By Björn Gustafsson, Alexander Vasiliev

ISBN-10: 3764399058

ISBN-13: 9783764399054

Our wisdom of items of advanced and strength research has been better lately by means of rules and structures of theoretical and mathematical physics, comparable to quantum box conception, nonlinear hydrodynamics, fabric technology. those are the various topics of this refereed number of papers, which grew out of the 1st convention of the eu technological know-how origin Networking Programme 'Harmonic and complicated research and functions' held in Norway 2007.

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Appl. : PS-3 integral equations and projective structures on Riemann surfaces. , 192:4, 479–514 (2001) Andrei Bogatyrev INM RAS, ul. ru Analysis and Mathematical Physics Trends in Mathematics, 49–76 c 2009 Birkh¨ auser Verlag Basel/Switzerland Generalized Hamilton–Jacobi Equation and Heat Kernel on Step Two Nilpotent Lie Groups Ovidiu Calin, Der-Chen Chang and Irina Markina Abstract. We study geometrically invariant formulas for heat kernels of subelliptic diﬀerential operators on two step nilpotent Lie groups and for the Grusin operator in R2 .

Therefore D(p+ ) + D(p− ) = (Ω). 38 A. 23) of the branching divisors, we see that each of the structures p± has the branching number two on the curve M . It remains to note that the pair of pants P(R3 ) are exactly “one half” of M . 2. 21) and total branching number two in the pants (see remark above). We can prove that p+ = p− everywhere. 26) where the deviation from equality means that there is a point where p+ = p− . 26) is four and the same number is deg(Ω) = 4g − 4. Therefore this pair of functions p± will give us the holomorphic vector W (p+ (y), p− (y)) in the pants which solves our Riemann monodromy problem.

Mathematics Subject Classiﬁcation (2000). 53C17, 53C22, 35H20. Keywords. Sub-Laplacian, Heat operator, H-type groups, action function, volume element. 1. Introduction Let us start with the Laplace operator on Rn , Δ= 1 2 n j=1 ∂2 . ∂x2j It is well known that the heat kernel for Δ is the Gaussian: |x−x0 |2 1 − 2t Pt (x, x0 ) = . n e (2πt) 2 Given a general second-order elliptic operator in n-dimensional Euclidean space, ΔX 1 = 2 n Xj2 + lower-order term, j=1 The ﬁrst author is partially supported by the NSF grant #0631541.

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