By Valentino Magnani

We receive an intrinsic Blow-up Theorem for normal hypersurfaces on graded nilpotent teams. This method permits us to symbolize explicitly the Riemannian floor degree by way of the round Hausdorff degree with admire to an intrinsic distance of the crowd, particularly homogeneous distance. We practice this outcome to get a model of the Riemannian coarea forumula on sub-Riemannian teams, that may be expressed by way of arbitrary homogeneous distances.We introduce the normal classification of horizontal isometries in sub-Riemannian teams, giving examples of rotational invariant homogeneous distances and rotational teams, the place the coarea formulation takes a less complicated shape. through a similar Blow-up Theorem we receive an optimum estimate for the Hausdorff measurement of the attribute set relative to C1,1 hypersurfaces in 2-step teams and we end up that it has finite Q − 2 Hausdorff degree, the place Q is the homogeneous size of the crowd.

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**Extra info for A Blow-up Theorem for regular hypersurfaces on nilpotent groups**

**Example text**

G. [1] and references cited therein). In the [1–6] the construction of nonlinear generalizations of the Schr¨odinger equation was based on the idea of symmetry and the following problems were solved: 1. Nonlinear Schr¨odinger equations, which are compatible with the Galilean relativistic principle, are described. 2. All nonlinear equations, which preserve nontrivial AG2 (1, n)-symmetry of the linear Schr¨odinger equation, are constructed. Let us adduce some nonlinear generalizations of the Schr¨odinger equation that have AG2 (1, n)-symmetry, namely: iUt + ∆U = λ1 |U |4/n U, [1, 2] (1) iUt + ∆U = λ1 |U |a |U |a U, |U |2 [3, 4] (2) iUt + ∆U = λ1 ∆|U |2 U, |U |2 [6] (3) where U = U (t, x) is an unknown diﬀerentiable complex function, Ut ≡ ∂U ∂t , ∆ ≡ √ ∂|U | ∂2 ∂2 ∗ + · · · + ∂x2 , x = (x1 , .

Note 2. If one put in the formulae (1) and (3) from (6) a = k1 , b = k2 and a = k1 , b = 0 respectively, then we get P (1, 3)-invariant systems of PDE constructed in [4]. Further, if we make in (6) the change of variables u1 = 1 (u + u∗ ) , 2 u2 = 1 (u − u∗ ) , 2i then we get the six classes of inequivalent PDE for complex ﬁeld invariant under the extended Poincar´e group. Equations of the form (3) are widely used in the quantum ﬁeld theory to describe at the classical level spinless charged mesons [7].

A, 1986, 113, 408–412. 11. , Properties of nonlinear Schr¨ odinger equations associated with diﬀeomorphism group representations, J. Phys. A: Math. , 1994, 27, 1771–1780. 12. , New exact solutions of the wave equation by reduction to the heat equation, J. Phys. A: Math. , 1995, 28, 5291–5304. 13. , Symmetry and non-Lie reduction of the nonlinear Schr¨ odinger equation, Ukr. Math. , 1993, 45, 539–551. 14. , Complex generalization of the Hamilton–Jacobi equation and its solutions, in Symmetry Analysis and Solutions of the Equations of Mathematical Physics, Kyiv, Institute of Mathematics, Ukrainian Acad.