A unified construction yielding precisely Hilbert and James sequences spaces

A UNIFIED CONSTR UCTION YIELDING PRECISEL Y HILBER T AND JAMES SEQUENCES SP A CES Du ˇ san Repo v ˇ s and P a vel V. Semenov A bs tr act . F ollo wing James’ approac h, we shall define the Banac h space J ( e ) for eac h v ector e = ( e 1 , e 2 , ..., e d ) ∈ R d with e 1 6 = 0. The construction immediate ly implies that J ( 1) c oincides with the Hi lbert s pace i 2 and that J (1; − 1) c oincides with the celebrated quasireflexive James space J . The results of this p aper sho w that, up to an isomorphism, there are only the following t wo p ossibili ties: (i) either J ( e ) is isomorphic to l 2 ,if e 1 + e 2 + . .. + e d 6 = 0 (ii) or J ( e ) is i somorphic to J . Such a dichoto my also holds f or every separable Orlicz sequence space l M . 0. In tro duction In infinite-dimensional analysis and top ology – in Banach space theory , tw o se- quences spaces – the Hilber t s pa ce l 2 and the James spa ce J – are certainly presented as a tw o principa lly o ppos ite ob jects. In fact, the Hilb ert spa ce is the ” simplest” Banach space with a max imally nice analytica l, geometrical and top ologica l pro p- erties. On the con trary , the pro perties of the James space are so unusual and unexp e cted that J is often called a ”space of co un ter examples” (see [3,5]). Let us list s ome of the James spa ce pro p erties: (a) J has the Sc hauder bas is , but admits no is omorphic em b edding in to a s pace with unconditiona l Schauder basis [1,3,4]; (b) J and its s econd conjugate J ∗∗ are s eparable, but dim( J ∗∗ /χ ( J )) = 1, where χ : J → J ∗∗ is the canonical embedding (see [1]); (c) in spite of (b), the spaces J and J ∗∗ are isometric with resp ect to an eq uiv a len t nor m (see [2]); (d) J and J ⊕ J are non-isomor phic and mor eov er, J a nd B ⊕ B are non-iso morphic for an arbitr ary w eakly complete B (see [3, 4]); (e) on J there exists a C 1 -function with b ounded suppor t, but there ar e no C 2 -functions wit h bounded support (see [7]); (f ) ther e exists a n infinite-dimensional manifold mo delled on J which ca nnot be homeo morphically embedded in to J (se e [4 ,7]); and (g) the group GL ( J ) of all inv ertible contin uous op erators o f J onto itself is homoto pically non-tr ivial with resp ect to the top ology gener ated by op erator’s norm (s e e [8]), but it is co n tr actible in p oint wise conv erg ency op era to r top ology (see [10 ] and the b o ok [3] for more references). In this pap er we shall define the Banach spa ce J ( e ) for each vector e = ( e 1 , e 2 , ..., e d ) ∈ R d with e 1 6 = 0. The construction immediately implies that J (1) = l 2 and 1991 Mathematics Subje c t Classific ati on . Primary: 54C60, 54C65, 41A65; Secondary: 5 4C55, 54C20. Key wor ds and phr ases. Hilb ert space;Banac h s pace; James sequence space; Inv ertible contin- uous op erator. Ty p eset b y A M S -T E X 1 2 DU ˇ SAN REPOV ˇ S AND P A VEL V. S EMENOV J (1; − 1 ) = J . Surpr isingly , there are only these t wo po ssibilities, up to a n isomor- phism. It app ears that J ( e ) is isomo rphic to l 2 , if e 1 + e 2 + ... + e d 6 = 0 (see Theor e m 5) and J ( e ) is is omorphic to J otherwis e (see Theorem 6). Such a dichotomy holds not only for the space l 2 (whic h is clearly defined by using the numerical function M ( t ) = t 2 , t ≥ 0), but a ls o for an a r bitrary Orlicz sequence space l M defined by an ar bitr ary Or licz function M : [0 ; + ∞ ) → [0 ; + ∞ ) with the so-c alled ∆ 2 -condition. Then there are a lso ex actly tw o possibilities for J ( e ): either J ( e ) is isomorphic to l M , or J ( e ) is isomo rphic to the James-Orlicz space J M (see [9]). F o r simplicity , we restr ict ours elv es b elow for M ( t ) = t 2 , t ≥ 0. 1. Preliminaries Let d b e a natural num b er and e = ( e 1 , e 2 , ..., e d ) ∈ R d a d -vector with e 1 6 = 0. Having in our formulae many brackets we shall cho ose the sp ecial notatio n a ∗ b for the usual scala r pr oduct of tw o elements a ∈ R d and b ∈ R d . A d -subset ω of N is defined b y setting ω = { n (1) < n (2 ) < ... < n ( d ) < n ( d + 1) < ... < n ( k d − 1) < n ( k d ) } ⊂ N for some natural k and then the subsets ω (1) = { n (1 ) < n (2) < ... < n ( d ) } , ω (2) = { n ( d +1) < n ( d +2) < ... < n (2 d ) } , ......, ω ( k ) = { n (( k − 1) d +1) < ... < n ( k d ) } are called the d -compo nen ts of the set d -set ω . F or each d -set ω and ea c h infinite s e quence of reals x = ( x ( m )) m ∈ N ∈ R N we denote x ( ω ) = ( x ( m )) m ∈ ω and x ( ω ; i ) = ( x ( m )) m ∈ ω ( i ) . Definition 1. F or e ach d ∈ N , e ∈ R d , x ∈ R N and d -set ω = { n (1) < ... < n ( k d ) } the ( e, ω ) -variation of x is define d by the e quality ( e, ω ) = v u u t k X i =1 ( e ∗ x ( ω ; i )) 2 = p ( e 1 x ( n (1)) + ... + e d x ( n ( d ))) 2 + ... + ( e 1 x ( n (( k − 1) d + 1)) + ... + e d x ( n ( k d ))) 2 Definition 2. F or e ach d ∈ N , e ∈ R d , x ∈ R N the e -variation of x is define d by the e qu ality || x || e = s u p { e ( x, ω ) : ω ar e d -subset of N } . Definition 3. The set of al l infinite se qu enc es of r e als ten ding to zer o with finite e -variation is denote d by J ( e ) . W e omit the routine verification of the following prop osition. Prop osition 4. ( J ( e ); || · || e ) is a Banach sp ac e for e ach e = ( e 1 , e 2 , ..., e d ) ∈ R d with e 1 6 = 0 .  Note that a restriction e 1 6 = 0 is purely technical. It av oids the case e = 0 ∈ R d and guarantees that k (1; 0; 0; ... ) k e > 0 . B elo w w e fix such a hypotesis. A U NIFIED CONSTRUCTION 3 Theorem 5. If e 1 + e 2 + ... + e d 6 = 0 , then J ( e ) and l 2 ar e isomorphic. Theorem 6. If e 1 + e 2 + ... + e d = 0 , then J ( e ) and J ar e isomorphic. Theorem 5 is prov ed in Section 2 a s the co rollary o f Lemmas 7 -10. W e believe that Lemma 9 is of interest indep endent ly of Theore m 5 a nd its proof. Theorem 6 is prov ed in Sectio n 3 as a corolla ry of Lemmas 1 1 -13. Lemma 11 rea lly str esses the impor tance o f eq ualit y e 1 + e 2 + ... + e d = 0. Lemma 13 is the most difficult t o prov e. In the last case so me s pecial combi- natorial Sublemma 14 is needed. Roughly sp eaking, it states that ea c h 2- subset of naturals a dmits a r epresentation as a union of at mos t N = [0 , 5 d ] + 2 of its 2-subsets whic h consist o f d separated pairs. It seems that this sta temen t is new and p ossibly interesting for geo metric co mbinatorics. F o r example o ne can try to find an a nalog o f Sublemma 1 4 for finite planar subsets. One more op en question conce r ns analog s of Theo rems 5 and 6 for spaces o f functions over the segment [0 ; 1 ]. The main o bstruction her e is that the James functional space J F has a non-sepa r able dual space [4]. Also, we b elieve that Theorems 5 and 6 ar e true for a generaliza tions of J in the spirit of results o f [6]. 2. Pro of of Theorem 5 Lemma 7. The inclusion op er ator id : l 2 → J ( e ) is wel l-define d and c ontinuous. Pro of. Let k · k 2 be the s tandard Euclidean norm. Fix any x = ( x 1 , x 2 , x 3 , ... ) ∈ l 2 and pick a n y d -set ω = ω (1 ) ∪ ω (2) ∪ ... ∪ ω ( k ) with d -comp onents ω (1) , ω (2) , ...ω ( k ). Then ( e ∗ x ( ω ; i )) 2 ≤ k e k 2 2 · k x ( ω ; i ) k 2 2 due to the Ca uc hy inequality . Hence, ( e ( x ; ω )) 2 = k X i =1 ( e ∗ x ( ω ; i )) 2 ≤ k e k 2 2 · ( k X i =1 k x ( ω ; i ) k 2 2 ) ≤ ( k e k 2 k x k 2 ) 2 and therefore k x k e = sup { e ( x ; ω ) : ω } ≤ k e k 2 k x k 2 = C k x k 2 .  Lemma 8. det    0 e 1 e 2 . . . e d e 1 0 e 2 . . . e d · · · · e 1 e 2 . . . e d 0    = ( − 1) d ( d Y i =1 e i )( d Y i =1 e i ) .  Lemma 9. F or e ach d ∈ N , e ∈ R d with e 1 6 = 0 and e 1 + e 2 + ... + e d 6 = 0 ther e exists a c onstant C = C e > 0 s u ch that for every se qu enc e of r e als x (1) , x (2) , ..., x ( d ) , x ( d + 1 ) the ine qu ality | d X i =1 e i x ( n ( i )) | ≥ C | x (1) | holds for some d-set 1 ≤ n (1) < ... < n ( d ) ≤ d + 1 . Pro of. The a ssertion is obvious for x (1) = 0. So let x (1) 6 = 0 and consider the case when a ll num b ers e 1 , e 2 , ..., e d are non-zero . Denote by L the linear mapping of 4 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEMENOV R d +1 int o itself defined by the matrix from Lemma 5. By this lemma, L : R d +1 → R d +1 is an isomorphism. Consider R d +1 with the max - no rm k ( x (1) , x (2) , ..., x ( d ) , x ( d + 1)) k = max {| x ( j ) | : 1 ≤ J ≤ d + 1 } , i.e. as the Ba nach space l d +1 ∞ of dimensio n d + 1. Define the constant C as the distance b etw een the origin a nd the L image of the set of all ele ments with the fir st co ordinate eq ual to ± 1 : C = dist(0; { L ( y (1) , y (2) , ..., y ( d ) , y ( d + 1) : y (1) = ± 1 } ) > 0 . Next, pic k x = ( x (1) , x (2) , ..., x ( d ) , x ( d + 1)) ∈ l d +1 ∞ with x (1) 6 = 0 and set y ( i ) = x ( i ) · ( x (1)) − 1 , i = 1 , 2 , ..., d, d + 1 . Then y (1) 6 = 0 and k L ( y (1) , y (2 ) , ..., y ( d ) , y ( d +1 )) k ∞ ≥ C , k L ( y (1) , y (2) , ..., y ( d ) , y ( d +1 )) k ∞ ≥ C | x (1) | . By definition of the max nor m and by the definition o f the isomorphis m L we see that | P d i =1 e i x ( n ( i )) | ≥ C | x (1) | , for some indices 1 ≤ n (1) < ... < n ( d ) ≤ d + 1. It is ea sy to chec k that for an arbitrar y e ∈ R d with e 1 6 = 0 and e 1 + e 2 + ... + e d 6 = 0 the cons tan t C e , works pr oper ly , where the vector e , consists of all non-zero co ordinates of the vector e .  Lemma 10. The inclusion op er ator I d : l 2 → J ( e ) is a su rje ction. Pro of. Supp ose to the co n tra ry that k x k e < ∞ but k x k 2 = ∞ for some x = ( x ( m )) m ∈ N ∈ R N . Due to the eq ua lit y ∞ X m =1 x 2 ( m ) = d +1 X i =1 ( ∞ X k =1 x 2 ( k ( d + 1) + i )) we see that for s ome 1 ≤ i ≤ d + 1 the series P ∞ k =1 x 2 ( k ( d + 1) + i ) is div ergent. So let C b e the constant from Lemma 9. Applying this lemma for each natura l k to the reals x ( k ( d + 1) + i ) , x ( k ( d + 1) + i + 1) , x ( k ( d + 1) + i + 2) , ..., x ( k ( d + 1 ) + i + d ) we find some d -set, say ω ( k ), suc h that | e ∗ x ( ω ( k )) | ≥ C | x ( k ( d + 1) + i ) | . Hence, ∞ X k =1 ( e ∗ x ( ω ( k ))) 2 ≥ C ∞ X k =1 x 2 ( k ( d + 1) + i ) = ∞ and this is why k x k e = ∞ .  Note that Theorem 5 implies that for e 1 + e 2 + ... + e d 6 = 0 it s uffices to define J ( e ) as the set of all sequence s with a finite e -v aria tion. In this situation the conv er g ence of co ordinates to zero is a corollar y o f finitenes s of the e -v ariation. A U NIFIED CONSTRUCTION 5 3. Pro of of Theorem 6 As it w as mentioned ab ov e we fir s t explain the reaso n for the a ppear ance of the restriction e 1 + e 2 + ... + e d = 0. Lemma 11. The inclusion op er ator I d : J (1; − 1 ) → J ( e ) is wel l-define d and c ontinuous. Pro of. F or a rbitrary reals t 1 , t 2 , ..., t d we see that | e 1 t 1 + e 2 t 2 + ... + e d t d | = | e 1 ( t 1 − t 2 ) + ( e 1 + e 2 ) t 2 + ... + e d t d | = = | e 1 ( t 1 − t 2 ) + ( e 1 + e 2 )( t 2 − t 3 ) + ( e 1 + e 2 + e 3 ) t 3 + ... + e d t d | = = | d − 1 X i =1 ( e 1 + e 2 + ... + e i )( t i − t i +1 ) | ≤ C d − 1 X i =1 | t i − t i +1 | = and ( e 1 t 1 + e 2 t 2 + ... + e d t d ) 2 ≤ ( C d − 1 X i =1 | t i − t i +1 | ) 2 ≤ C 2 ( d − 1) d − 1 X i =1 ( t i − t i +1 ) 2 where C = max {| e 1 + e 2 + ... + e i | : 1 ≤ i ≤ d − 1 } . Now pic k a n y d -set ω = ω (1) ∪ ω (2) ∪ ... ∪ ω ( k ) with d -comp onents ω (1) , ω (2) , ..., ω ( k ). Making the estimates above we see that ( e ( x ; ω )) 2 = k X j =1 ( e 1 x ( n (( j − 1) d + 1)) + e 2 x ( n (( j − 1) d + 2 )) + ... + e d x ( n ( j d ))) 2 ≤ ≤ C 2 ( d − 1 ) k X j =1 d − 1 X j =1 ( x ( n (( j − 1) d + i )) − x ( n (( j − 1 ) d + i + 1))) 2 ≤ C 2 ( d − 1 ) k x k 2 J (1; − 1) according to the definition of one o f equiv alent norms in the James s pace J = J (1; − 1 ), see [1, 3]. Hence k x k J ( e ) ≤ C √ d − 1 k x k J (1; − 1) .  The following lemma g iv e s a c ha nce to pass from an ar bitr ary vector e = ( e 1 , e 2 , ..., e d ) ∈ R d to the spe c ial( d + 1)-v ector u d = (1 , − 1 , 0 , 0 , ... 0) ∈ R d +1 . Lemma 12. The inclusion op er ator I d : J ( e ) → J ( u d ) is wel l-define d and c ontin- uous. Pro of. Fix x ∈ J ( e ) a nd pick any ( d + 1)-set ω = ω (1) ∪ ω (2) ∪ ... ∪ ω ( k ) with ( d + 1)-comp onents ω (1) , ω (2) , ..., ω ( k ). F o r e a c h co mponent ω ( j ) = { n (( j − 1)( d + 1) + 1) < n (( j − 1( d + 1) + 2) < ... < n ( j ( d + 1 )) } let ω , ( j ) = ω ( j ) \ { n (( j − 1)( d + 1) + 1) } and ω , ( j ) = ω ( J ) \ { n (( j − 1)( d + 1) + 2) } . Then ω , = ω , (1) ∪ ω , (2) ∪ ... ∪ ω , ( k ) and ω ,, = ω ,, (1) ∪ ω ,, (2) ∪ ... ∪ ω ,, ( k ) are tw o d - sets with d -comp onents ω , (1) , ..., ω , ( k ) a nd with d -components ω ,, (1) , ω ,, (2) , ..., ω ,, ( k ). Consider for s implicity the case j = 1. Then e 1 ( x ( n (2))) − x ( n (1))) = ( e 1 x ( n (2))) + e 2 x ( n (3))) + ... + e d x ( d + 1)))) − 6 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEMENOV − ( e 1 x ( n (1))) + e 2 x ( n (3))) + ... + e d x ( d + 1)))) = e ∗ x ( ω , ; 1) − e ∗ x ( ω ,, ; 1) and ( x ( n (2))) − x ( n (1))) 2 ≤ 2 e 2 1 (( e ∗ x ( ω , ; 1)) 2 + ( e ∗ x ( ω ,, ; 1)) 2 ) . Having such an estimate for ea c h j = 2 , 3 , ..., k and summarizing all inequalities we see that ( u d ( x ; ω )) 2 ≤ 2 e 2 1 ( e ( x ; ω , )) 2 + ( e ( x ; ω ,, )) 2 ) ≤ 4 e 2 1 k x k 2 e = C 2 k x k 2 e . Passing to the supremum ov er all ( d + 1)- sets, we finally obtain k x k u d ≤ C k x k e .  So our fina l lemma shows that dep e ndence on d ∈ N can in fact b e eliminated and we can retur n to the o r iginal vector (1; − 1) = u 1 . T ogether with L e mmas 11 and 12 it co mpletes the pro of of the theorem. Lemma 1 3. The inclusion op er ator I d : J ( u d ) → J ( u 1 ) is wel l-define d and c on- tinuous. Pro of. Firs t, we need the following pur ely co m bina torial sublemma. W e will tem- po rarily sa y that a 2- set ∆ = { d (1) < d (2) < . . . < d (2 s − 1) < d (2 s ) } ⊂ N is d − dispe rsed if s = 1 , or if s > 1 and d (2 j + 1) ≥ d (2 j ) + d for all j = 1 , 2 , . . . , d − 1 .  Sublemma 1 4. Every 2-set ω = { n 1 < n 2 < ... < n 2 k − 1 < n 2 k } c an b e de c om- p ose d into a u nion of at most [0,5 d]+2 p airwise disjoi nt, d − disper sed 2 − subse ts . Pro of of sublemm a. Induction on k . The initial step k = 1 is trivial. So let ω , = ( n 1 ; n 2 ) ∪ ( n 3 ; n 4 ) ∪ ... ∪ ( n 2 k − 1 ; n 2 k ) ∪ ( n 2 k +1 ; n 2 k +2 ) = ω ∪ ( n 2 k +1 ; n 2 k +2 ) . By induction h yp othesis we have that ω = ∆ 1 ∪ ... ∪ ∆ m , m ≤ [0 , 5 d ] + 1 for so me [0 , 5 d ] + 2 pair wis e disjoint, d -disp ersed 2 -subsets ∆ 1 , ..., ∆ m . There a re exactly t wo p ossibilities: a) Inequa lit y max ∆ i ≤ n 2 k +1 − d − 1 holds for some 1 ≤ i ≤ m . Then one can sim- ply a dd the pair ( n 2 k +1 ; n 2 k +2 ) to ∆ i . Clearly the 2- sets ∆ i = ∆ i ∪ ( n 2 k +1 ; n 2 k +2 ) is also d -disper sed and ω , = ∆ 1 ∪ ... ∪ ∆ i − 1 ∪ ∆ i ∪ ∆ i +1 ∪ ... ∪ ∆ m , m ≤ [0 , 5 d ] + 1 . Hence, in this case the num b er of items in the decomp osition of ω , int o d -dis pers e d 2-subsets is the same a s for ω . b) Inequalities max ∆ i ≥ n 2 k +1 − d are true for all 1 ≤ i ≤ m . This means that in each 2-subset ∆ i its max imal pair int ersects with the seg ment [ n 2 k +1 − d ; n 2 k +1 − 1]. But all ∆ i consist of a pairwise dis jo in t, linearly or dered pairs. Therefor e on this A U NIFIED CONSTRUCTION 7 segment of the fixed length d can in general, b e placed either at mos t [0 , 5 d ] pair s , or at most [0 , 5( d − 1)] pairs and additionally one yet maximal element of so me ∆ i . Hence in this ca se m ≤ max { [0 , 5 d ] , 1 + [0 , 5( d − 1)] } ≤ [0 , 5 d ] + 1 . This implies that one ca n simply conside r the pair ( n 2 k +1 ; n 2 k +2 ) as an a dditional, separate item in the deco mpositio n of ω , int o union of d -disp ersed 2 -subsets.  Let us return to the pr o of o f Lemma 1 3. The main adv antage o f a d -disp ersed 2- set ∆ is that one can ”extend” it up to a ( d + 1)- set ∇ by adding the ( d − 1) na tural nu mbers which immediately follow d (2 j ) to each 2-co mponent { d (2 j − 1); d (2 j ) } of ∆. Namely , ∆(1) = { d (1); d (2) } ⇒ ∇ (1) = { d (1); d (2); d (2) + 1; d (2) + 2 ; ... ; d (2) + d − 1 } ∆(2) = { d (3); d (4) } ⇒ ∇ (2) = { d (3); d (4); d (4) + 1; d (4) + 2 ; ... ; d (4) + d − 1 } ∆( s ) = { d (2 s − 1); d (2 s ) } ⇒ ∇ ( s ) = { d (2 s − 1); d (2 s ); d (2 s ) + 1; ... ; d (2 s ) + d − 1 } . Clearly max ∇ (1) < min ∇ (2) < max ∇ (2) < min ∇ (3) < ...max ∇ ( s − 1) < min ∇ ( s ) and that is why the sets ∇ (1 ) , ∇ (2) , ..., ∇ ( s ) really are ( d + 1 ) − c omponents of their union ∇ . So for eac h x = ( x ( m )) m ∈ N ∈ R N we hav e ( u 1 ( x, ∇ )) 2 = s X j =1 ( x ( d (2 j )) − x ( d (2 j − 1))) 2 = = s X j =1 ( x ( d (2 j )) − x ( d (2 j − 1) + 0 · x ( d (2 j ) + 1 ) + ... + 0 · x ( d (2 j ) + d − 1)) 2 = = ( u d ( x, ∇ )) 2 and finally for an ar bitrary 2 -set ω = { n 1 < n 2 < ... < n 2 k − 1 < n 2 k } w e obtain ( u 1 ( x, ω )) 2 = k X i =1 ( x ( n (2 i )) − x ( n (2 i − 1))) 2 = m X j =1 ( X i ∈ ∆ j ( x ( n (2 i )) − x ( n (2 i − 1))) 2 ) = = m X j =1 ( u 1 ( x, ∆ j )) 2 = m X j =1 ( u d ( x, ∇ j )) 2 ≤ m k x k 2 J ( u d ) . Hence the inclusion op erator id : J ( u d ) → J ( u 1 ) is a well-defined mapping and its norm doe s not excee d the constant p [0 , 5 d ] + 2 .  Ac knowledgemen ts The fir st author was supp orted by the Slovenian Rese a rch Agency gr a n ts No. P1-02 92-010 1 -04 a nd Bl-R U/05-0 7/7. The seco nd a utho r was supp orted by the RFBR grant No. 05-0 1-0099 3. 8 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEMENOV References [1] R.C. 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Edelshtein, Homotopic al typ e of line ar gr oup of two classes of Banach sp ac es 4:3 (1970), F unct . Anal. Appl., 61–72. [9] P . V. Semeno v, James-Orlicz sp ac e , Russian Math. Surveys 34:4 (1979), 209–210 (in Russian). [10] P . V. Semenov, A c ounter e x ample t o a Ge o ghe gan-West pr oblem , Pro c. Amer. Math. Soc. 124 (1996), 939–943. In st it ut e of M a th em a t ic s, P h ys ics an d M e ch ani cs , an d U n iv e rs it y o f L ju bl jan a, P . O . B ox 2 96 4 , L jub lja na , Sl ove ni a 1 0 0 1 E-mail addr ess : dusan.re povs@gues t.arnes.si D epar t men t o f M a th em a t ic s, M o sc ow C it y P edag og ic al U niv er sit y, 2 -nd S el sk o - kh oz y as tv en ny i pr. 4 , M o sc ow , Ru ss ia 1 2 92 2 6 E-mail addr ess : pavels@o rc.ru