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In the case when tangent manifold is not a differentiable manifold (and hence when the transition from tangent space to cotangent space is degenerate), Hamiltonian system will be fulfilled only on pseudostructures. The Poincare invariant will be also fulfilled only on pseudostructures, namely, on integral curves. That is, the Poincare invariant will be a closed inexact exterior form. In the directions normal to integral curves the differential ds, which corresponds to the Poincare invariant, will be discontinuous.

In the case when the Lagrangian manifold is differentiable, the Hamiltonian systems can be described by pseudogroups, in particular, by Lie pseudogroups. However, the group theory is not sufficient for describing a behavior of Hamiltonian systems and Lagrange equation for real physical processes.

References [1] Arnold V. I. Mathematical methods of classical mechanics.-Moscow, 2003 (in Russian).

Billiard scattering on rough surfaces Plakhov Alexander (Portugal) University of Aveiro, Department of Mathematics plakhov@mat.ua.pt Billiard scattering on smooth surfaces can be described by the simple law: angle of incidence = angle of reflection. Now consider a rough surface: there are “microscopic” hollows that distort the motion of billiard particles. The question is: How to describe the reflection law in this case We study this question in the two-dimensional case. The notion of rough surface is defined. A reflection law is identified with a joint distribution on the (two-dimensional) parameter space: (angle of incidence, angle of reflection). The main result to be reported is characterization of the set of all possible reflection laws. We will also discuss applications of this result to the problems of minimal and maximal resistance and to studying physical phenomena like Magnus effect (lateral deflection of a spinning ball).

Some geodetics particularity in a spherically symmetrical space.

Popov N. N. (Russia) Computer Centre of the Russian Academy of Sciences mark00@comtv.ru Einstein noted in one of his works that the Newtonian law of gravitation describes the gravitational phenomena as incompletely as the Coulombian law of electrostatics and magnetostatics, the electromagnetic ones.

The fact that the Newtonian law is still considered satisfactory for the calculation of the celestial bodies motion, can be explained by the order of velocities and accelerations.

Proceeding from the principle of inert and gravitational mass equivalence in the general theory of relativity, Einstein reduced the task of deriving equations for a material point moving in a gravitational field, to the problem of a test body moving along a geodesic in a curved spacetime. Specifically, in case the gravitational field is created by an only point mass M, the problem is reduced to a solution of differential equations of geodesics in a spherically symmetrical space described by the Schwarzschild metric.

Starting with early works of Einstein and Schwarzschild and followed by later works, the solution of geodetics equations in the spherically symmetrical space described by Schwarzschild metrics a drds2 = 1 - dt2 - - r2(d2 + sin2 d2), a r 1 r has required the following time normalization:

a dt 1 - = 1, (1) r ds where a is the Schwarzschild radius, t the world time, s the proper time along the geodesic line. Normalization (1) leads to asymptotic coincidence of the world time t and proper time s at infinity. It means physically that the test body velocity turns to zero at infinity. With the fulfilment of condition (1) the test body acceleration coincides with the Newton gravitational acceleration d2r a = dt2 2rat distances r a. Obviously, the rejection of (1) leads to gravitational acceleration depending on the test body initial speed at infinity v according to the following equation d2r a a 3 a 1 dr = - 1 - +, a dt2 2r2 r 2 r2 a - dt r 2 dr a a where = v + 1 -.

dt r - a r c Apparently, if the test body velocity surpasses, the gravitational attraction turns into gravitational repulsion.

Moduli Spaces of Higher Spin Surfaces Pratoussevitch A. (UK) University of Liverpool annap@liv.ac.uk We describe the moduli space of m-spin structures on a Riemann surface. We show that any connected component of this moduli space is homeomorphic to a quotient of the vector space Rd by a discrete group action.

Birational rigidity and singularities of linear systems Pukhlikov A. V. (Russia) Steklov Mathematical Institute, University of Liverpool pukh@mi.ras.ru 1. Definition. (i) The variety V is said to be birationally superrigid, if for any movable linear system on V the equality cvirt() = c(, V ) holds.

(ii) The variety V (respectively, the Fano fiber space V/S) is said to be birationally rigid, if for any movable linear system on V there exists a birational self-map Bir V (respectively, a fiber-wise birational self-map Bir(V/S)), providing the equality cvirt() = c(, V ).

Here c(, V ) is the threshold of canonical adjunction, cvirt() is the virtual threshold of canonical adjunction. The importance of the property of birational (super)rigidity can be seen from the following fact.

2. Proposition. Let V be a primitive Fano variety, V a Fano variety with Q-factorial terminal singularities and Picard number one, that is, Pic V Q = QKV, : V V a birational map.

(i) Assume that V is birationally rigid. Then on V there are no structures of a rationally connected fiber space.

(ii) Assume that V is birationally rigid. Then V and V are (biregularly) isomorphic (although the map itself is, generally speaking, not an isomorphism).

(iii) Assume that V is birationally superrigid. Then is a biregular isomorphism. In particular, the groups of birational and biregular selfmaps of the variety V coincide: Bir V = Aut V.

3. Here are some examples of Fano varieties, for which birational superrigidity is known.

(i) A smooth three-dimensional quartic V = V4 P4 is birationally superrigid: this follows immediately from the arguments of [1].

(ii) A generic complete intersection Vd1·····dK PM+k of index one (that is, d1 + · · · + dk = M + k) and dimension M 4 is birationally superrigid for V 2k + 1 [2,3].

(iii) More examples are given by iterated Fano double covers [4] and Fano cyclic covers [5].

4. Conjecture. A smooth Fano complete intersection of index one and dimension 4 in a weighted projective space is birationally rigid, of dimension 5 birationally superrigid.

5. The only known method of proving birational (super)rigidity is the method of maximal singularities that reduces the problem of (super)rigidity of a variety V to studying movable linear systems on V + with a maximal singularity (that is, a prime divisor E V satisfying the Noether-Fano inequality E() > na(E), where n = c(, V )), see [1,2,4].

References [1] Iskovskikh V.A. and Manin Yu.I., Three-dimensional quartics and counterexamples to the Lroth problem, Math. USSR Sbornik. (1971), no. 1, 140-166.

[2] Pukhlikov A.V., Birational automorphisms of Fano hypersurfaces, Invent. Math. 134 (1998), no. 2, 401-426.

[3] Pukhlikov A.V., Birationally rigid Fano complete intersections, Crelle J. fr die reine und angew. Math. 541 (2001), 55-79.

[4] Pukhlikov A.V., Birationally rigid iterated Fano double covers. Izvestiya: Mathematics. 67 (2003), no. 3, 555-596.

[5] Pukhlikov A.V., Birational geometry of algebraic varieties with a pencil of Fano complete intersections. Manuscripta Mathematica.

121 (2006), 491-526.

Singularities in relaxation oscillations and geometric control theory Remizov A. O. (Portugal) University of Porto aremizov@fc.up.pt I. Consider the slow-fast system dx dy = F (x, y) + f(x, y, ), = G(x, y) + o(), (1) dt dt where x X = Rm, y Y = Rn, (R1, 0), and m 1, n 3. The slow motion field of (1) is the projection of the vector field dx dy = f(x, y, 0), = G(x, y), dt dt onto the tangent bundle of the slow surface S : F (x, y) = 0 along the fibers of the bundle : E Y with the base Y and coordinates x on the fibers, where E = X Y. This projection is defined at the points where T E = T S T (here T E, T S and T are the tangent spaces to E, S and the fiber of ), i.e., at the points where the slow surface S is transversal to the bundle. In coordinates it can be written as the vector field - dxi F Fi dy = - G,, = G(x, y), i = 1,..., m, (2) dt x y dt F defined on S \ K, where K : = 0 is the set of the points where x T E = T S T, i.e., it is the locus of the projection S Y along X (triangle brackets mean the standard scalar product). For the study of the slow motion at the points of K consider the field F Fi, i = 1,..., m, i = - G,, = G (3) y x where dot means differentiation with respect to the new variable, con F. Note, that the components nected with time t by the relation = x of (3) belong to the local ideal I the ring of smooth germs) generated (in F Fi by the germs: and G,, i = 1,..., m.

x y II. Totally singular extremals of the affine-control system dx = f0(x) + uf1(x), x Rn, u R1, (4) dt with n 3 and piecewise smooth locally bounded controls u(t) U are the extremals of (4) such that the Hamiltonian H of (4) attains the maximum (or minimum) in the interior of U, i.e., the extremals satisfy the identity h1(x, p) = 0, where hi(x, p) = fi(x), p, p (Rn) is the covector, H = h0 + uh1. Double differentiation of the previous identity gives h01 = 0 and h001 - uh110 = 0, where hi1,...,ik = {hi1, hi2,...,ik}, and {, } means Poisson brackets. The identity h001 - uh110 = 0 allows to express u through variables (x, p) at all points where h110 = 0, hence the totally singular extremals of (4) are the trajectories of the vector field dx h001 dp h0 h001 h= f0 + f1, = - -, (5) dt h110 dt x h110 x defined on S \ K, where S : h1 = h01 = 0 is an invariant manifold of the field (5), and K S : h110 = 0. For the study of the field (5) at the points of K consider the field h0 h = h110f0 + h001f1, = -h110 - h001. (6) x x where dot means differentiation with respect to the new variable, connected with time t by the relation = h110. Note that all components of the field (6) belong to the local ideal I generated by h110 and h001.

III. The aim of the talk is to study singular points of the vector fields (3) and (6). Under some natural assumptions the germs of these fields at the singular points have a common property: among their components there exist m + 1 (m = 1 for (6)) germs that generate the local ideal I (in the ring of smooth germs) containing all others component. Hence the fields (3) and (6) can be written in general form:

m+ = vi, i = 1,..., m + 1, = ijvi, j = 1,..., l, (7) i j i=where the local ideal is I = (v1,..., vm+1). The spectrum of the linear part of (7) at the singular point is (1,..., m+1, 0,..., 0). Suppose that Rei = 0, i = 1,..., m+1, then the set of the singular points v1 = 0,..., vm+1 = c of (7) is the central manifold W of codimension m + 1. We will consider finite-smooth normal forms of the fields (7) at the singular points. Note that some similar results for the case m = 1 (the local ideal I has two generators) were obtained before by J. Sotomayor, M. Zhitomirskiy (in the non-resonant case) and S. Voronin (for analytical classification).

Research was supported under grant SFRH/BPD/26138/2005.

Homogenization of some hydrodynamics problems with rapidly oscillating data Sandrakov G. V. (Ukraine) Kyiv National University sandrako@mail.ru Let be a small positive parameter and (u, p) be a Hopf’s solution of the initial-boundary value problem for unsteady Navier–Stokes equations u - u + u ·u + p = F in (0, T ), t divu = 0 in (0, T ), (1) u t=0 = 0 in, u = 0 on (0, T ), where F = F (t, x, x/), F (t, x, y) L2(0, T ; L2(; Lper(Y )/R)n), Rn is a bounded domain with a smooth boundary, T is a positive number, and 2 n 4. Here, a subscript per means 1-periodicity with respect to y Rn and Y = [0, 1]n is a periodicity cell. Thus, by definition F (t, x, y) is 1-periodic in y, F (t, x, y)dy = 0 for a. e. (t, x) (0, T ), and the Y restriction of F (t, x, y) to Y is an element of L2(0, T ; L2(; L(Y ))n).

Theorem. Let xF L1(0, T ; L2(; Lper(Y )/R)nn) and (u, p) is a solution of problem (1). Then, there are positive 0 and 0 such that u 2 + u 2 C(2 + 2-1), L(0,T ;L2()n) L2(0,T ;L2()nn) and p W -1,(0,T ;L2()/R) C( + 2-1-n/4), where C is a constant independent of and whenever 0 < 0 and 0 < 0.

Methods of homogenization are used to prove of the theorem (see [1]).

Similar theorems for equations (1) and the linearized equations will be discussed also, for example, when F (t, x, y) dy = 0.

Y References [1] Sandrakov G. V. The influence of viscosity on oscillations in some linearized problems of hydrodynamics // Izvestiya: Math. – 2007.

– V. 71, N 1. – P. 97–148.

The global theory of real corank 1 singularities and its applications to the contact geometry of space curves Sedykh Vyacheslav D. (Russia) Russian State Gubkin University of Oil and Gas sedykh@mccme.ru Let Mm and Nn be real C-smooth closed (compact without boundary) manifolds of dimensions m and n, respectively, where l = n - m 0.

Consider a stable smooth mapping f : M N.

Assume that f is a mapping of corank 1 that is it can have only singularities of types Aµ. We recall that f has a singularity of type Aµ at a given point x M if its local algebra at x is isomorphic to the R-algebra R[[t]]/(tµ+1) of truncated polynomials in one variable of degree at most µ.

The multi-singularity of f at a point y N is the unordered set of singularities of f at points from f-1(y). Multi-singularities of the mapping f are classified by elements A = Aµ1 + · · · + Aµp of the free additive Abelian semigroup A generated by the symbols A0, A1, A2,.... The p number codim A = (l + 1) µi + pl is called the codimension of a l i=multi-singularity of type A.

The mapping f can have only multi-singularities of codimension at most n. The set Af of points y N, where f has a multi-singularity of type A A, is a smooth submanifold of codimension codim A in N. The l Euler characteristic (Af ) of the manifold Af is the alternating sum of its Betti numbers.

We find a complete system of universal linear relations between the Euler characteristics of the manifolds of multi-singularities of mappings under consideration. Namely, we prove that for any A A such that codim A n - 1 (mod 2), the Euler characteristic (Af) is a linear coml bination l (Af ) = KA (X)(Xf) (1) X of the Euler characteristics (Xf), where the summation is carried over all X A such that codim X n (mod 2) and codiml X > codiml A.

l The universality of the relation (1) means that all its coefficients do not depend on f and on the topology of the manifolds M, N. We show that l every coefficient KA (X) is a rational number depending only on A, X, and on the parity of the number l. Moreover, we produce a combinatorial l algorithm for the calculation of the numbers KA (X).

The completeness of the system of relations (1) in the simplest case m < n means the following. Let Wm,n be the class of all stable smooth corank 1 mappings of smooth closed m-dimensional manifolds into smooth closed manifolds of dimension n. Then any universal linear relation with real coefficients between the Euler characteristics of manifolds of multi-singularities of mappings f Wm,n is a linear combination of the relations of the form (1) over all A A such that codiml A n-1 (mod 2) and codiml A < n.

We apply these results to the contact geometry of space curves. In particular, we obtain multidimensional generalizations of the Bose theorem on supporting circles of a plane curve and multidimensional generalizations of the Freedman theorem on the number of triple tangent planes of a curve in 3-space.

References [1] V. D. Sedykh, On the topology of singularities of the set of supporting hyperplanes of a smooth submanifold in an affine space, J. London Math. Soc. (2) 71 (2005), no. 1, 259–272.

[2] V. D. Sedykh, The topology of corank 1 multi-singularities of stable smooth mappings of equidimensional manifolds, C. R. Acad. Sci.

Paris, Sr I Math. 340 (2005), no. 6, 441–444.

[3] V. D. Sedykh, Corank 1 singularities of stable smooth mappings and special tangent hyperplanes to a space curve, Mat. Zametki 78 (2005), no. 3, 413–427.

[4] V. D. Sedykh, On the topology of symmetry sets of smooth submanifolds in Rk. In: Singularity Theory and its Applications, Adv. Stud.

Pure Math. 43, Math. Soc. Japan, Tokyo, 2006, 401–419.

[5] V. D. Sedykh, Resolution of corank 1 singularities of the range of a stable smooth mapping into a space of the grater dimension, Izv.

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