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

Differential and difference equations have long played important roles in the history of theoretical models. The oscillation theory as a part of the qualitative theory of these types of equations has been developed rapidly in the past thirty years. The extensive application prospect facilitates the development of this field. Our current book tends to center around the relevant oscillation of second and third order functional differential and difference equations, neutral differential and difference equations and some applications on partial delay equations. The book stresses the similarty of the techniques used in studying oscillation of differential and difference equations and brings the reader to the forefront of current research in...

ISBN: 978-3-6392-3869-3
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets (also spelled "Oseledec") in 1965 and reported at the International Mathematical Congress in Moscow in 1966. A conceptually different proof of the multiplicative ergodic theorem was found by M. S. Raghunathan. The theorem has been extended to semisimple Lie groups by V. A. Kaimanovich and further generalized in the works of David Ruelle, Gregory Margulis, Anders Karlsson, and F. Ledrappier. The multiplicative ergodic theorem is stated in terms of matrix cocycles of a dynamical system....

ISBN: 978-6-1312-9705-2
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High Quality Content by WIKIPEDIA articles! Ostrowski's theorem, due to Alexander Ostrowski, states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value. Since | | is non-Archimedean, |n| ? 1 for all integers n. Also as | | is non-trivial, there exists an integer n such that |n| < 1 and n = p_1^{e_1} ldots p_r^{e_r} by integer factorization. From this, we can deduce |p| < 1 for some prime p. Suppose for contradiction p, q are distinct primes with |p|, |q| < 1. Pick e, f such that |p|e, |q|f < 1 and write 1 = rpe + sqf for some integers r, s by Bezout's identity. But then 1 = |rpe + sqf| < max(|r|, |s|) ? 1, which is a desired...

ISBN: 978-6-1312-9837-0
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High Quality Content by WIKIPEDIA articles! In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence. The result is named after the mathematicians Alexander Ostrowski and Jacques Hadamard. Let 0 < p1 < p2 < ... be a sequence of integers such that, for some ? > 1 and all j ? N, frac{p_{j + 1}}{p_{j}} > lambda. Let (?j)j?N be a sequence of complex numbers such that the power series f(z) = sum_{j in mathbf{N}} alpha_{j} z^{p_{j}} has...

ISBN: 978-6-1312-9728-1
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High Quality Content by WIKIPEDIA articles! Otto Eduard Neugebauer (May 26, 1899 – February 19, 1990) was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences in antiquity and into the Middle Ages. By studying clay tablets he discovered that the ancient Babylonians knew much more about mathematics and astronomy than had been previously realized. The National Academy of Sciences has called Neugebauer "the most original and productive scholar of the history of the exact sciences, perhaps of the history of science, of our age." Neugebauer began as a mathematician, turned first to Egyptian and Babylonian mathematics, and then took up the...

ISBN: 978-6-1310-9717-1
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High Quality Content by WIKIPEDIA articles! In mathematics, Out(Fn) is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory. A point of the outer space is essentially an R-Graph X homotopy equivalent to a bouquet of n circles together with a certain choice of a free homotopy class of a homotopy equivalence from X to the Bouquet of n circles. An R- Graph is just a weighted Graph with weights in R. The sum of all weights should be 1 and all weights should be positive. To avoid disambiguity (and to get a finite dimensional space) it is furthermore required, that the valency of each vertex should be at least 2. A more descriptive view avoiding the homotopy equivalence f...

ISBN: 978-6-1312-9932-2
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, in particular in measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. A general theory of outer measures was first introduced by Caratheodory to provide a basis for the theory of measurable sets and countably additive measures. Caratheodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Caratheodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff...

ISBN: 978-6-1313-0015-8
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High Quality Content by WIKIPEDIA articles! Outer Billiards is a dynamical system based on a convex shape in the plane. Classically, this system is defined for the Euclidean plane but one can also consider the system in the hyperbolic plane or in other spaces that suitably generalize the plane. Outer billiards differs from a usual dynamical billiard in that it deals with a discrete sequence of moves outside the shape rather than inside of it. Let P be a convex shape in the plane. Given a point x0 outside P, there is typically a unique point x1 (also outside P) so that the line segment connecting x0 to x1 is tangent to P at its midpoint and a person walking from x0 to x1 would see P on the right. The map F: x0 -> x1 is called the outer...

ISBN: 978-6-1313-0000-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BCE, in India in 100 CE, and in Arabia in 800 CE, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day. Данное издание...

ISBN: 978-6-1313-0079-0
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High Quality Content by WIKIPEDIA articles! Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. Most, if not all, of the objects studied in finite mathematics are countable sets, such as integers, finite graphs, and formal languages. Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages. In some mathematics curricula, finite mathematics courses cover discrete mathematical concepts for business,...

ISBN: 978-6-1303-0594-9
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs. Outliers can occur by chance in any distribution, but they are often indicative either of measurement error or that the population has a heavy-tailed distribution. In the former case one wishes to discard them or use statistics that are robust to outliers, while in the latter case they indicate that the distribution has high kurtosis and that one should be very cautious in using tool or intuitions that assume a normal distribution. A...

ISBN: 978-6-1303-3900-5
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High Quality Content by WIKIPEDIA articles! Arithmetic is the oldest branch of mathematics, used by almost everyone. Its tasks range from the simple act of counting to advanced science and business calculations. The traditional arithmetic operations are addition, subtraction, multiplication, and division, although more advanced operations such as percentages, square root, exponentiation, and logarithmic functions are also a part of this subject. (Note: number theory is sometimes referred to as "higher arithmetic", but this term should not be confused with "arithmetic"). Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд...

ISBN: 978-6-1311-2102-9
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A few decades ago it might have been said that combinatorics is little more than a way to classify poorly-understood problems, and some standard remedies. Great progress has been made since 1960. This page is complementary to the list of graph theory topics: graph theory being the part of combinatorial mathematics that is most like a separate discipline. In general, combinatorics is as much about problem solving than theory building. Since combinatorial mathematics is effectively the environment for the study of data structures in computer science, there are very many topics that arise there. The same could be...

ISBN: 978-6-1313-0094-3
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High Quality Content by WIKIPEDIA articles! In technical drawing an oval (from Latin ovum, 'egg') is a figure constructed from two pairs of arcs, with two different radii (see image on the right). The arcs are joined at a point, in which lines tangential to both joining arcs lie on the same line, thus making the joint smooth. Any point of an oval belongs to an arc with a constant radius (shorter or longer), whereas in an ellipse the radius is continuously changing. The shape of an egg is approximately that of half each a prolate (long) and roughly spherical (potentially even slightly oblate/short) ellipsoid joined at the equator, sharing a principal axis of rotational symmetry, as illustrated above. Although the term egg-shaped usually...

ISBN: 978-6-1313-0086-8
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High Quality Content by WIKIPEDIA articles! When q is even, the situation is completely different. In this case, sets of q + 2 points, no three of which collinear, may exist in a finite projective plane of order q and they are called hyperovals; these are maximal arcs of degree 2. Given an oval there is a unique tangent through each point, and if q is even Qvist (1952) showed that all these tangents are concurrent in a point p outside the oval. Adding this point (called the nucleus of the oval) to the oval gives a hyperoval. Conversely, removing one point from a hyperoval immediately gives an oval. Every nonsingular conic in the projective plane, together with its nucleus, forms a hyperoval. For each of these sets, there is a system of...

ISBN: 978-6-1312-9976-6
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The flexibility of the overcomplete frame is one of its key advantages when used in expressing a signal or approximating a function. However, because of this redundancy, a function can have multiple expressions under an overcomplete frame [6]. When the frame is finite, the decomposition can be expressed as f=Ax , where f is the function we want to approximate, A is the matrix containing all the elements in the frame, and x is the coefficients of f under the representation of A. Without any other constraint, the frame will choose to give x with minimal norm in L2(R). Based on this, some other properties may also be considered when solving the equation, such as sparsity. So different researchers...

ISBN: 978-6-1313-0409-5
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. The terminology can be described in terms of the concept of counting constants. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom. Therefore the critical case occurs when the number of equations and the number of independent variables are equal. For every degree of freedom, there exists a corresponding restraint. The overdetermined case occurs when the system has been overconstrained—that is, when the number of equations outnumbers the number of the unknowns....

ISBN: 978-6-1303-3916-6
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In statistics, overfitting occurs when a statistical model describes random error or noise instead of the underlying relationship. Overfitting generally occurs when a model is excessively complex, such as having too many degrees of freedom, in relation to the amount of data available. A model which has been overfit will generally have poor predictive performance, as it can exaggerate minor fluctuations in the data. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1304-8798-0
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! American football, known in the United States simply as football and often as gridiron or tackle football outside the United States, is a competitive team sport known for combining strategy with physical play. The objective of the game is to score points by advancing the ball into the opposing team's end zone. The ball can be advanced by carrying it (a running play) or by throwing it to a teammate (a passing play). Points can be scored in a variety of ways, including carrying the ball over the opponent's goal line, catching a pass thrown over that goal line, kicking the ball through the goal posts at the opponent's end zone, or tackling an opposing ball carrier within his end zone. The winner...

ISBN: 978-6-1313-0455-2
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation. It has applications that include statistics, computer vision, image and signal processing, electrical engineering, and differential equations. The convolution can be defined for functions on groups other than Euclidean space. In particular, the circular convolution can be defined for periodic functions (that is, functions on the circle), and the discrete convolution can be defined for functions on the set of integers. These...

ISBN: 978-6-1313-0442-2
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation. It has applications that include statistics, computer vision, image and signal processing, electrical engineering, and differential equations. The convolution can be defined for functions on groups other than Euclidean space. In particular, the circular convolution can be defined for periodic functions (that is, functions on the circle), and the discrete convolution can be defined for functions on the set of integers. These...

ISBN: 978-6-1313-0435-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general. Matrices consisting of only one column or row define the components of vectors, while higher-dimensional (e.g., three-dimensional) arrays of numbers define the components of a generalization of a vector called a tensor. Matrices...

ISBN: 978-6-1313-0431-6
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Two overlapping confidence intervals have been used in the past to conduct statistical inferences about two normal population means. Several authors have examined the shortcomings of Overlap procedure and have determined that such a method distorts the significance level of testing the equality of two normal population means and reduces the statistical power of the test. Nearly all results for small samples in Overlap literature have been obtained either by simulation or by inaccurate formulas. This book will present exact formulas for the % overlap that two independent confidence intervals can have, but the null hypothesis of equality of two population means or variances must be rejected at a level of significance for any sample sizes....

ISBN: 978-3-6391-7704-6
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, in the field of combinatorics, a polar space of rank n (n >= 3), or projective index n–1, consists of a set P, conventionally the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms : * Every subspace, together with its own subspaces, is isomorphic with a partial geometry PG(d,q) with –1 Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1313-0560-3
Издательство: Книга по требованию
Дата выхода: июль 2011