochronous and thus his pendulum clock would keep different time depending on how far the pendulum swung. After determining the correct path, Christiaan Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve. These attempts proved to not be...

ISBN: 978-6-1311-7285-4
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High Quality Content by WIKIPEDIA articles! Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates. The taxicab metric is also known as rectilinear distance, L1 distance or ell1 norm, city block distance, Manhattan distance, or Manhattan length, with corresponding variations in the name of the geometry. The latter names allude to the grid layout of most streets on the island of Manhattan, which causes the shortest path a car could take between two points in the city to have length equal to the points' distance in taxicab...

ISBN: 978-6-1303-2013-3
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In calculus, Taylor's theorem gives a sequence of approximations of a differentiable function around a given point by polynomials (the Taylor polynomials of that function) whose coefficients depend only on the derivatives of the function at that point. The theorem also gives precise estimates on the size of the error in the approximation. The theorem is named after the mathematician Brook Taylor, who stated it in 1712, though the result was first discovered 41 years earlier in 1671 by James Gregory. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения...

ISBN: 978-6-1303-4880-9
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High Quality Content by WIKIPEDIA articles! In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is named after the English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после...

ISBN: 978-6-1303-3547-2
Издательство: Книга по требованию
Дата выхода: июль 2011