СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ Siberian Electronic Mathematical Reports

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1 S e MR ISSN СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ Siberian Electronic Mathematical Reports Том 5, стр (08) УДК DOI /semi MSC 30C65 THE COEFFICIENT OF QUASIMÖBIUSNESS IN PTOLEMAIC SPACES V.V. ASEEV Abstract. In ptolemaic spaces the class of η-quasimöbius mappings f : X Y with control function η(t) = C max{t α, t /α } may be completely characterized by the inequality K ( + log P (ft ))/( + log P (T )) K for all tetrads T X where P (T ) denotes the ptolemaic characteristic of a tetrad. The number K has properties quite similar to those of coefficients of quasiconformality, so the concept of K- quasimöbius mapping may be introduced. In particular, the stability theorem is proved for ( + )-quasimöbius mappings in R n. Keywords: ptolemaic space, Möbius mapping, quasimöbius mapping, (power) quasimöbius mapping, quasisymmetric mapping, stability theorem.. Tetrads The tetrad in the semimetric space (X, ρ) is a quadruple T = (x, x, x 3, x 4 ) of mutually distinct points. The absolute cross-ratio of a tetrad T is R(T ) = R(x, x, x 3, x 4 ) := ρ(x, x )ρ(x 3, x 4 ) ρ(x, x 3 )ρ(x, x 4 ). (.) The semimetric space (X, ρ) is called ptolemaic if the Ptolemy inequality ρ(x, x )ρ(x 3, x 4 ) + ρ(x, x 4 )ρ(x, x 3 ) ρ(x, x 3 )ρ(x, x 4 ) (.) Aseev, V.V., The coefficient of quasimöbiusness in Ptolemaic spaces. c 08 Aseev V.V. The work is supported by the program of fundamental scientific researches of the SB RAS No...., project No Received June, 8, 07, published March, 6,

2 THE COEFFICIENT OF QUASIMÖBIUSNESS IN PTOLEMAIC SPACES 47 holds for any quadruple (x, x, x 3, x 4 ) of point in X. Ptolemaic characteristic of a tetrad in (X, ρ) is P (T ) = P (x, x, x 3, x 4 ) := ρ(x, x )ρ(x 3, x 4 ) + ρ(x, x 4 )ρ(x, x 3 ) ρ(x, x 3 )ρ(x, x 4 ) The inequality P (T ) is true for any tetrad T in ptolemaic space (X, ρ). For some general properties of ptolemaic spaces see [, 3, pp.78-80].. Quasimöbius mappings. (.3) Let η : [0, + ) [0, + ) be a homeo-morphism of [0, + ) R onto itself. An injective mapping f : X Y in semimetric spaces (X, ρ), (Y, σ) is called η- quasimöbius if the estimate R(fT ) η(r(t )) holds for any tetrad T in X. In that case we write f η-qm and call η the control function for f. For the definition and basic properties of quasimöbius mappings, as well as for their connections with quasiconformal mappings, see []. In particular, µ : X Y is a möbius mapping if R(fT ) = R(T ) for each tetrad T in X. The case where the control function η(t) is of the form η(t) = C max{t /α, t α }, with C, α (.) defines a special important subclass of quasimöbius mappings, so called (power) quasimöbius mappings. The function (.) will be called (power) control function. The inverse function to (.) is η (t) = min{(t/c) α, (t/c) /α }, and η(t) := η(/t) = C min{tα, t /α }, (.) η (t) := η (/t) = max{(ct)α, (Ct) /α } C α max{t α, t /α }. (.3) For tetrads T = (x, x, x 3, x 4 ) and T = (x, x 3, x, x 4 ) as well as for their images ft = (y, y, y 3, y 4 ) and ft = (y, y 3, y, y 4 ), where y j = f(x j ), we have the equalities R(T ) = R(T ), R(fT ) = R(fT ). If f : X Y is η-quasimöbius then the estimate R(fT ) η(r(t )) holds, which is equivalent to ( ) R(fT ) η. R(T ) That is the inequality R(fT ) (η(/r(t )) = η(r(t )) holds for any tetrad T in X. It follows that f : X Y being η-quasimöbius with (power) control function (.) the two-sided estimate η(r(t )) R(fT ) η(r(t )) (.4) holds for any tetrad T in X, and the mapping f : f(x) X is also (power) quasimöbius with the (power) control function C α max{t α, t /α } = C α η(t).

3 48 V.V. ASEEV The notion of (power) control function was initially introduced in [3] for quasisymmetric mappings and then in [] for qiasimöbius mappings. The complete characterization for the class of all metric spaces where every quasisymmetric embedding has a power control function (.) was obtained in [4, Th. 6.] in terms of upper sets introduced by D.A. Trotsenko. In particular, this class contains all uniformly perfect metric spaces. 3. Distortion of ptolemaic characteristic As a special case of more general properties presented in [5, Propositions 3.3, 3.4] we state the following Lemma. Let f : X Y be a (power) quasimöbius mapping in ptolemaic spaces (X, ρ), (Y, σ) with the (power) control function η(t) from (.). Then the following inequality α C [P (T )]/α P (ft ) C[P (T )] α. (3..) holds for every tetrad T in X. Proof. Given a tetad T = (x, x, x 3, x 4 ) in X we denote y j = f(x j ) for j =,, 3, 4. Then η-quasimöbius property of f together with the ptolemaic condition P (T ) implies the right-hand inequality in (3..): P (ft ) = σ(y, y ) σ(y 3, y 4 ) σ(y, y 3 ) σ(y, y 4 ) + σ(y, y 4 ) σ(y, y 3 ) σ(y, y 3 ) σ(y, y 4 ) η(p (T )) = C[P (T )]α. One of items in the sum ρ(x, x ) ρ(x 3, x 4 ) ρ(x, x 3 ) ρ(x, x 4 ) + ρ(x, x 4 ) ρ(x, x 3 ) ρ(x, x 3 ) ρ(x, x 4 ) = P (T ) must be P (T )/. So the left-hand inequality in (.4) implies the estimate ( ) ( ) ( ) ρ(x, x ) ρ(x 3, x 4 ) ρ(x, x 4 ) ρ(x, x 3 ) P (T ) P (ft ) η + η η. ρ(x, x 3 ) ρ(x, x 4 ) ρ(x, x 3 ) ρ(x, x 4 ) Then the expression (.) for η(t) together with the ptolemaic inequality P (T ) leads to left-hand part in the desired estimate (3..): P (ft ) { (P ) α ( ) } /α (T ) P (T ) C min, min{p (T )α, P (T ) /α } [P (T )]/α α = C α. C Theorem. Let f : X Y be an injective mapping in ptolemaic spaces (X, ρ), (Y, σ). Then (i) f being a (power) η-quasimöbius with η(t) = C max{t α, t /α } the estimate K holds for any tetrad T in X; (ii) if the estimate + log(p (ft )) + log(p (T )) K := α( + log(cα )) (3..) K + log(p (ft )) + log(p (T )) K := +, (3..3)

4 THE COEFFICIENT OF QUASIMÖBIUSNESS IN PTOLEMAIC SPACES 49 holds for any tetrad T in X then f is η-quasimöbius with (power) control function η(t) = (9e ) max{t +, t + }. (3..4) Proof. (i) Let T be a tetrad in X. Then by the right-hand of (3..) we have and consequently log(p (ft )) log(c) + α log(p (T )), + log(p (ft )) + log(p (T )) log(c) + α log(p (T )) + + log(p (T )) + log(p (T )) log(c) + α α( + log(c)) α( + log(c α )) = K. Thus the right-hand inequality in (3..) has been obtained. The mapping f : f(x) X is (power) η -quasimöbius with the (power) control function η (t) = C α max{t α, t /α } (see ). Applying the right-hand inequality in (3..) to the mapping f with C α instead of C and to a tetrad ft in f(x) we obtain the inequality + log(p (T )) + log(p (ft )) α( + log(cα )) = K which gives us the left-hand estimate in (3..). The proof of (i) is complete. (ii) It follows from (3..3) that e K K [P (T )] K for any tetrad T in X. That is e P (ft ) e K [P (T )] K + [P (T )] + P (ft ) e [P (T )] +. (3..5) Given a tetrad T = (x, x, x 3, x 4 ) in X and it s image ft = (y, y, y 3, y 4 ) in Y where y j = f(x j ), j =,, 3, 4, we need in the estimate R(fT ) η(r(t )). It has been proved in [5, Corollary.9] that any ptolemaic metric four-point set A may be embedded into the extended complex plane by möbius transformations which do not change the characteristics R and P of all tetrads in A. Then considering the restriction f T we may assume without loss of generality that tetrads T and f T are in the extended complex plane, and T = (0, z,, ), ft = (0, w,, ). The required estimate R(fT ) η(r(t )) is equivalent to the inequality w η(). (3..6) For tetrads T and ft we have p := P (T ) = + z, q := P (ft ) = w + w. The right-hand estimate in (3..5) means that Considering the tetrad T = (0,, z, ) in T we have p := P (T ) = + z By the left-hand inequality in (3..5) we have q e p +. (3..6) q e ; q := P (ft ) = + p + + w w. (3..7).

5 50 V.V. ASEEV Now the equality ( + q )/( + q ) = w together with (3..6)-(3..7) leads to the estimate w = + q + e p + + q + e e p + e p + + p + e e + p + + p + + p + p e + p which means that Case. Let. Then p p = + p + e p = e + + (p p ) p +, w e (p p ) p +. (3..8) ( + z )( + z ) and p +. So in this case we obtain the desired estimate Case. Let. Then and p p = ( + ) 3 w e 3 < (9e ) +. (3..9) ( + z )( + z ) ( p + = ( + z ) + z So in this case we also obtain the desired estimate ) + +. w (9e ) + + = (9e ) +. Thus w (9e ) max{ +, + } = η(), and (ii) has been proved. 4. Coefficient of quasimöbiusness ( + ) 9 Theorem makes it possible to measure the (power) quasimöbius property with just a one number instead of the control function and justifies the following concept. Definition. Given K the injective mapping f : X Y in ptolemaic spaces (X, ρ), (Y, σ) is said to be K-quasimöbius if the two-sided estimate K + log(p (ft )) + log(p (T )) K, (4..) holds for any tetrad T in X. The minimal number K ensuring the ineqiality (4..) for every tetrads in X may be regarded to as the coefficient of quasi-möbiusness of the mapping f, quite analogous to the notion of coefficient of quasiconformality. This analogy appears in the following elementary basic properties. Proposition. If f : X Y is K-quasimöbius then f : f(x) X is also K-quasimöbius.

6 THE COEFFICIENT OF QUASIMÖBIUSNESS IN PTOLEMAIC SPACES 5 Proposition. If f : X Y is K -quasimöbius and g : f(x) Z is K - quasimöbius then g f : X Z will be K-quasimöbius with K = K K. Proposition 3. Every -quasimöbius mapping f : X Y is a möbius mapping. Proof. It follows from the Definition that -quasimöbius mapping preserves the ptolemaic characteristic of tetrads. Then by [5, Theorem.3] it preserves the absolute cross-ratio for every tetrad in X. Thus f is a möbius mapping. The notion of the coefficient of quasimöbiusness allows to consider various extremal problems in the class of (power) quasimöbius mappings just similar to these presented in [6] for quasiconformal mappings. Moreover, if X and Y consist of the same finite number of points the mapping of X onto Y with minimal coefficient of quasimöbiusness may be found by computer calculation. 5. Connection with [s]-characteristic of quasimöbiusness In order to consider the problems of approximation and stability the following number characterictic [s] was first introduced in [7] for quasisymmetric and then in [] for quasimöbius mappings. Definition. Let X,Y be semimetric spaces, and s [0, ]. An η-quasimöbius (or η-quasisymmetric) mapping f : X Y is called [s]-quasi-möbius (respectively, [s]-quasisymmetric) if η(t) t s whenever t /s. Proposition 4. Let s (0, ], C e, log( + s ) log(c/s), (5..) and η(t) = C max{t +, t + }. Then η(t) t s whenever t /s. Proof. In case t /s we have t t t[c t ] s [(C/s) ] s [( + s ) ] = s, as desired. In case t (0, ] we have t t φ(t) := C t + t. Since the function φ(t) has unique point of extremum ( C t 0 = + ) + it is increasing in [0, ], so that as desired. [ ] + C = ( + ) / [ ] + C > e t t φ() = C ( + s ) s, Now we can specify the rusult in [8, Lemma 4.] on [s]-characteristic of the inverse mapping.

7 5 V.V. ASEEV Theorem. Let s (0, ]. If the injective mapping f : X Y in ptolemaic spaces is K-quasimöbius with K = + and log( + s ) log(9e /s), s (0, ], then both mappings f and f are [s]-quasimöbius in the sense of Definition. Proof. Since both f and f are K-quasimöbius (see Proposition ), it follows from Theorem (ii) that they are η-quasimöbius with η(t) = C max{t +, t + } where C = 9e > e. Then Proposition 4 means that both f and f are [s]- quasimöbius. 6. Quasisymmetry in chordal metric It is well known that quasi-möbius mapping in bounded metric spaces is quasisymmetric one. Lemma ([], Theorem 3.). Let (X, ρ) and (Y, σ) be bounded metric spaces of diameters d(x) and d(y ) respectively. Let for a given ω-quasimöbius mapping f : X Y there exist points a, a, a 3 X such that ρ(a i, a j ) δ ; σ(f(a i ), f(a j )) δ for every distinct i, j {,, 3}. Then f is η-quasisymmetric with the control function η(t) = d(y ) ( ) d(x) ω t. (6..) δ δ Proof. Let be given mutually distinct points x, y, z X. By the triangle inequality in X and Y we can find an index j {,, 3} such that ρ(x, a j ) δ/ ; σ(f(z), f(a j )) δ/. Then the ω-quasimöbius property for the tetrad T = (y, x, z, a j ) gives the desired estimate: ( ) δ σ(f(x), f(y)) ρ(x, y) d(x) R(T ) ω(r(ft )) ω. d(y ) σ(f(z), f(y)) ρ(z, y) δ In the special case where X, Y R n are equipped with chordal metric and f is a (power) quasimöbius mapping we need in some more precise estimate for the control function η(t) of quasisymmetry. The proof of the following lemma is based on elementary routine estimations and has been placed in Appendix at the end of the article. Lemma 3. Let the set A R n to contain the points 0, e 0, where e 0 = which are fixed points for an ω-quasimöbius mapping f : A R n with the control function ω(t) = C max{t +, t + }. Then for any x A in every one of the following two situations () a = e 0 and ( x 4C / or x /(4C )); () (a = and 0 x 4C /) or (a = 0 and x /(4C ))

8 THE COEFFICIENT OF QUASIMÖBIUSNESS IN PTOLEMAIC SPACES 53 the inequalities C σ(f(a), a) σ(x, a) + are true with the constant C = 6 C 5 e. σ(f(x), a) σ(x, a) + C (6..) Theorem 3. Let the set A R n contain the points 0, e 0, where e 0 = which are fixed points for an ω-quasimöbius mapping f : A R n with the control function ω(t) = C max{t +, t + }. Then f is η-quasisimmetric mapping in chordal metric in R n with the control function where C = 9 C e 4. η(t) = C max{t +, t + } Proof. Given distinct points x, y, x A let us denote their images under f by x, ỹ, z respectively. It should be proved the inequality ( ) { σ( x, ỹ) σ(x, y) (σ(x, ) + ( ) } σ( z, ỹ) η y) σ(x, y) + = C max,. σ(z, y) σ(z, y) σ(z, y) If a {0, e 0, } and a x, a z then ω-quasimöbius property of f produces the inequality { (σ(x, ) + ( ) } σ( x, ỹ)σ( z, a) y)σ(z, a) σ(x, y)σ(z, a) σ( z, ỹ)σ( x, a) + C max, ; σ(z, y)σ(x, a) σ(z, y)σ(x, a) where σ( x, ỹ) σ( z, ỹ) C max T = σ(z, a)+ σ( z, a) { (σ(x, ) + y), σ(z, y) + σ( x, a) ; T = σ(z, a) ( ) σ(x, y) + σ(z, y) σ(z, a) + σ( z, a) } max{t, T }, σ( x, a) σ(z, a) +. So we have to obtain estimate max{t, T } (C) where C is the constant from Lemma 3. In case ( x /(4C ) and /(4C )) we put a = 0 and use (6..) both for x and z in Lemma 3, the situation (). In case ( x 4C / and 4C /) we put a = and use (6..) both for x and z in Lemma 3, the situation (). In case (( x 4C / and ](4C ) or x /(4C ) and 4C /)) we put a = e 0 and use both for x and z the inequality (6..) from Lemma 3, the situation (). Thus we obtain the desied estimate for T and T in all possible cases. 7. Stability We shall use the following stability theorem for [s]- quasi-symmetric mappings in R n which had been obtained by J. Partanen in his dissertation in 99. It is more convenient for our purposes to formulate it with R n+ instead of R n. Theorem 4 ([8], Theorem.6). Let A R n+ be compact and B A have at least two distinct points. Then there ezists a function λ(s; B, A ) which 0 as s 0, such that for any [s]-quasisymmetric mapping f : A R n+ which is

9 54 V.V. ASEEV identical on B there exists an euclidean isometry h : R n+ R n+ identical on B such that max f(x) h(x) λ(s; x A A, B). (7..) The estimate function λ in this theorem essentially depends on metric properties of the set A and may be detailed in some special cases, see [9]. We shall prove the following version of stability theorem for K-quasi-möbius mappings in R n. Theorem 5. Let the set A R n contain points 0, e 0, where e 0 =. Then for a given δ > 0 there exists 0 > 0 with the following property: For every K- quasimöbius mapping f : A R n with fixed points 0, e 0, and K = + where 0 there exists an euclidean isometry h with fixed points 0, e 0 such that max σ(f(x), h(x)) δ, x A where σ(.,.) denotes the chordal distance in R n. Proof. By Theorem (ii) the K-quasimöbius mapping f : A R n with K = + is (power) η-qusimöbius with the control function η(t) = (9e ) max{t +, t + }. Since f( ) = it is η-quasisymmetric in euclidean metric in R n. Then Theorem 3 says that f is also η -quasisymmetric in chordal metric with the distortion function η (t) = (C 3 ) max{t +, t + } where C3 = 9 9 e 6. Since and max{t +, t + } max{t +, t + } the mapping f is η -quasisymmetric in chordal metric with the control function η (t) = C3 max{t +, t + }. The stereographic projection π : Rn S R n+, π(0) = 0 is the isometry of the space R n equipped with chordal metric to the sphere S R n+ equipped with euclidean metric. So we may identify R n as a sphere S in R n+. We denote e = π(e 0 ), p = π( ). Then the mapping g = π f π : A = π(a) S is η -quasisymmetric in euclidean metric in R n+ and is identical on the set B = {0, e, p} π(a) S. As the function λ(s, A, B) in the Theorem 4 tends to 0 as s 0 we can find for a given δ > 0 such s 0 > 0 that λ(s 0, A, B) δ. Then we can find 0 such that log( + s 0) log(c 3 /s 0 ) (7..) for all < 0. In this case we have by Proposition 4 the estimate η (t) t s 0 to be valid for all 0 t s 0. That means that g is [s 0 ]-quasisymmetric in euclidean metric in R n+. By Theorem 4 there exists an euclidean isometry h : R n+ R n+ with fixed points 0, e, p such that max g(y) y Σ(A) h (y) λ(s 0, A, B) δ. The isometry h with fixed points 0, p is identical on the whole line through these points, so the center y 0 of sphere S is also a fixed point for h. It means that h (S) = S and h = π h π is an isometry in chordal metric in R n. Hence h being a möbius mapping with fixed points 0, e 0, is mere an euclidean isometry in R n with fixed line through 0 and e 0. Next we have the equality g(y) h (y) = σ(f(π (y)), h(π (y))) = σ(f(x), h(x)) for all x A with y = π(x). Thus max g(y) y A h (y) = max σ(f(x), h(x)) δ x A

10 THE COEFFICIENT OF QUASIMÖBIUSNESS IN PTOLEMAIC SPACES 55 provided Appendix Here we present the proof to Lemma 3. Case (). Since the mapping j(x) = x/ x (j(0) =, j( ) = 0, j(e 0 ) = e 0 ) preserves chordal distances the mapping g = j f j satisfies on j(a) the same conditions as f. Then for each α { +, /( + )} the equality σ(g(j(x)), e 0 ) σ(j(x)), e 0 ) α = σ(j(f(x)), e 0) σ(j(x), e 0 ) α = σ(f(x), e 0) σ(x, e 0 ) α holds for all x A. Thus in the sutuation () it suffices to proove () for the case x 4C /. Applying to x := f(x) the ω-quasimöbius property we have: ( x C C ) 4C C C ; + x ln ( x ) + x ( + x ) ln + x x + x x ; x e 0 ( + x ) + x + x e ; (0.) x ( x ) = x ( / x ) (/) = ; x e 0 ( x ) + x + x (e ). (0.) ln + x ( x ) x ( x ) = ( ) + ( + x + x x e 0 x ( + ) ( + x x e 0 ln ( + x ) + x ) + x ( / x ) 4C (/) ; ) + (e ) ; (0.3) x ( + x ) 8C ; ( + x x + ) + e. (0.4) Thus using the inequalities ( ) ( ), ( ) + = ( ) + /( ) and (0.)-(0.4) we obtain the desired estimate C (e 3 ) σ( x, e 0) σ(x, e 0 ) + Case (). Since j is the chordal isometry the equality σ( x, e 0) σ(x, e 0 ) + (e3 ) C σ(g(j(x)), 0) σ(j(f(x)), 0) σ(f(x), ) = = σ(j(x), 0) α σ(j(x), 0) α σ(x, ) α. holds. So in the situation () it suffices to prove (6..) for the case a =.

11 56 V.V. ASEEV If x 4C /, the ω-quasimöbius property together with the inequality / produces the following estimates (σ(x, )) + = ( + x ) + + x C ( + x ) + + x + (C ) x + x (C ) x + x + x + + x + + x + (C ) x x + (C ) x (6 C 5 e ) ; ( + x ) + ( + x ) + + x (+) = σ(x, ) + + x C C x ( + x ) (+) (C ) x (6 C 5 e ). Thus we obtain the estimates (6..) in the case a =, x 4C /: C (6 C 5 e ) σ(x, ) + σ(x, ) + 6 C 5 e ) = C If a =, x then x + x C x +, and σ(x, ) + = ( + x ) + + x ( ) C + x C + x x (C ) + x x ( + x ) (C ) (C e ) C ; σ(x, ) + = + x ( + x ) + (C ) + x x + + x (C ) x (C ) Thus (6..) is true in the case a =, x. At last, in the case a =, 0 x we have σ(x, ) + σ(x, ) + = = + C x + + x ( + x ) + (C e ) C. ( + x ) + + x ( + ) + e () e C ; + x ( + x ) +. C + x + C ( + ) (Ce) C. So (6..) is true in the case a =, 0 x as well. Now the lemma 3 has been comletely proved. The author is thankful to referee for his(her) critical remarks on the content of this article.

12 THE COEFFICIENT OF QUASIMÖBIUSNESS IN PTOLEMAIC SPACES 57 References [] M.L. Blumenthal, Theory and Applications of Distance Geometry, Oxford: Clarendod Press, 953. MR [] J. Väisälä, Quasimöbius maps, J. Anal. Math., 44 (984/85), MR08095 [3] P. Tukia, J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math., 5: (980), MR [4] D.A. Trotsenko, J. Väisälä, Upper sets and quasisymmetric maps, Ann. Acad. Sci. Fenn. Ser. A I Math., 4: (999), MR74387 [5] V.V. Aseev, A.V. Sychev, A.V. Tetenov, Möbius-invariant metrics and generalized angles in ptolemeic spaces, Sib. Math. J., 46: (005), MR493 [6] F.W. Gehring, J. Väisälä, The coefficients of quasiconformality of domains in space, Acta Math., 4: (965), 70. MR [7] P. Tukia, J. Väisälä, Extension of embeddings close to isometries or similarities, Ann. Acad. Sci. Fenn. Ser. A I Math., 9 (984), MR07540 [8] J. Partanen, Invariance theorems for the bilipschitz and quasisymmetric extension properties, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissert., 80 (99), 40. MR [9] P. Alestalo, D.A. Trotsenko, On mappings that are close to a similarity, Math. Rep., Buchar., 5(65):4 (03), MR Vladislav Vasilyevich Aseev Sobolev Institute of Mathematics, pr. Koptyuga, 4, , Novosibirsk, Russia address: btp@math.nsc.ru