China
Africa
Explore chapters and articles related to this topic
Applications
Published in James K. Peterson, Basic Analysis I, 2020
Let πn={ti}i=0n be the uniform partition of the interval [0, L] with ‖πn‖=L/n. Let ϕi be the C∞ bump function defined on [ti, ti+1] centered at the midpoint (ti, ti+1)/2 of height 1. Let f be any continuous function on [0, L].
A review on the impacts of connected vehicles on pavement management systems
Published in International Journal of Pavement Engineering, 2023
Mohammad Saleh Entezari, Amir Golroo
In the next study on using CVs for pavement monitoring, Bridgelall et al. (2017b) inspected the relative sensitivity of RIF and IRI to variations in profile wavelengths. Besides, using a bump function for the simulation of dominant profile wavelengths was investigated. The case study was conducted as follows. First, five concrete or asphalt blocks in the MnRoad laboratory were selected. Then, a certified profiler was used to traverse the blocks. A smartphone was mounted on the vehicle in order to collect the required data for calculating RIF and IRI at the same time. The following results were produced by this research project: Relative roughness and the RIF/IRI factor have an inverse proportional relationship. In other words, as roughness increases, the RIF/IRI factor decreases.IRI and RIF have a proportional relationship. After 49 traversals the error magnitude of the proportional relationship reaches below 5%.Both indices are capable of characterising roughness from spatial wavelengths up to two metres. But, RIF maintains its sensitivity even in the wavelengths beyond.
Estimates of fundamental solution for Kohn Laplacian in Besov and Triebel-Lizorkin spaces
Published in Applicable Analysis, 2023
Tongtong Qin, Der-Chen Chang, Yongsheng Han, Xinfeng Wu
A class of singular integral operators T is initially given as a mapping from functions with compact supports, to with a distribution kernel The following properties hold: If have disjoint supports, then If φ is a normalized bump function associated to a ball of radius r, then and If then and for Properties (I-1) through (I-3) also hold with x and y interchanged. That is, these properties also hold for the adjoint operator defined by
A generalized multidirectional mean value inequality and dynamic optimization
Published in Optimization, 2019
The proof of this result in [6] makes use of a powerful nonlocal fuzzy sum rule developed by the same author. In this paper we use optimization techniques which are related to the original proof in [1] to extend this result to cover Banach spaces admitting globally Lipschitz, β-smooth bump functions. Recall that a bump function is a function which is supported on a nonempty bounded set. Every Banach space which admits a β-smooth norm admits a globally Lipschitz, β-smooth bump function. Indeed, the reader may check that for a β-smooth norm, the function is such a bump function. It was shown in [18], however, that there exist compact Hausdorff spaces X with the property that the Banach space admits a Fréchet-differentiable bump function but does not admit even a Gâteaux-differentiable renorm. Thus the collection of Banach spaces covered by Theorem 2.2 is strictly larger than that in [6].