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The Heat Kernel
Published in M. W. Wong, Partial Differential Equations, 2022
We call the function kt,t>0, the heat kernel or the Weierstrass kernel of the Laplacian Δ on ℝn. We also denote kt*f by etΔf for t>0. The operator etΔ,t>0, is known as the heat semigroup for the Laplacian.
On the p-adic Navier–Stokes equation
Published in Applicable Analysis, 2020
Andrei Yu. Khrennikov, Anatoly N. Kochubei
The Vladimirov operator , , of fractional differentiation, is defined first as a pseudo-differential operator with the symbol : where we show arguments of functions and their direct/inverse Fourier transforms. There is also a hypersingular integral representation giving the same result on but making sense on much wider classes of functions (e.g. bounded locally constant functions): The Cauchy problem for the heat-like equation is a model example for the theory of p-adic parabolic equations. If ψ is regular enough, for example, , then a classical solution is given by the formula where is, for each t, a probability density and The ‘heat kernel’ can be written as the Fourier transform See [8] for various series representations and estimates of the kernel .
Manufacturability analysis for additive manufacturing using a novel feature recognition technique
Published in Computer-Aided Design and Applications, 2018
Yang Shi, Yicha Zhang, Steven Baek, Wout De Backer, Ramy Harik
Heat Kernel Signature (HKS) is a concise and efficient pointwise shape descriptor developed in computer vison field in recent years. It inherits important properties from the heat kernel, which can fully describe the shape of a surface. Heat kernel signature is directly related to the Gaussian curvature on a surface, and is also closely related to diffusion maps and diffusion distances [14], which means it can describe not only the shape but also the positon of a point on a given domain. In other words, heat kernel signature is able to present the topologic and geometric characteristic of a feature. In this paper, we will illustrate a novel feature recognition technique based on heat kernel signature and apply it to manufacturability analysis for additive manufacturing.
Well-posedness of the fractional Ginzburg–Landau equation
Published in Applicable Analysis, 2019
Xian-Ming Gu, Lin Shi, Tianhua Liu
When and , the kernel is the classical Poisson kernel and heat kernel, respectively. When , is a nonnegative and non-increasing radial function, and satisfies the dilation relation