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List Of Things Named After Pafnuty Chebyshev
{{Short description, none Mathematics * Chebyshev center * Chebyshev constants * Chebyshev cube root * Chebyshev distance * Chebyshev equation * Chebyshev's equioscillation theorem * Chebyshev filter, a family of analog filters in electronics and signal processing * Chebyshev function in number theory * Chebyshev integral * Chebyshev iteration * Chebyshev method * Chebyshev nodes * Chebyshev polynomials and the "Chebyshev form" **Chebyshev norm **Discrete Chebyshev polynomials **Discrete Chebyshev transform * Chebyshev rational functions *Chebyshev–Gauss quadrature * Chebyshev–Markov–Stieltjes inequalities * Chebyshev's bias * Chebyshev's inequality in probability and statistics ** Chebyshev–Cantelli inequality **Multidimensional Chebyshev's inequality *Chebyshev pseudospectral method * Chebyshev space * Chebyshev's sum inequality * Chebyshev's theorem (other) Mechanics * Chebyshev linkage, a straight line generating linkage * Chebyshev's Lambda Mechanism and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Center
In geometry, the Chebyshev center of a bounded set Q having non-empty Interior (topology), interior is the center of the minimal-radius ball enclosing the entire set Q, or alternatively (and non-equivalently) the center of largest inscribed ball of Q. In the field of parameter estimation, the Chebyshev center approach tries to find an estimator \hat x for x given the feasibility set Q , such that \hat x minimizes the worst possible estimation error for x (e.g. best worst case). Mathematical representation There exist several alternative representations for the Chebyshev center. Consider the set Q and denote its Chebyshev center by \hat. \hat can be computed by solving: : \min_ \left\ with respect to the Euclidean distance, Euclidean norm \, \cdot\, , or alternatively by solving: : \operatorname \max_ \left\, x - \hat x \right\, ^2. Despite these properties, finding the Chebyshev center may be a hard numerical optimization problem. For example, in the second represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Chebyshev Transform
In applied mathematics, the discrete Chebyshev transform (DCT), named after Pafnuty Chebyshev, is either of two main varieties of DCTs: the discrete Chebyshev transform on the 'roots' grid of the Chebyshev polynomials of the first kind T_n (x) and the discrete Chebyshev transform on the 'extrema' grid of the Chebyshev polynomials of the first kind. Discrete Chebyshev transform on the roots grid The discrete chebyshev transform of u(x) at the points is given by: : a_m =\frac\sum_^ u(x_n) T_m (x_n) where: : x_n = -\cos\left(\frac (n+\frac)\right) : a_m = \frac \sum_^ u(x_n) \cos\left(m \cos^(x_n)\right) where p_m =1 \Leftrightarrow m=0 and p_m = 2 otherwise. Using the definition of x_n , : a_m =\frac \sum_^ u(x_n) \cos\left(\frac(N+n+\frac) \right) : a_m =\frac \sum_^ u(x_n) (-1)^m\cos\left(\frac(n+\frac) \right) and its inverse transform: : u_n =\sum_^ a_m T_m (x_n) (This so happens to the standard Chebyshev series evaluated on the roots grid.) : u_n =\s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linkage (mechanical)
A mechanical linkage is an assembly of systems connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for example, and are called joints. A linkage modeled as a network of rigid links and ideal joints is called a kinematic chain. Linkages may be constructed from open chains, closed chains, or a combination of open and closed chains. Each link in a chain is connected by a joint to one or more other links. Thus, a kinematic chain can be modeled as a graph in which the links are paths and the joints are vertices, which is called a linkage graph. The movement of an ideal joint is generally associated with a subgroup of the group of Euclidean displacements. The number of parameters in the subgroup is called the degrees of freedom (DOF) of the joint. Mechanical linkages are usually designed to tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Linkage
In kinematics, Chebyshev's linkage is a four-bar linkage that converts rotational motion to approximate linear motion. It was invented by the 19th-century mathematician Pafnuty Chebyshev, who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight-line motion (a straight line mechanism). This was also studied by James Watt in his improvements to the steam engine, which resulted in Watt's linkage. – Cross link straight-line mechanism Equations of motion The motion of the linkage can be constrained to an input angle that may be changed through velocities, forces, etc. The input angles can be either link ''L''2 with the horizontal or link ''L''4 with the hori ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev's Theorem (other) .
{{mathdab ...
Chebyshev's theorem is any of several theorems proven by Russian mathematician Pafnuty Chebyshev. * Bertrand's postulate, that for every ''n'' there is a prime between ''n'' and 2''n''. * Chebyshev's inequality, on the range of standard deviations around the mean, in statistics * Chebyshev's sum inequality, about sums and products of decreasing sequences * Chebyshev's equioscillation theorem, on the approximation of continuous functions with polynomials * The statement that if the function \pi(x)\ln x/x has a limit at infinity, then the limit is 1 (where is the prime-counting function). This result has been superseded by the prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev's Sum Inequality
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if :a_1 \geq a_2 \geq \cdots \geq a_n \quad and \quad b_1 \geq b_2 \geq \cdots \geq b_n, then : \sum_^n a_k b_k \geq \left(\sum_^n a_k\right)\!\!\left(\sum_^n b_k\right)\!. Similarly, if :a_1 \leq a_2 \leq \cdots \leq a_n \quad and \quad b_1 \geq b_2 \geq \cdots \geq b_n, then : \sum_^n a_k b_k \leq \left(\sum_^n a_k\right)\!\!\left(\sum_^n b_k\right)\!. Proof Consider the sum :S = \sum_^n \sum_^n (a_j - a_k) (b_j - b_k). The two sequences are non-increasing, therefore and have the same sign for any . Hence . Opening the brackets, we deduce: :0 \leq 2 n \sum_^n a_j b_j - 2 \sum_^n a_j \, \sum_^n b_j, hence :\frac \sum_^n a_j b_j \geq \left( \frac \sum_^n a_j\right)\!\!\left(\frac \sum_^n b_j\right)\!. An alternative proof is simply obtained with the rearrangement inequality, writing that :\sum_^ a_i \sum_^ b_j = \sum_^ \sum_^ a_i b_j =\sum_^\sum_^ a_i b_ = \sum_^ \sum_^ a_i b_ \leq \s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Space
In approximation theory, a Haar space or Chebyshev space is a finite-dimensional subspace V of \mathcal C(X, \mathbb K), where X is a compact space and \mathbb K either the real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ... or the complex numbers, such that for any given f \in \mathcal C(X, \mathbb K) there is exactly one element of V that approximates f "best", i.e. with minimum distance to f in supremum norm. References Approximation theory {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Pseudospectral Method
The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. Unlike the Legendre pseudospectral method, the Chebyshev pseudospectral (PS) method does not immediately offer high-accuracy quadrature solutions. Consequently, two different versions of the method have been proposed: one by Elnagar et al., and another by Fahroo and Ross. The two versions differ in their quadrature techniques. The Fahroo–Ross method is more commonly used today due to the ease in implementation of the Clenshaw–Curtis quadrature technique (in contrast to Elnagar–Kazemi's cell-averaging method). In 2008, Trefethen showed that the Clenshaw–Curtis method was nearly as accurate as Gauss quadrature. This breakthrough result opened the door for a covector mapping theorem for Chebyshev PS methods. A complete mathematical theory for Chebyshev PS methods was fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multidimensional Chebyshev's Inequality
In probability theory, the multidimensional Chebyshev's inequality is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount. Let X be an N-dimensional random vector with expected value \mu=\operatorname and covariance matrix : V=\operatorname X - \mu) (X - \mu)^T \, If V is a positive-definite matrix, for any real number t>0: : \Pr \left( \sqrt > t\right) \le \frac N Proof Since V is positive-definite, so is V^. Define the random variable : y = (X-\mu)^T V^ (X-\mu). Since y is positive, Markov's inequality holds: : \Pr\left( \sqrt > t\right) = \Pr( \sqrt > t) = \Pr(y > t^2) \le \frac. Finally, :\begin \operatorname &= \operatorname X-\mu)^T V^ (X-\mu)\ pt&=\operatorname \operatorname ( V^ (X-\mu) (X-\mu)^T )\ pt&= \operatorname ( V^ V ) = N \end. Infinite dimensions There is a straightforward extension of the vector version of Chebysh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantelli Inequality
In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. The inequality states that, for \lambda > 0, : \Pr(X-\mathbb ge\lambda) \le \frac, where :X is a real-valued random variable, :\Pr is the probability measure, :\mathbb /math> is the expected value of X, :\sigma^2 is the variance of X. Applying the Cantelli inequality to -X gives a bound on the lower tail, : \Pr(X-\mathbb le -\lambda) \le \frac. While the inequality is often attributed to Francesco Paolo Cantelli who published it in 1928, it originates in Chebyshev's work of 1874. When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. The Chebyshev inequality has "higher moments versions" and "vector versions", and so does the Cantelli inequality ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev's Inequality
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/''k''2 of the distribution's values can be ''k'' or more standard deviations away from the mean (or equivalently, at least 1 − 1/''k''2 of the distribution's values are less than ''k'' standard deviations away from the mean). The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers. Its practical usage is similar to the 68–95–99.7 rule, which applies only to normal distributions. Chebyshev's inequality is more general, stating th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev's Bias
In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4''k'' + 3 than of the form 4''k'' + 1, up to the same limit. This phenomenon was first observed by Russian mathematician Pafnuty Chebyshev in 1853. Description Let π(''x''; ''n'', ''m'') denote the number of primes of the form ''nk'' + ''m'' up to ''x''. By the prime number theorem (extended to arithmetic progression), :\pi(x;4,1)\sim\pi(x;4,3)\sim \frac\frac. That is, half of the primes are of the form 4''k'' + 1, and half of the form 4''k'' + 3. A reasonable guess would be that π(''x''; 4, 1) > π(''x''; 4, 3) and π(''x''; 4, 1) < π(''x''; 4, 3) each also occur 50% of the time. This, however, is not supported by numerical evidence — in fact, π(''x''; 4, 3) > π(''x''; 4, 1) occurs much more frequently. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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