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By Bulgaria) International Workshop on Complex Structures, Vector Fields (6th 2002 Varna, Dimiev S., Sekigawa K.
It is a number of John von Neumann's papers and excerpts from his books which are so much attribute of his task. The booklet is prepared via the explicit topics - quantum mechanics, ergodic thought, operator algebra, hydrodynamics, economics, desktops, technology and society. The sections are brought by way of brief explanatory notes with an emphasis on fresh advancements in accordance with von Neumann's contributions. An total photograph is equipped via Ulam's 1958 memorial lecture. Facsimilae and translations of a few of his own letters and a newly accomplished bibliography according to von Neumann's personal cautious compilation are additional actual Analytic virtually advanced Manifolds (L. N. Apostolova); Involutive Distributions of Codimension One in Kahler Manifolds (G. Ganchev); 3 Theorems on Isotropic Immersion (S. Maeda); at the Meilikhson Theorem (M. S. Marinov); Curvature Tensors on nearly touch Manifolds with B-Metric (G. Nakova); advanced buildings and the Quark Confinement (I. B. Pestov); Curvature Operators within the Relativity (V. Videv, Y. Tsankov); On Integrability of virtually Quaternionic Manifolds (A. Yamada); and different papers
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Example text
We obtain the following basic classes of involutive distributions A on M D^o (V a »j)y = 0, 9 = 0, y), Tl — 1 4) Jv, 0 = 0, (V x ij)y = 0, 5) ^ 2 ® ^3 Tl/ 7) ^i ffi Ja 1. , 3 Involutive distributions of codimension one in Kahler manifolds Let (M, g, J, ry) be a 2n-dimensional Kahler manifold with unit 1 form rj If the distribution A of r\ is involutive then the covanant derivative Vrj satisfies the equalities (10) Let now ( V, 5, J, 17) be a Hermitian vector space with unit 1 form 77 The unit vector corresponding to r\ and the nullity space of r\ are denoted by £ and A respectively Further fj denotes the 1 -form corresponding to the vector J£ and AO denotes the vector space of vectors perpendicular to £ and J£ We denote by C, the linear space of all tensors L over V of type (0 2) satisfying the properties ( 10) o) = L(y0,x0), Let {e,} x = l 2(n - 1) be an orthonormal basis for the space A0 and go be the restriction of the scalar product onto A0 Then we consider the trace of the tensor L on AO Here is the inverse matrix of INVOLUTIVE DISTRIBUTIONS OF COD1MENSION ONE M Further we associate the following scalars and 1 forms with the tensor L 6(X] = L ( £ , X ) - p f j ( X ) , 8*(X) = It follows from (11) that 9(x0)=L(t,x0), 0( J£) = 0 ( 0 = 0 , The subgroup of the unitary group U(n) preserving the structure (g, J, 77) is the group U(n) x 7(2) where 7(2) denotes the unit matrix of order 2 Taking into account the representations X = x0 + fj(X)J£ + r,(X)t, Y = y0 + ij(Y)Jt we introduce the following tensors (projection operators) o) - Li(X,Y) = L2(X,Y) = L3(X,Y) = L(Jx0,Jy0) 2 tr0i _ L(x0,yo)+L(Jx0,Jyo) )= tr0L -p'fj(X)fi(Y), L1(X,Y)=Pn(X)fj(Y) These tensors determine the following subspaces of C £t = {Le£\L = Ll}, i = l, ,7 For any tensor L £ £ we have L = LI + + L7 We note that the tensors g(xo,yo) fj(X)rj(Y) and rj(X)fj(Y) are the invariant under the action of U(n - 1) x 1(2) tensors in the space £ The action of U(n 1) x 7(2) on the space C\ @ £2 0 £3 reduces to the action of U(n - 1) Taking into account that the decomposition C\ ® £3 ® £3 is irreducible under the action of 17 (n — 1) we obtain Proposition 3 1 (Decomposition into basic classes) £ = £i® ®£ 7 , where the factors £,(« = !
With L > n where C L = (C L ) 0 © (CL)! [15] Definition 1 ([13]) A function / Cz,m ™ -> CL is called a superholomorphic or superdifferentiable function if there exist /M € ~H(Cm, C) holomorphic functions such that where Mn - { (/ui, , pn) 1 < Mi < < Mn < « } [8] As it follows in [8] for each /x in ML and Keywords superholomorphic functions superforms supervector space complex superholomorphic supermamtolds Mathematics Subject Classification (1991) 58ASO 24 OKA S THEOREM 25 a typical element b of CL may be expressed as 6= where the coefficients V are complex numbers With the norm on CL defined by ii&ii = E 1^ CL is a Banach algebra [12] Let M be a Hausdorff topological space [13] (a) An (m, n) chart on M over CL is a pair (U, ip) with [/ an open set of M and i/j a homeomorphism of U onto an open subset of CL™ n (b) An (m,n) superholomorphic structure on M over CL is a collection {({/«, Va) | a € A} of (m, n) charts on M such that (/) M = U a gAf^a and (//) for each pair a fl in A the mapping ip/j o ^a"1 is a superholomorphic function of ^a(Uar\U/3) ontoipp(Uaf~lUp) (III) the collection {(f/ Q , i/}a) | a e A} is a maximal collection of open charts for which (/) and (//) hold Definition 2 ([13]) An (m, n) dimensional complex superholomorphic supermanifold over CL is a Hausdorff topological space M with an (m,n) holomor phic structure over CL Example 1 CL™ ™ is an (m, n) dimensional complex superholomorphic super manifold For a given (m, n)-dimensional supermamfold M, there is a natural projection onto an underlying m-dimensional conventional manifold M0 A supermamfold is said to be simply connected if that is the case for its underlying manifold Definition 3 ([4]) A subset M' of a complex superholomorphic supermamfold M of dimension (m, n) is called a complex superholomorphic sub-supermamfold of dimension (m',n') m > m' n > n' if M' is contained in the union of a set {(U, VO} of charts each of which has the property that for all (z, C) € U n M' V(*,0 = O*1.
3 Involutive distributions of codimension one in Kahler manifolds Let (M, g, J, ry) be a 2n-dimensional Kahler manifold with unit 1 form rj If the distribution A of r\ is involutive then the covanant derivative Vrj satisfies the equalities (10) Let now ( V, 5, J, 17) be a Hermitian vector space with unit 1 form 77 The unit vector corresponding to r\ and the nullity space of r\ are denoted by £ and A respectively Further fj denotes the 1 -form corresponding to the vector J£ and AO denotes the vector space of vectors perpendicular to £ and J£ We denote by C, the linear space of all tensors L over V of type (0 2) satisfying the properties ( 10) o) = L(y0,x0), Let {e,} x = l 2(n - 1) be an orthonormal basis for the space A0 and go be the restriction of the scalar product onto A0 Then we consider the trace of the tensor L on AO Here is the inverse matrix of INVOLUTIVE DISTRIBUTIONS OF COD1MENSION ONE M Further we associate the following scalars and 1 forms with the tensor L 6(X] = L ( £ , X ) - p f j ( X ) , 8*(X) = It follows from (11) that 9(x0)=L(t,x0), 0( J£) = 0 ( 0 = 0 , The subgroup of the unitary group U(n) preserving the structure (g, J, 77) is the group U(n) x 7(2) where 7(2) denotes the unit matrix of order 2 Taking into account the representations X = x0 + fj(X)J£ + r,(X)t, Y = y0 + ij(Y)Jt we introduce the following tensors (projection operators) o) - Li(X,Y) = L2(X,Y) = L3(X,Y) = L(Jx0,Jy0) 2 tr0i _ L(x0,yo)+L(Jx0,Jyo) )= tr0L -p'fj(X)fi(Y), L1(X,Y)=Pn(X)fj(Y) These tensors determine the following subspaces of C £t = {Le£\L = Ll}, i = l, ,7 For any tensor L £ £ we have L = LI + + L7 We note that the tensors g(xo,yo) fj(X)rj(Y) and rj(X)fj(Y) are the invariant under the action of U(n - 1) x 1(2) tensors in the space £ The action of U(n 1) x 7(2) on the space C\ @ £2 0 £3 reduces to the action of U(n - 1) Taking into account that the decomposition C\ ® £3 ® £3 is irreducible under the action of 17 (n — 1) we obtain Proposition 3 1 (Decomposition into basic classes) £ = £i® ®£ 7 , where the factors £,(« = !