High Quality Content by WIKIPEDIA articles! In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, compact 4-manifold M has a spin structure (or, equivalently, the second Stiefel-Whitney class w2(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952. Rokhlin's theorem can be deduced from the fact that the third stable homotopy group of spheres ?S3 is cyclic of order 24; this is Rokhlin's original approach. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1329-7052-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, compact 4-manifold M has a spin structure (or, equivalently, the second Stiefel-Whitney class w2(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952. The intersection form is unimodular by Poincare duality, and the vanishing of w2(M) implies that the intersection form is even. By a theorem of Cahit Arf, any even unimodular lattice has signature divisible by 8, so Rokhlin's theorem forces one extra factor of 2 to divide the signature. Данное...

ISBN: 978-6-1312-6845-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In calculus, a branch of mathematics, Rolle's theorem essentially states that a differentiable function, which attains equal values at two points, must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero. A version of the theorem was first stated by the Indian astronomer Bh?skara II in the 12th century The first known formal proof was offered by Michel Rolle in 1691, which used the methods of differential calculus. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1329-7082-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In calculus, a branch of mathematics, Rolle's theorem essentially states that a differentiable function, which attains equal values at two points, must have a point somewhere between them where the slope of the tangent line to the graph of the function is zero. A version of the theorem was first stated by the Indian astronomer Bh?skara II in the 12th century. The first known formal proof was offered by Michel Rolle in 1691, which used the methods of differential calculus. The name "Rolle's theorem" was first used by M.W.Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1312-6680-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In 1965, in his studio in Warsaw, Opalka began painting a process of counting - from one to infinity. Starting in the top left-hand corner of the canvas and finishing in the bottom right-hand corner, the tiny numbers are painted in horizontal rows. Each new canvas, which the artist calls a 'detail', takes up counting where the last left off. Each 'detail' is the same size (196 x 135 cm), the dimension of his studio door in Warsaw. All details have the same title, "1965 / 1 – "; the idea does not date although the artist has pledged his life to its execution: 'All my work is a single thing, the description from number one to infinity. A single thing, a single life.' Over the years there have...

ISBN: 978-6-1311-7392-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Many letters of the Latin alphabet, both capital and small, are used in mathematics, science and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, physical entities. Certain letters, when combined with special formatting, take on special meaning. Below is an alphabetical list of the letters of the alphabet with some of their uses. The field in which the convention applies is mathematics unless otherwise noted. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1329-7130-2
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. The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however the figure resulting from removing six singular points is one.The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each axis which terminate in pinch points. The entire surface has tetrahedral...

ISBN: 978-6-1329-7291-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors. It greatly reduced the time needed to perform the basic operations of Roman arithmetic using Roman numerals. As Karl Menninger says on page 315 of his book, "For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1312-6663-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The Romanian numbers are the system of number names used in Romanian to express counts, quantities, ranks in ordered sets, fractions, multiplication, and other information related to numbers. In Romanian grammar, the words expressing numbers are a separate part of speech, called numeral (plural: numerale), along with nouns, verbs, etc. (Note that English numeral and Romanian numeral have different meanings; also, Romanian num?r only partially overlaps in meaning with English number.) Nevertheless, these words play the same roles in the sentence as they do in English: adjective, pronoun, noun, and occasionally others. This article focuses on the mechanism of naming numbers in Romanian and the...

ISBN: 978-6-1312-6650-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The Romanian numbers are the system of number names used in Romanian to express counts, quantities, ranks in ordered sets, fractions, multiplication, and other information related to numbers.The symbols for numbers in Romanian texts are the same as those used in English, with the exception of using the comma as the decimal separator and the period or the space (ideally a narrow space) for grouping digits by three in large numbers. For example, in Romanian 1,5 V means one and a half volts, and 1.000.000 or 1 000 000 means one million. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд...

ISBN: 978-6-1329-7314-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Ingarden was born in Krakow, Austria-Hungary, on February 5, 1893. He first studied mathematics and philosophy in Lwow under Kazimierz Twardowski, then moved to Gottingen to study philosophy under Edmund Husserl. He was considered by Husserl to be one of his best students and accompanied Husserl to Freiburg, where in 1918 Ingarden submitted his doctoral dissertation with Husserl as director. Ingarden then returned to Poland, where he spent his academic career after obtaining his doctorate. For a long period he had to support himself by secondary-school teaching. In 1925 he submitted his Habilitationschrift, Essentiale Fragen, to Kazimierz Twardowski at Lwow University (now Lviv in Ukraine)....

ISBN: 978-6-1311-7368-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! This result confirmed that semialgebraic sets in Rn form what is now known as an o-minimal structure on R. These are collections of subsets Sn of Rn for each n >= 1 such that we can take finite unions and complements of the subsets in Sn and the result will still be in Sn, moreover the elements of S1 are simply finite unions of intervals and points. The final condition for such a collection to be an o-minimal structure is that the projection map on the first n coordinates from Rn+1 to Rn must send subsets in Sn+1 to subsets in Sn. The Tarski–Seidenberg theorem tells us this holds if Sn is the set of semialgebraic sets in Rn. Данное издание не является...

ISBN: 978-6-1311-7406-3
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however the figure resulting from removing six singular points is one. The simplest construction is as the image of a sphere centered at the origin under the map f(x,y,z) = (yz,xz,xy). This gives us an implicit formula of x2y2 + y2z2 + z2x2 ? r2xyz = 0. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд...

ISBN: 978-6-1312-6657-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In numerical analysis, Romberg's method (Romberg 1955) generates a triangular array consisting of numerical estimates of a definite integral by using Richardson extrapolation (Richardson 1910) repeatedly on the trapezium rule. Romberg's method is a Newton–Cotes formula -- it evaluates the integrand at equally-spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. If it is possible to evaluate the integrand at unequally-spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. Данное издание не является...

ISBN: 978-6-1329-7330-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Sir Ronald Aylmer Fisher, FRS (17 February 1890 – 29 July 1962) was an English statistician, evolutionary biologist, eugenicist and geneticist. He was described by Anders Hald as "a genius who almost single-handedly created the foundations for modern statistical science," and Richard Dawkins described him as "the greatest of Darwin's successors". It was Fisher who referred to the growth rate r (used in equations such as the logistic function) as the Malthusian parameter, as a criticism of the writings of Thomas Robert Malthus. Fisher referred to "...a relic of creationist philosophy..." in observing the fecundity of nature and deducing (as Darwin did) that this therefore drove natural...

ISBN: 978-6-1303-3096-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged...

ISBN: 978-6-1329-7350-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no...

ISBN: 978-6-1312-6199-2
Издательство: Книга по требованию
Дата выхода: июль 2011

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
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The root mean square deviation (RMSD) is the measure of the average distance between the backbones of superimposed proteins. In the study of globular protein conformations, one customarily measures the similarity in three-dimensional structure by the RMSD of the C? atomic coordinates after optimal rigid body superposition.A widely used way to compare the structures of biomolecules or solid bodies is to translate and rotate one structure with respect to the other to minimize the RMSD. Coutsias, et al. presented a simple derivation, based on quaternions, for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors.They proved that the...

ISBN: 978-6-1329-7374-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The root mean square deviation (RMSD) or root mean square error (RMSE) is a frequently-used measure of the differences between values predicted by a model or an estimator and the values actually observed from the thing being modeled or estimated. RMSD is a good measure of precision. These individual differences are also called residuals, and the RMSD serves to aggregate them into a single measure of predictive power. For an unbiased estimator, the RMSE is the square root of the variance, known as the standard error. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после...

ISBN: 978-6-1304-8269-5
Издательство: Книга по требованию
Дата выхода: июль 2011

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 depends on the quantity limsup_{nrightarrowinfty}sqrt[n]{|a_n|}, where an are the terms of the series, and states thats the series converges absolutely if this quantity is less than 1 but diverges if it is greater than 1. It is particularly useful in connection with power series. The terms of this series would then be given by an = cn(z – p)n. One then applies the root test to the an as above. Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series...

ISBN: 978-6-1312-6091-9
Издательство: Книга по требованию
Дата выхода: июль 2011

ISBN: 978-6-1312-6135-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Finding roots of certain functions, especially polynomials, frequently requires the use of specialised or approximation techniques (for example, Newton's method). The concept of complex numbers was developed to handle the roots of quadratic or cubic equations with negative discriminants (that is, those leading to expressions involving the square root of negative numbers).Every real polynomial of odd degree has at least one real number as a root. Many real polynomials of even degree do not have a real root, but the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real...

ISBN: 978-6-1329-7388-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! There are n different nth roots of unity: zk (k = 1, 2, 3, ..., n) where z is any primitive nth root of unity. These n roots are distributed evenly over the unit circle as can be seen in the plot on the right-hand side of the three 3rd roots of unity. There are ?(n) primitive nth roots of unity. If z is any fixed primitive nth root of unity, then the primitive nth roots of unity are those zk where 1 ? k ? n and k and n are coprime. The notion of root of unity also applies to any algebraic ring with 1, namely a root of unity is any element of finite multiplicative order. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1311-3028-1
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, 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
Издательство: Книга по требованию
Дата выхода: июль 2011