By Bulgaria) International Workshop on Complex Structures, Vector Fields (6th 2002 Varna, Dimiev S., Sekigawa K.

ISBN-10: 9812384529

ISBN-13: 9789812384522

This can be a number of John von Neumann's papers and excerpts from his books which are such a lot attribute of his job. The booklet is geared up by means of the explicit topics - quantum mechanics, ergodic idea, operator algebra, hydrodynamics, economics, pcs, technological know-how and society. The sections are brought by means of brief explanatory notes with an emphasis on contemporary advancements in line with von Neumann's contributions. An total photograph is equipped by means of Ulam's 1958 memorial lecture. Facsimilae and translations of a few of his own letters and a newly accomplished bibliography in response to von Neumann's personal cautious compilation are additional genuine Analytic virtually complicated 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 virtually touch Manifolds with B-Metric (G. Nakova); complicated 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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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.

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.

J£ = 0*(Jx0), x0,y0 e A0 Thus any mvolutive distribution A generates 1 ) a symmetric tensor field of type (0 2) on A0 (5) (V zo 17)3/0 = (Vyo^zo, x0,y0eAo, 2) two 1 forms on AO (6) 9(x0) = g(V^,x0), 0*(x0)=g(VjtJ£,x0), x0 e A 0 , 3) two functions on M (7) P= ff(V^,JO, P*=ff(Vj4J£,0 Then the condition characterising an mvolutive distribution A becomes dr) = rjf\d + pr}Af) In the next section we give the background of our considerations 2 Involutive distributions in a Riemanman manifold Let (M, g, rf) be a Riemanman manifold with unit 1 form rj If the distribution A is mvolutive then the covanant derivative Vr/ has the properties (8) (Vjc»j)£ = 0, X Let now ( V, 3, r/) be an n-dimensional Euclidean space with inner product g and unit 1 form 77 The unit vector field corresponding to 77 and the nullity space of 77 are denoted by £ and A respectively We consider the linear space C of all tensors L of type (0 2) over V which have the properties implied by (8) (9) L(X,£) = 0, X&V, L(x,y) = L(y,x), We associate the following 1 form with the tensor L x,y e A INVOLUTIVE DISTRIBUTIONS OF COD1MENSION ONE 33 It follows from (9) that 9 is a 1-form on A 6(x)=L(t,x), a r e A, 0(0 = 0 The subgroup of O(n) preserving the structure (g, 77) is the group O(n- 1) x 1 This group acts on the linear space £ in the standard way [4 6] a(L)(X,Y) = L(a-1X,a~1Y), L e £, a e O(n - 1) x 1 The scalar product g in V induces a scalar product <, > in £ in the usual way where {ei, , e n } is an orthonormal basis of V Taking into account that we consider the following tensors L3(X,Y)=n(X)0(Y) These tensors determine the following subspaces of £ £1 { L e £ L = Li}, £2 { L e £ | L £3 {L e £ I L = L3} For any L € £ we have L = LI + L2 + £3 It is easy to check that the subspaces £1 £2 £3 are mutually orthogonal and invariant under the action of the group O(n - 1) x 1 The action of O(n - 1) x 1 on the space C\ 9 £2 comsides with the action of O(n - 1) Taking into account that the decomposition £1 9 £2 is irreducible under the action of 0(n — 1) we obtain Proposition 2 1 (Decomposition into basic classes) £ = £1 9 £2 9 £3 where C,\ , £2 and £3 are mutually orthogonal invariant under the action of 0(n - 1) x 1 and irreducible factors 34 G GANCHEV The basic subspaces L\ £2 and £3 generate further 23 = 8 invariant sub spaces of £ £0, £1, £ 2 , £3, £2® £3, £i®£ 3 , £ i ® £ 2 , £, the space £Q being the zero subspace of £ i e L — 0 Now let (M,g,rf) be a Riemannian manifold with unit 1 form 77 and set V(p) = TPM p e M and consider the corresponding linear space £(p) over V(p) p € M According to (8) and (9) we can set L(p) — Vrj Applying the above considerations to the tensor L(p) = VT?

### Trends in complex analysis, differential geometry, and mathematical physics : proceedings of the 6th International Workshop on Complex Structures and Vector Fields : St. Konstantin, Bulgaria, 3-6 September 2002 by Bulgaria) International Workshop on Complex Structures, Vector Fields (6th 2002 Varna, Dimiev S., Sekigawa K.

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