Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semi-simple Lie algebras. We show the construction of a root system from a semi-simple Lie algebra and conversely, the construction of a semi-simple Lie algebra from a reduced crystallographic root system. Let g be a semi-simple complex Lie algebra. Let further h be a Cartan subalgebra of g, i.e. a maximal abelian subalgebra. Then h acts on g via simultaneously diagonalizable linear maps in the adjoint representation. For in h* define mathfrak{g}_lambda := {ainmathfrak{g}: [h,a]=lambda(h)atext{ for all...

ISBN: 978-6-1312-6107-7
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In control theory, the root locus is the locus of the roots of the characteristic equation of the closed loop transfer function as the loop gain of the feedback system is increased from zero to infinty. Traditionally the root locus is constructed with the loop gain K as the variable. However, any system parameter can be used to construct a root locus. The root locus is a useful tool for analyzing the transient response, as well as the stability of a single input single output dynamic systems. A system is stable if all of its poles are in the left-hand side of the s-plane or inside the unit circle of the z-plane. Данное издание не является оригинальным....

ISBN: 978-6-1303-1375-3
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High Quality Content by WIKIPEDIA articles! In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.It is particularly useful in connection with power series.The root test was developed first by Cauchy and so is sometimes known as the Cauchy root test or Cauchy's radical test. Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1329-7412-9
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High Quality Content by WIKIPEDIA articles! In mathematics, the root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids. Vit can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values. It is a special case of the power mean with the exponent p = 2. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения...

ISBN: 978-6-1304-8251-0
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High Quality Content by WIKIPEDIA articles! A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. Such an x is called a root of the function f. This article is concerned with finding scalar, real or complex roots, approximated as floating point numbers. Finding integer roots or exact algebraic roots are separate problems, whose algorithms have little in common with those discussed here. Finding a root of f(x) ? g(x) = 0 is the same as solving the equation f(x) = g(x). Here, x is called the unknown in the equation. Conversely, any equation can take the canonical form f(x) = 0, so equation solving is the same thing as computing (or finding) a root of a function....

ISBN: 978-6-1312-6069-8
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High Quality Content by WIKIPEDIA articles! The Rootare-Prenzlow Equation is named for Estonian-American scientist Hillar Rootare and American scientist Carl Prenzlow, first published in their 1967 paper, "Surface Areas from Mercury Porosimetry Measurements," Rootare, H.M. , and Prenzlow, C.F., 71 J. Phys. Chem. p. 2733 (1967). The equation first formulated a means to calculate cumulative surface areas of porous solids based on data taken in mercury porosimetry testing. Rootare and Spencer later devised a computer programme to carry out automated calculations, "A Computer Program for Pore Volume and Pore Area Distribution," Rootare Данное издание не является оригинальным. Книга...

ISBN: 978-6-1311-5456-0
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High Quality Content by WIKIPEDIA articles! In knot theory each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are called ideal knots and ideal links respectively.The ropelength of a knot curve C is defined as the ratio {scriptstyle L, =, operatorname{Len}(C)/ tau(C)}, where ?(C) is the thickness of the link defined by C.In our terms we are asking if there is a knot with ropelength 12. This question has been answered, and it was shown to be impossible. However, the search for the answer has spurred a lot of research on both theoretical and computational ground. It has been...

ISBN: 978-6-1312-6058-2
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728. the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole with the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth. Данное издание не...

ISBN: 978-6-1329-7472-3
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High Quality Content by WIKIPEDIA articles! In mathematics, a rose (also known as a bouquet of circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are closely related to free groups.A rose is a wedge sum of circles. That is, the rose is the quotient space C/S, where C is a disjoint union of circles and S a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, and one edge for each circle. This makes it a simple example of a topological graph. Данное издание не является оригинальным. Книга печатается...

ISBN: 978-6-1312-6026-1
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High Quality Content by WIKIPEDIA articles! In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. Up to similarity, these curves can all be expressed by a polar equation of the form !,r=cos(ktheta). If k is an integer, the curve will be rose shaped with 2k petals if k is even, and k petals if k is odd. When k is even, the entire graph of the rose will be traced out exactly once when the value of changes from 0 to 2 . When k is odd, this will happen on the interval between 0 and . (More generally, this will happen on any interval of length 2 for k even, and for k odd.) Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1312-6039-1
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a rose (also known as a bouquet of circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are closely related to free groups.A rose is a wedge sum of circles. That is, the rose is the quotient space C/S, where C is a disjoint union of circles and S a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, and one edge for each circle. This makes it a simple example of a topological graph. Данное издание не является оригинальным. Книга печатается...

ISBN: 978-6-1329-7501-0
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematical optimization, the Rosenbrock function is a non-convex function used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Rosenbrock's banana function. The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult. It is defined by f(x, y) = (1-x)^2 + 100(y-x^2)^2 .quad. It has a global minimum at (x,y) = (1,1) where f(x,y) = 0. A different coefficient of the second term is sometimes given, but this does not affect the position of the global minimum. Two variants are commonly encountered. One is the sum of N / 2 uncoupled 2D...

ISBN: 978-6-1312-5603-5
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematical optimization, the Rosenbrock function is a non-convex function used as a performance test problem for optimization algorithms introduced by Rosenbrock (1960). It is also known as Rosenbrock's valley or Rosenbrock's banana function.The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1329-7528-7
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematical logic, Rosser's trick is a method for proving Godel's incompleteness theorems without the assumption that the theory being considered is ?-consistent (Smorynski 1977, p. 840; Mendelson 1977, p. 160). This method was introduced by J. Barkley Rosser in 1936, as an improvement of Godel's original proof of the incompleteness theorems that was published in 1931.While Godel's original proof uses a sentence that says (informally) "This sentence is not provable", Rosser's trick uses a formula that says "If this sentence is provable, there is a shorter proof of its negation". Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1329-7567-6
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the rotation number is an invariant of homeomorphisms of the circle. It was first defined by Henri Poincare in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincare later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number. The rotation number is invariant under topological conjugacy, and even topological semiconjugacy: if f and g are two homeomorphisms of the circle and hcirc f = gcirc h for a continuous map h of the circle into itself (not necessarily homeomorphic) then f and g have the same rotation numbers. It was used by Poincare and Arnaud Denjoy for topological classification of...

ISBN: 978-6-1312-5686-8
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High Quality Content by WIKIPEDIA articles! Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has three rotational symmetries. More examples may be seen below. Formally, rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of E+(m). Данное издание не является оригинальным....

ISBN: 978-6-1303-1677-8
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the rotation number is an invariant of homeomorphisms of the circle. It was first defined by Henri Poincare in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincare later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number. An important property of the rotation number is that it is continuous when viewed as a map from the group of homeomorphisms of the circle into the circle. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1329-7680-2
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High Quality Content by WIKIPEDIA articles! In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming. A rotation and the above-mentioned transformations are isometries; they leave the distance between any two points unchanged after the transformation. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1303-1557-3
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High Quality Content by WIKIPEDIA articles! In linear algebra, a rotation matrix is any matrix that acts as a rotation in Euclidean space. In three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in 2 or 3 dimensions, rotation matrices can be defined for n-dimensional space. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1303-1605-1
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In geometry a rotation representation expresses the orientation of an object relative to a coordinate reference frame. This concept extends to classical mechanics where rotational kinematics is the science of describing the purely rotational motion of an object. According to Euler's rotation theorem the general displacement of a rigid body with one point fixed is described by a rotation about some axis. Furthermore, such a rotation may be uniquely described by a minimum of three parameters. However, for various reasons, there are several ways to represent such a rotation. Many of these definitions use more than the necessary minimum of three parameters, although any such representation still...

ISBN: 978-6-1303-1591-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. For example, the function f(x,y) = x2 + y2 is invariant under rotations of the plane around the origin. For a function from a space X to itself, or for an operator that acts on such functions, rotational invariance may also mean that the function or operator commutes with rotations of X. An example is the two-dimensional Laplace operator ? f = ?xx f + ?yy f: if g is the function g(p) = f(r(p)), where r is any rotation, then (? g)(p) = (? f)(r(p)); that is, rotating a function merely rotates its Laplacian....

ISBN: 978-6-1305-1631-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A rotation of axes is a form of Euclidean transformation in which the entire xy-coordinate system is rotated in the counter-clockwise direction with respect to the origin (0, 0) through a scalar quantity denoted by ?. With the exception of the degenerate cases, if a general second-degree equation has a Bxy term, then Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 represents one of the 3 conic sections, namely, an ellipse, hyperbola, or parabola. If a locus is defined on the xy-coordinate system as left(x, yright), then it is denoted as left(xcos theta + ysin theta, -xsin theta + ycos thetaright) on the rotated x'y'-coordinate system. Likewise, if a locus is defined on the x'y'-coordinate system as...

ISBN: 978-6-1312-5711-7
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In combinatorial mathematics, rotation systems encode embeddings of graphs onto orientable surfaces, by describing the circular ordering of a graph's edges around each vertex. A more formal definition of a rotation system involves pairs of permutations; such a pair is sufficient to determine a multigraph, a surface, and a 2-cell embedding of the multigraph onto the surface. Every rotation scheme defines a unique 2-cell embedding of a connected multigraph on a closed oriented surface (up to orientation preserving topological equivalence). Conversely, any embedding of a connected multigraph G on an oriented closed surface defines a unique rotation system having G as its underlying multigraph....

ISBN: 978-6-1312-5736-0
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Rotating calipers is a computational algorithm developed by Michael Shamos in 1978 for determining all antipodal pairs of points and vertices on a convex polygon or convex hull. The term "rotating calipers" was later coined in 1983 by the computer scientist Godfried Toussaint. The name comes from the analogy of rotating a spring-loaded caliper around the outside of a convex polygon. Every time one blade of the caliper lies flat against an edge of the polygon, it forms an antipodal pair with the point or edge touching the opposite blade. The algorithm runs in O(n) time. Applicable problems: Diameter (maximum width) of a convex polygon, Width (minimum width) of a convex polygon, Maximum distance...

ISBN: 978-6-1312-5667-7
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Rotation around a fixed axis is a special case of rotational motion. It does not involve rotation around more than one axis, and cannot describe such phenomena as wobbling or precession. The kinematics and dynamics of rotation around a fixed axis of a rigid object are mathematically much simpler than those for rotation of a rigid body; they are entirely analogous to those of linear motion along a single fixed direction, which is not true for rotation of a rigid body. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1303-1670-9
Издательство: Книга по требованию
Дата выхода: июль 2011