Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
5,763
result(s) for
"Chebyshev"
Sort by:
New General Variants of Chebyshev Type Inequalities via Generalized Fractional Integral Operators
In this study, new and general variants have been obtained on Chebyshev’s inequality, which is quite old in inequality theory but also a useful and effective type of inequality. The main findings obtained by using integrable functions and generalized fractional integral operators have generalized many existing results as well as iterating the Chebyshev inequality in special cases.
Journal Article
Form finding in elastic gridshells
Elastic gridshells comprise an initially planar network of elastic rods that are actuated into a shell-like structure by loading their extremities. The resulting actuated form derives from the elastic buckling of the rods subjected to inextensibility. We study elastic gridshells with a focus on the rational design of the final shapes. Our precision desktop experiments exhibit complex geometries, even from seemingly simple initial configurations and actuation processes. The numerical simulations capture this nonintuitive behavior with excellent quantitative agreement, allowing for an exploration of parameter space that reveals multistable states. We then turn to the theory of smooth Chebyshev nets to address the inverse design of hemispherical elastic gridshells. The results suggest that rod inextensibility, not elastic response, dictates the zeroth-order shape of an actuated elastic gridshell. As it turns out, this is the shape of a common household strainer. Therefore, the geometry of Chebyshev nets can be further used to understand elastic gridshells. In particular, we introduce a way to quantify the intrinsic shape of the empty, but enclosed regions, which we then use to rationalize the nonlocal deformation of elastic gridshells to point loading. This justifies the observed difficulty in form finding. Nevertheless, we close with an exploration of concatenating multiple elastic gridshell building blocks.
Journal Article
A comparison between different discretization techniques for the Doyle-Fuller-Newman Li+ battery model
This work performs a numerical comparison between different discretization techniques applied to the Doyle-Fuller-Newman (DFN) Lithium-ion battery model. More specifically, the central difference approximation, Crank-Nicolson, and Chebyshev discretization methods applied to the DFN model are compared. These methods are contrasted in terms of accuracy, stability, and computational times, providing the reader with several insights regarding the selection of discretization techniques according to the type of application to be carried out, highlighting the pros and cons of the analyzed methods.
Journal Article
On the average value of \\(\\pi (t)-\\operatorname {\\textrm {li}}(t)\\)
We prove that the Riemann hypothesis is equivalent to the condition \\(\\int _{2}^x\\left (\\pi (t)-\\operatorname {\\textrm {li}}(t)\\right )\\textrm {d}t<0\\) for all \\(x>2\\). Here, \\(\\pi (t)\\) is the prime-counting function and \\(\\operatorname {\\textrm {li}}(t)\\) is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function \\(\\theta (t)\\) and discuss the extent to which one can make related claims unconditionally.
Journal Article
(Spectral) Chebyshev collocation methods for solving differential equations
Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named
Hamiltonian Boundary Value Methods (HBVMs)
. Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli (
Numer. Algo.
,
27
, 119–130
2021
). In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods.
Journal Article
Study of distance metrics on k - nearest neighbor algorithm for star categorization
Classification of stars is essential to investigate the characteristics and behavior of stars. Performing classifications manually is error-prone and time-consuming. Machine learning provides a computerized solution to handle huge volumes of data with minimal human input. k-Nearest Neighbor (kNN) is one of the simplest supervised learning approaches in machine learning. This paper aims at studying and analyzing the performance of the kNN algorithm on the star dataset. In this paper, we have analyzed the accuracy of the kNN algorithm by considering various distance metrics and the range of k values. Minkowski, Euclidean, Manhattan, Chebyshev, Cosine, Jaccard, and Hamming distance were applied on kNN classifiers for different k values. It is observed that Cosine distance works better than the other distance metrics on star categorization.
Journal Article
A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems
ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational
advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the
capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental
power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named
computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of
them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but
there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating
high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional
functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required
parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the
prescribed approximation accuracy
eBook
Approximation Properties of Chebyshev Polynomials in the Legendre Norm
In this paper, we present some important approximation properties of Chebyshev polynomials in the Legendre norm. We mainly discuss the Chebyshev interpolation operator at the Chebyshev–Gauss–Lobatto points. The cases of single domain and multidomain for both one dimension and multi-dimensions are considered, respectively. The approximation results in Legendre norm rather than in the Chebyshev weighted norm are given, which play a fundamental role in numerical analysis of the Legendre–Chebyshev spectral method. These results are also useful in Clenshaw–Curtis quadrature which is based on sampling the integrand at Chebyshev points.
Journal Article
The Design and Simulation of a Fifth-order Chebyshev Low-Pass Filter
Chebyshev filter is a design classification of filters. It uses the Chebyshev transfer function, and has many filter types, such as high-pass, low-pass, band-pass, high-resistance, band-stop and so on. Compared with the Butterworth filter, the transition band of the Chebyshev filter is very narrow, but the internal amplitude-frequency characteristics are unstable [1][2]. Chebyshev filter achieves complex transmission zeros to improve the group delay characteristics in the pass band and reduce the signal distortion. In the process of designing a five-order Chebyshev filter, an active filter network needs to be established, and each stage of the low-pass filter composed of an amplifier is connected in the same cascade to form a multi-order Chebyshev low-pass filter. To design a Chebyshev filter, a Sallen-key structure is needed, and the transfer function is determined according to the position and size of the different resistors and capacitors placed, and a Chebyshev filter with different functions is constructed. The Multisim Simulation software is used to simulate the design results, and the frequency response amplitude curve in the image is found to be fluctuating with equal ripple in the pass or stop band.
Journal Article