High Quality Content by WIKIPEDIA articles! The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic. The works are Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics and Sophistical Refutations.The order of the works is not chronological (which is now hard to determine) but was deliberately chosen by Theophrastus to constitute a well-structured system. Indeed, parts of them seem to be a scheme of a lecture on logic. The arrangement of the works was made by Andronicus of Rhodes around 40 BC.Aristotle's Metaphysics has some points of overlap with the works making up the Organon but is not traditionally considered part of it; additionally there...

ISBN: 978-6-1313-0284-8
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High Quality Content by WIKIPEDIA articles! In surgery theory, a branch of mathematics, an orientation character on a group ? is a group homomorphism. Given a manifold M, one takes ? = ?1M (the fundamental group), and then ? sends an element of ? to ? 1 if and only if the class it represents is orientation-reversing. This map ? is trivial if and only if M is orientable. The orientation character is an algebraic structure on the fundamental group of a manifold, which captures which loops are orientation reversing and which are orientation preserving. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после...

ISBN: 978-6-1313-0279-4
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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a "clockwise" orientation for all loops in the manifold. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1303-3875-6
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High Quality Content by WIKIPEDIA articles! An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces). In comparison, an ordinary (i.e., non-oriented) matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered. All oriented matroids have an underlying matroid. Thus, results on ordinary matroids can be applied to oriented matroids. However, the converse is false; some matroids cannot become an oriented matroid by orienting an underlying structure...

ISBN: 978-6-1313-0274-9
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En mathematiques, une orientation est une convention a fixer pour l'objet etudie, dont la formulation depend de la nature de cet objet. Elle se revele necessaire par exemple, pour regler des problemes de signes. Si l'espace possede une orientation canonique, l'orientation d'espaces vectoriels ou de figures geometriques en general releve d'une convention arbitraire. Tous les objets concernes n'admettent pas forcement une orientation : si tel est le cas, ils sont dits orientables. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1326-3297-5
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High Quality Content by WIKIPEDIA articles! In mathematics and physics, the notion of orientation entanglement is sometimes used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be simply connected. Consider the following example. A coffee cup is suspended in a room by a pair of elastic rubber bands fixed to the walls of the room. The cup is rotated by its handle through a full twist of 360°, so that the handle is brought all the way around the central vertical axis of the cup and back to its original position. Note that after this rotation, the cup has been returned to its original orientation, but that its orientation with respect to...

ISBN: 978-6-1313-0277-0
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High Quality Content by WIKIPEDIA articles! The proof of Godel's completeness theorem given by Kurt Godel in his doctoral dissertation of 1929 (and a rewritten version of the dissertation, published as an article in 1930) is not easy to read today; it uses concepts and formalism that are outdated and terminology that is often obscure. The version given below attempts to faithfully represent all the steps in the proof and all the important ideas, while restating the proof in the modern language of mathematical logic. This outline should not be considered a rigorous proof of the theorem. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1303-2611-1
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High Quality Content by WIKIPEDIA articles! In mathematics, the Ornstein–Uhlenbeck operator can be thought of as a generalization of the Laplace operator to an infinite-dimensional setting. The Ornstein–Uhlenbeck operator plays a significant role in the Malliavin calculus.Often, when working on Rn, one works with respect to Lebesgue measure, which has many nice properties. However, remember that the aim is to work in infinite-dimensional spaces, and it is a fact that there is no infinite-dimensional Lebesgue measure. Instead, if one is studying some separable Banach space E, what does make sense is a notion of Gaussian measure; in particular, the abstract Wiener space construction makes sense. Данное издание не...

ISBN: 978-6-1313-0244-2
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High Quality Content by WIKIPEDIA articles! In mathematics, the Ornstein–Uhlenbeck process (named after Leonard Ornstein and George Eugene Uhlenbeck), also known as the mean-reverting process, is a stochastic process rt given by the following stochastic differential equation: dr_t = theta (mu-r_t),dt + sigma, dW_t,, where ? > 0, ? and ? > 0 are parameters and Wt denotes the Wiener process. The Ornstein–Uhlenbeck process is one of several approaches used to model (with modifications) interest rates, currency exchange rates, and commodity prices stochastically. The parameter ? represents the equilibrium or mean value supported by fundamentals; ? the degree of volatility around it caused by shocks, and ? the rate by which these...

ISBN: 978-6-1305-0140-2
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High Quality Content by WIKIPEDIA articles! The Ornstein–Uhlenbeck process is one of several approaches used to model (with modifications) interest rates, currency exchange rates, and commodity prices stochastically. The parameter ? represents the equilibrium or mean value supported by fundamentals; ? the degree of volatility around it caused by shocks, and ? the rate by which these shocks dissipate and the variable reverts towards the mean. The Ornstein–Uhlenbeck process is an example of a Gaussian process that has a bounded variance and admits a stationary probability distribution, in contrast to the Wiener process; the difference between the two is in their "drift" term. For the Wiener process the drift term is constant, whereas...

ISBN: 978-6-1313-0243-5
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High Quality Content by WIKIPEDIA articles! Oronce Fine (or Fine. In Latin, Orontius Finnaeus or Finaeus; Italian: Oronzio Fineo) (December 20, 1494-August 8, 1555) was a French mathematician and cartographer. Born in Briancon, the son and grandson of physicians, he was educated in Paris (College de Navarre), and obtained a degree in medicine in 1522. He was imprisoned in 1524, probably for practicing judicial astrology. In 1531, he was appointed to the chair of mathematics at the College Royal (the present College de France), founded by Francis I of France, where he taught until his death. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1303-9074-7
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High Quality Content by WIKIPEDIA articles! An orthogonal wavelet is a wavelet where the associated wavelet transform is orthogonal. That is the inverse wavelet transform is the adjoint of the wavelet transform. If this condition is weakened you may end up with biorthogonal wavelets.A necessary condition for the existence of a solution to the refinement equation is that some power (1+Z)A, A>0, divides the polynomial a(Z):=a_0+a_1Z+dots+a_{N-1}Z^{N-1} (see Z-transform). The maximally possible power A is called polynomial approximation order (or pol. app. power) or number of vanishing moments. It describes the ability to represent polynomials up to degree A-1 with linear combinations of integer translates of the scaling function....

ISBN: 978-6-1312-9583-6
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High Quality Content by WIKIPEDIA articles! In mathematics, orthogonal coordinates are defined as a set of d coordinates q = (q1, q2, ..., qd) in which the coordinate surfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant. For example, the three-dimensional Cartesian coordinates (x, y, z) is an orthogonal coordinate system, since its coordinate surfaces x=constant, y=constant, and z=constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. Данное издание не...

ISBN: 978-6-1303-1639-6
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High Quality Content by WIKIPEDIA articles! In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns (or rows) are orthogonal unit vectors (i.e., orthonormal). Because the columns are unit vectors in addition to being orthogonal, some people use the term orthonormal to describe such matrices. An orthogonal matrix Q is necessarily square and invertible, with inverse Q?1 = QT. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation. Данное издание не является оригинальным. Книга печатается...

ISBN: 978-6-1303-1564-1
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High Quality Content by WIKIPEDIA articles! The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A and B and asked to find an orthogonal matrix R which most closely maps A to B. Specifically, R = argmin_Omega |AOmega-B|_F quadmathrm{subject to}quad Omega^T Omega=I, where |cdot|_F denotes the Frobenius norm. There are a number of related problems to the classical orthogonal Procrustes problem. One might generalize it by seeking the closest matrix in which the columns are orthogonal, but not necessarily orthonormal. Alternately, one might constrain it by only allowing rotation matrices with determinant 1 (special orthogonal matrices) In this case, one can...

ISBN: 978-6-1312-9593-5
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High Quality Content by WIKIPEDIA articles! In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1,...,vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1,...,uk} that generate the same subspace as the vectors v1,...,vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span. In addition, if we want the resulting vectors to all be unit vectors, then the procedure is called orthonormalization. Colloquially, orthogonalization is the process of...

ISBN: 978-6-1312-9598-0
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High Quality Content by WIKIPEDIA articles! In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If M is a manifold equipped with a metric g, then an orthonormal frame at a point P of M is an ordered basis of the tangent space at P consisting of vectors which are orthonormal with respect to the bilinear form gP. A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since these lie in Euclidean spaces to...

ISBN: 978-6-1312-9643-7
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Orthogonal Polynomials. Binomial type, Polynomial sequence, Hermite polynomials, Laguerre polynomials, Jacobi polynomials, Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Generalized Fourier series, Sheffer sequence, Appell sequence, Umbral calculus, Secondary measure, Approximation theory Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1300-9061-6
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! An orthographic projection is a map projection of cartography. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective (or azimuthal) projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the Orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges, but distances are preserved along parallels (Snyder 1987, 1993). Orthographic projection was called "analemma" by the Greeks (see Ptolemy). The current name was adopted in 1613 by Francois...

ISBN: 978-6-1312-9615-4
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High Quality Content by WIKIPEDIA articles! In geometry, an orthocentric system is a set of four points in the plane one of which is the orthocenter of the triangle formed by the other three. If four points form an orthocentric system, then each of the four points is the orthocenter of the other three. These four possible triangles will all have the same nine-point circle. Consequently these four possible triangles must all have circumcircles with the same circumradius. The center of this common nine-point circle lies at the centroid of the four orthocentric points. The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points...

ISBN: 978-6-1313-0239-8
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High Quality Content by WIKIPEDIA articles! In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. In 2- or 3-dimensional Euclidean space, two vectors are orthogonal if their dot product is zero, i.e. they make an angle of 90° or ?/2 radians. Hence orthogonality of vectors is a generalization of the concept of perpendicular. In terms of Euclidean subspaces, the orthogonal complement of a line is the plane perpendicular to it, and vice versa. Note however that there is no correspondence with regards to perpendicular planes, because vectors in subspaces start from the origin. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1303-0057-9
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High Quality Content by WIKIPEDIA articles! In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. More generally the orthogonal group of a non-singular quadratic form over F is the group of linear operators preserving the form (the above group O(n,F) is then the orthogonal group of the sum-of-n-squares quadratic form). The Cartan–Dieudonne theorem describes the structure of the orthogonal group. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения...

ISBN: 978-6-1303-1616-7
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High Quality Content by WIKIPEDIA articles! In mathematics, in the field of ordinary differential equations, a non trivial solution to an ordinary differential equation F(x,y,y', dots, y^{(n-1)})=y^{(n)} quad x in [0,+infty) is called oscillating if it has an infinite number of roots, otherwise it is called non-oscillating. The differential equation is called oscillating if it has an oscillating solution. In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. Ordinary differential equations are distinguished from partial differential equations, which involve partial derivatives of several...

ISBN: 978-6-1312-9682-6
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Дата выхода: июль 2011

ISBN: 978-3-8383-6028-7
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The concept of a linewidth is borrowed from laser spectroscopy. The linewidth of a laser is a measure of its phase noise. The spectrogram of a laser is produced by passing its light through a prism. The spectrogram of the output of a pure noise-free laser will consist of a single infinitely thin line. If the laser exhibits phase noise, the line will have non-zero width. The greater the phase noise, the wider the line. The same will be true with oscillators. The spectrum of the output of a noise-free oscillator have energy at each of the harmonics of the output signal, but the bandwidth of each harmonic will be zero. If the oscillator exhibits phase noise, the harmonics will not have zero...

ISBN: 978-6-1312-9688-8
Издательство: Книга по требованию
Дата выхода: июль 2011