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"Chebyshev"
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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
Two dimensional pseudo-Chebyshev wavelets and their application in the theory of approximation of functions belonging to Hölders class
For the first time in 2022, the authors introduced the notion of pseudo-Chebyshev wavelets in the context of one dimension. Continuing the study in advance sense, in this article, two dimensional pseudo Chebyshev wavelets are introduced. Two dimensional pseudo Chebyshev wavelet expansion of a function of two variable is defined and verified. This research paper introduces a novel algorithm based on the two dimensional pseudo Chebyshev wavelet method to address computation problems in approximation theory. The methods are illustrated by an example and compared with prominent Chebyshev wavelet methods to demonstrate the validity and applicability of the results. The error analysis and convergence analysis of a functions in the Hölder classes have been studied by this wavelets. More over the error of approximation of functions of Holder’s class have been estimated by an orthogonal projection operators of its two dimensional pseudo Chebyshev wavelets. The results of this paper are the significant developments in wavelet analysis.
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
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
The improved mayfly optimization algorithm with Chebyshev map
The mayfly optimization (MO) algorithm was just proposed recently, simulation experiments proved that it was capable to optimize both the benchmark functions and the real problems we met. In this paper, the MO algorithm would be improved with Chebyshev map, simulation experiments were carried out and results showed that the improved algorithm would indeed increase the capability.
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
(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
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
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