lecture18

lecture18 - May 26, 2010 Janine LoBue Analytic Methods and...

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Janine LoBue Analytic Methods and the Heat Kernel 1 Introduction In this lecture, we employ some standard analytic methods from spectral geometry to better un- derstand the heat kernel. In particular, we determine upper and lower bounds for the difference between the heat kernel and the steady-state vector. 2 Review of Facts about the Heat Kernel Recall that the heat kernel with temperature t and seed s is defined to be the vector k t,s = e - t X k 0 t k k ! sW k = se - t ( I - W ) . We have shown the following useful properties of the heat kernel: H t = e - t L ∂t H t = -LH t Tr ( H t ) = X x H t ( x,x ) = X i e - λ i t For a subset of the vertices S , define the function f S ( u ) = ± d u volS u S 0 else . Recall that ∂t k t,f S ( S ) = 1 volS X u v ( k t/ 2 ,u ( S ) - k t/ 2 ,v ( S )) 2 . Since this is a sum of squares, in particular we have that ∂t k t,f S ( S ) is non-negative. 1
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lecture18 - May 26, 2010 Janine LoBue Analytic Methods and...

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