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The Gaussian Kinematic Formula

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2019)

Abstract

Our aim in this chapter is to show you how to prove the Gaussian kinematic formula, (1.3.3). That is, that for \( f:M\rightarrow\mathbb{R}^d, \)with component random fields \(f_1,\ldots,f_d\) which are smooth, zero mean, unit variance, and Gaussian, and where M and \(D\subset\mathbb{R}^d\) are nice enough,

$$\mathbb{E}\{{\mathcal{L}_i}(A(f,M,D))\}=\sum\limits_{j=0}^{{\rm dim}M-i }\left[\begin{array}{ll}i+j\cr j\end{array}\right](2\pi)^{-j/2}\mathcal{L}_{i+j}(M)\mathcal{M}_j^i(D).$$
((4.0.1))

Keywords

  • Gaussian Process
  • Euler Characteristic
  • Isotropic Case
  • Spectral Moment
  • Coarea Formula

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Correspondence to Robert J. Adler .

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© 2011 Springer-Verlag Berlin Heidelberg

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Adler, R.J., Taylor, J.E. (2011). The Gaussian Kinematic Formula. In: Topological Complexity of Smooth Random Functions. Lecture Notes in Mathematics(), vol 2019. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19580-8_4

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