8-th International Workshop on Complex structures and vector fields

Sofia, 21-26 August 2006

Abstracts

August 18, 2006

AUGUST 21 Differential Geometry

8:00 – 9:15  REGISTRATION
9:15 – 9:45  OPENING

(Dodunekov, J. Stamenov, S. Dimiev, V. Gerdjikov)

9:45 – 10:30 Yasuo Matsushita Almost Kähler-Einstein structures on 8dimensional Walker manifolds

10:30 – 10:50 Coffee break

10:50 - 11:35 Peter Popivanov On the hypoellipticity of complex valued vector

fields

11:35 - 12:15 Toshiaki Adachi Killing helices on a complex projective space and canonical magnetic fields on geodesic spheres

Lunch time

15:30 - 16:10 Vestislav Apostolov Generalized Kähler manifolds, commuting

complex structures and split tangent bundles

16:10 - 16:50 Rossen Dandoloff Anholonomy of a moving space curve and the Schrdinger equation

16:50 - 17:10 Coffee break

17:10 - 17:50 Hideya Hashimoto On almost Hermitian structures of 6-

dimensional submanifolds in the octonions

17:50 - 18:30 Norio Ejiri Complex submanifolds and Lagrangian submanifolds associated with minimal surfaces in tori

AUGUST 22 Mathematical Physics and PDE

9:00 - 9:45 Julian Lawrinowicz Clusters and fractals in mathematical

physics

9:45 - 10:30 Vladimir Georgiev Nonexistence of nonzero resonances for Schrdinger operators with singular perturbation

10:30 10:50 Coffee break

10:50 - 11:30 Vladimir Gerdjikov Hamiltonian aspects of soliton equations

with deep reductions

11:30 - 12:00 Tihomir Valchev Bäcklund transformations and Riemann-Hilbert problem for N-wave equations with additional symmetries

Lunch time 15:30 - 16:10 Georgi Grahovski The Caudrey-Beals-Coifman systems and the

gauge group action

16:10 - 16:50 Nikolay Kostov The Manakov model as two moving interacting curves

16:50 - 17:10 Coffee break

17.10 - 17.50 Stoil Donev Integrability, curvature and description of photon-like

objects

17:50 - 18:30 Tadashi Sugiyama Characterization of totally umgilic immersions by curves of order 2

AUGUST 23 Differential Geometry

9:00 - 9:45  Kouei Sekigawa Some critical almost Kähler structures

9:45 - 10:25 Sadahiro Maeda Characterization of parallel isometric immersions of space forms into space forms in the class of isotropic immersions

10:25 - 10:45 Coffee break

10:45 - 11:15 Milen Hristov On the real hypersurfaces of Hermitian manifolds

11:15 - 11:45 Galya Nakova On some non-integrable almost contact manifolds

with Norden metric of dimension 5

11:45 - 12:25 Bozhidar Iliev Normal frames and linear transports along paths in line bundles. Applications to classical electrodynamics

Lunch time

15:30 - 16:00 Velichka Milousheva On the geometric structure of hypersur-

faces of conullity two in Euclidean space

16:00 - 16:40 Mancho Manev On quasi-Kähler manifolds with Norden metric

16:40 - 17:00 Coffee break

17:00 - 17:30 Dobrinka Gribacheva Lie groups as four-dimensional Rieman-

nian product manifolds

17:30 - 18:00 Marta Teofilova Lie groups as several complex manifolds with Norden metric

Conference dinner

AUGUST 24 EXCURSION

AUGUST 25 Complex Analysis

9:00 - 9:40 Vladimir Balan KCC and linear stability for the Parkinson

tremor model

9:40 - 10:20 Stancho Dimiev Bi-complex analytic pseudo-Euclidean geometry and applications

10:20 - 10:40 Coffee break

10:40 - 11:10 Nikolai Nikolov On the definition of Kobayashi-Buseman pseu-

dometric

11:10 - 11:40 Petar Stoev Differential forms of bi-complex variables
11:40 - 12:10 Petya Furlinska Polynomials of bi-complex variables

Lunch time

15:30 - 16:00 Branimir Kiradjiev Quadratic forms on the bi-complex algebras

16:00 - 16:30 Lilya Apostolova Real analyticity of the hyper-Kähler almost Kähler manifolds

16:30 - 16:50 Coffee break

16:50 - 17:20 Assen Kyuldjiev Symmetries of the Manev problem and its real

Hamiltonian form

17:20 - 18:00 Rossen Ivanov Camassa-Holm equation as a geodesic flow of the right invariant metric

AUGUST 26 Complex Analysis and Applied Mathematics

9:00 - 9:40 Hiroshi Matsuzoe Information geometry and affine differential

geometry

9:40 - 10:20 Vesna Manova Distributional boundary values of the functions in the Djarbjashian classes

10:20 - 10:40 Coffee break

10:40 - 11:20 Mihail Konstantinov Effects of finite arithmetic – myths and

realities

11:20 - 11:50 Angel Ivanov, George Venkov Global existence of the solution

to the Hartree equation

11:50 - 12:20 Ilona Zasada Phase transitions in binary alloy thin films

12:20 - 12:50 Marin Marinov Homogenization Of Nonlinear Parabolic Operators Of High Order

Lunch time 15:30 - 16:00 Vanya Markova Application of Unscented and Extended Kalman

Filtering for Estimating Quaternion Motion

16:00 - 16:30 Peace Nwaobilor Non-oscillating trajectories of vector fields and

Zariskis local uniformization

16:30 - 17:00 Clement Ijeh Centroidal Voronoi tessellation based algorithms for vector fields visualization and segmentation

17:00 - 17:20 Coffee break

17:20 - 17:50 Oladipupo Ipadeola Anisotropic depth migration on complex-

structure land data

17:50 - 18:20 Isaak Ibenme Dislocations in complex materials
18:20 - 18:50 CLOSING

Killing helices on a complex projective space and canonical magnetic fields on geodesic spheres

Toshiaki ADACHI Department of Mathematics, Nagoya Institute of Technology Nagoya 466-8555, Japan e-mail: adachi@nitech.ac.jp

We study the moduli space of helices of low orders which are generated by some Killing vector fields on a complex projective space. We call such helices Killing. For a real space form the moduli space, the set of congruence classes, of helices has a “building structure”, but for a complex projective space the moduli space of all Killing helices has a quite different structure.

In my talk, we first point out that the moduli space of circles has a lamination structure. In view of this structure we find that the moduli space of circles is a disjoint union of the moduli space of trajectories for Kähler magnetic fields and the moduli space of other circles. In order to clarify the structure of the moduli space of helices on a complex projective space, we study in the second stage special helices of proper order 4, and construct an embedding of the moduli space of circles other than trajectories for Kähler magnetic fields into the moduli space of helices of proper order 4. At the last stage, we point out some Killing helices of proper order 4 on a complex projective space are extrinsic shapes of trajectories for canonical magnetic fields on its geodesic spheres.

Generalized Kähler manifolds, commuting complex structures, and split tangent bundles.

\underline{Vestislav Apostolov}^{1}, M. Gualtieri^{2}

1UQAM - Université du Québec Montréal, Case postale 8888, succursale Centre-ville Montréal (Québec), H3C 3P8 Canada 2Department of Mathematics, Massachusetts Institute of Technology 77 Massachusetts avenue, Cambridge, MA 02139-4307, USA e-mails: apostolo@math.uqam.ca, mgualt@math.mit.edu

I will discuss generalized Kähler manifolds for which the induced complex structures commute, and then relate the existence problem of such manifolds to a conjecture by Beauville concerning Kähler manifolds with splittable holomorphic tangent bundle.

Real analyticity of the hyperkähler almost Kähler manifolds

Lilia N. Apostolova Institute of mathematics and informatics, acad. G. Bontchev str., 1113 Sofia, Bulgaria e-mail: liliana@math.bas.bg

Using the techniques in the paper by Kutzschebauch and Loose [1] it is proved that a smooth manifold equipped with two symplectic-homotopic symplectic structures ω_{0} and ω_{1} admits a real analytic structure such that ω_{0} and ω_{1} are real analytic ones. Then the result is used to prove that each hyperkähler almost Kähler manifolds with respect to two fundamental forms obtain a real analytic almost Kähler structure C^{∞} equivalent to the given hyperkähler almost Kähler structure.

References

[1] F. Kutzschebauch, F. Loose. Real analytic structures on a symplectic manifold. PAMS, (10) 128, 3009-3016 (2000).

KCC and linear stability for the Parkinson tremor model

Vladimir Balan, University Politehnica of Bucharest Faculty of Applied Sciences, Department of Mathematics I, RO-060042, Bucharest, Romania. e-mail: vbalan@mathem.pub.ro

The paper investigates the KCC (structural) stability of the brain-stimulated Parkinson tremor dynamical system. After a brief presentation of the known linear stability results of the model, are determined and studied the KCC-invariants. It is emphasized that the spectral properties of the second KCC-invariant provide practical information for the behavior of the investigated model parameters.

The Finslerian spray and the local nonlinear connection of the second-order differential system associated to the basic SODE are evidentiated, and integrability issues are discussed.

Selected References

References

[1] P.L. Antonelli, Equivalence problem for systems of second order ordinary differential equations, Encyclopedia of Math., Kluwer Acad. Publ, 2000.

[2] P.L. Antonelli, R. Ingarden, M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Acad. Publ, FTPH vol. 58, 1993.

[3] P.L.Antonelli, S. Rutz, V.S. Sabau, A transient-state analysis of Tyson’s model for the cell division cycle by means of KCC-theory, Open System & Information Dynamics, 9 (2002), 3, 223-238.

[4] V. Balan and I.R. Nicola, Linear and structural stability of a cell division process model, IJMMS, USA, 2005, 1-15.

[5] I. R. Nicola and V. Balan, Jacobi stability for dynamical systems with applications to biology, Proc. of the 3-rd International Colloquium ”Mathematics in Engineering and Numerical Physics” (MENP-3), 7-9 October 2004, Bucharest, Romania, BSG Proceedings 12, Geometry Balkan Press, Bucharest, 195-201.

Anholonomy of a moving space curve and the Schrödinger equation as a moving space curve

Rossen Dandoloff

Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, F-95302 Cergy-Pontoise Cedex, France e-mail: rossen.dandoloff@u-cergy.fr

The time evolution of a space curve is associated with a geometric phase. This phase arises because of the path dependence of the rotation of the natural Frenet-Serret triad as one moves along the curve. The corresponding non rotational triad undergoes Fermi-Walker parallel transport. The geometric phase, or the Fermi-Walker phase is associated with a second geometric phase, the so called ”incompatibility” phase whose presence is linked to the Schrodinger equation. We present a general condition for the evolution of a space curve in terms of geometric phases.

Double-complex analytic pseudo-Euclidean geometry and applications

\underline{Stancho Dimiev}^{1}, Slavka Slavova^{2}, Claude Surry^{3}

1Institute of mathematics and informatics, acad. G. Bontchev str., 1113 Sofia, Bulgaria 2Dobrich Pedagogical College block 12 Dobrotica, 9300 Dobrich, Bulgaria 3Font Romeu, France e-mails: sdimiev@math.bas.bg, slavova@dobrich.net,

Double-complex analyticity can be developed on double-complex algebras, which are complex 2-dimensional vector spaces endowed with multiplication law. The first example of a double-complex algebra (non-commutative) with a rich set of analytic functions was developed by Fueter (1935), for regular quaternion functions f(q) ∈ ℍ, q ∈ ℍ, q = t + xi + yj + zk, t, x, y, z ∈ ℝ, via the Cauchy-Riemann approach. If we write q = v + jw, j^{2} = −1, v, w ∈ ℂ (v = t + ix, w = y − iz), and f(q) = g(v, w) + jh(v, w), the so called Cauchy-Riemann-Fueter equations hold, namely

\frac{∂g}{∂v̄} = \frac{∂h}{∂w̄} , \frac{∂g}{∂w} = −\frac{∂h}{∂v},

according to Fueter’s notations .

Since there is no other double-complex division algebras than ℍ, we search non - division double-complex algerbras with a rich set of holomorphic functions. Here we stress on the non-division commutative algebra C(1, j) of double-complex numbers α = z +jw, j^{2} = i, z, w, i ∈ ℂ. It is to remark that C(1, j) is a complexification of the real algebra of even anti-cycles numbers R(1, j, j^{2}, j^{3}), j^{4} = −1.

According to the first author a double-complex function f(α) = f_{0}(z, w) + jf_{1}(z, w) is holomorphic iff it satisfy the following Cauchy-Riemann type equations

\frac{∂f_{0}}{∂z} = \frac{∂f_{1}}{∂w} , \frac{∂f_{0}}{∂w} = j \frac{∂f_{1}}{∂z} ,

f_{0} called the even part of f and f_{1} the odd one. The functions f_{0} and f_{1} are holomorphic functions of two complex variables. The set of zero divisors z + jw in C(1, j) is an analytic set in ℂ × ℂ satisfying the equation z^{2} − iw^{2} = 0. Some interconnection between holomorphic double-complex functions and holomorphic functions of two complex variables is demonstrated, developed by the second author, P. Stoev and B. Kiradjiev.

A geometric point of view can be developed introducing a pseudo-scalar product on C(1, j), namely ⟨z + jw, u + jv⟩ := zu − iwv, and ⟨z + jw, z + jw⟩ = z^{2} − iw^{2}. Then the isotropic cone of C(1, j) is defined by just the same equation as for the set of zerodivisors. In real coordinates (z = x + iy, w = ξ + iη ) the isotropic cone coincides with the intersection the following two real 3-dimensional surfaces in ℝ^{4}:

x^{2} − y^{2} + 2ξη = 0, ξ^{2} − η^{2} + 2xy = 0.

Introducing appropriates real parameters s and t, one proves that the mentioned isotropic cone is represented by a family of couples of central curves of hyperbolic type parametrized by the points (s, t) such that st = −1. The obtained geometry can be considered as an analogous of pseudoeclidean geometry on ℂ defined by the pseudo-scalar product ⟨x + iy, ξ + iη⟩ := xξ − yη, and ⟨x + iy, x + iy⟩ = x^{2} − y^{2}. In our case we take the natural neutral metric defined by |α|^{2} = |z|^{2} − |w|^{2}, α = z + iw, and we consider the inverse image of the so called ”absolute plane” O|z||w|.

The third author will present applications in mechanics.

Integrability, curvature and description of photon-like objects

\underline{Stoil Donev}, Maria Tashkova Institute for Nuclear Research and Nuclear Energy, 72 blvd.Tzarigradsko chaussee, 1784 Sofia, Bulgaria e-mails: sdonev@inrne.bas.bg, mtashkova@inrne.bas.bg

This paper aims to present and realize a general idea for description of finite physical objects of photon-like nature making use of the mathematical concepts and structures connected with the Frobenius integrability/nonintegrability. We consider some distribution Δ_{o} of vector fields on a manifold, then separate some integrable subdistribution Δ ⊂ Δ_{o}, representing the integrity of the object considered. The curvatures of all nonintegrable subdistributions of Δ are interpreted as generators of processes of internal energy-momentum exchange, i.e. of the internal dynamics of the object. The curvatures of distributions including vector fields from Δ_{o} and Δ are interpreted as generators of interaction of the physical object with the outside world. A basic example of photon-like objects is considered in detail.

Complex submanifolds and Lagrangian submanifolds associated with minimal surfaces in tori

Norio Ejiri Department of Mathematics, Meijo University, Tempaku, Nagoya, 468-8502 Japan e-mail: ejiri@ccmfs.meijo-u.ac.jp

We consider some energy function withe parameter on the Siegel upper half spce H_{γ} and the restriction to complex manifolods in H_{γ}. We prove that the catastrophe set of the function on complex submanifold are analytic set by using another complex structure different from the standard complex structure of the product of complex spaces and parameter spaces. Moreover the irreducible components containg a nondegenerate critical point give Lagrangian cones in complex Euclidean space (the cotangent bundle of parameter spaces). As example, we have analytic sets associated with minimal surfaces in flat tori and the irreducible component containg minimal surfaces with only Killing Jacobi fields gives a Lagrangian cone.

Polynomials of double-complex variables

Petia Furlinska Plovdiv, Bulgaria e-mail: petiaf@abv.bg

The structure of zero-sets of the polynomials of double-complex variables is studied in some particular cases.

Nonexistence of nonzero resonances for Schrödinger operators with singular perturbation

Vladimir Georgiev Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy e-mail: georgiev@dm.unipi.it

The existence of non - trivial solutions to the equation

−Δu + ia_{j}(x)\frac{∂u}{∂x_{j}} + b(x)u = λu, x ∈ R^{n}, λ > 0,  (1)

is a typical obstacle to establish dispersive properties of the time evolution group associated with the perturbed Laplace operator

−Δ + ia_{j}(x)\frac{∂}{∂x_{j}} + b(x).  (2)

The main result of the work establishes that there are no resonances embedded in the positive real line for the family of operators (2) under suitable short - range assumptions at infinity and possible local singularities of the coefficients a_{j}(x) and b(x).

Hamiltonian aspects of soliton equations with deep reductions

Vladimir Gerdjikov Institute for Nuclear Research and Nuclear Energy 72 Tzarigradsko chaussee, 1784 Sofia, Bulgaria e-mail: gerjikov@inrne.bas.bg

By soliton equations we mean nonlinear evolution equations (NLEE) in two-dimensional space-time allowing Lax represenation [L(λ), M(λ)] = 0. More specifically we consdier the NLEE related to the Lax operators of the form:

L(λ)ψ(x, t, λ) ≡ i\frac{dψ}{dx} + (q(x, t) − λJ) ψ(x, t, λ) = 0,

where q(x, t) = [J, Q(x, t)]. Here Q(x, t) is a generic element of the simple Lie algebra 𝔤 vanishing fast enough for x → ±∞ and J is a constant element of the Cartan subalgebra 𝔥 ⊂ 𝔤.

The soliton equations integrable by applying the inverse scattering method (ISM) [1] to L(λ) possess a hierarchy of Hamiltonian structures, i.e. for each such equation one can construct a sequence of Hamiltonians H^{(k)} and symplectic forms Ω^{(k)}, k = 1, 2, …, such that each pair (H^{(k)}, Ω^{(k)}) generates the same NLEE for all k. The phase space M_{J} ≡ {q(x, t)} can be viewed as the co-adjoint orbit of a certain Kac-Moody algebra passing through λJ.

Important classes of NLEE with physical applications are the N-wave equations [1] and the multicomponent nonlinear Schrödinger equations (MNLS) [2]. They are naturally related to the homogeneous and symmetric spaces respectively. We study the reductions of these NLEE following [3]. The reductions allow us to impose on q(x, t) a number of algebraic constraints compatible with the dynamics. Thus we are able to derive new soliton equations with fewer coefficient functions.

The effects of the reductions on the phase space M_{J} and on the hierarchies of the Hamiltonian structures are outlined [4]. In doing this we are using two important tools: a) the equivalence of the isnverse scattering problem for L(λ) to a Riemann-Hilbert problem and b) the generalized Fourier transforms [5] that allow us to linearize the NLEE.

We conslude with the class of soliton equations containing the affine Toda field theory models [3, 6]. They are related to L(λ) with ℤ_{h} reduction where h is the Coxeter number of 𝔤. We derive for them the so-called symplectic basis which allows us to obtain explicitly their action-angle variables [7]. Finally we briefly comment on the limiting case h → ∞.

References

[1] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. I. Pitaevskii. Theory of solitons: the inverse scattering method. (Plenum, N.Y.: Consultants Bureau, 1984).

[2] S. V. Manakov. J. Exp. Theor. Phys. 65 505-516 (1973); A.P. Fordy, P.P. Kulish. Comm. Math. Phys. 89, no.4, 423–443 (1983).

[3] A. V. Mikhailov. Physica D, 3D, n. 1/2, 73–117, 1981.

[4] V. S. Gerdjikov, G. G. Grahovski, R. I. Ivanov and N. A. Kostov. Inverse Problems 17, 999-1015 (2001); V. S. Gerdjikov, G. G. Grahovski, N. A. Kostov. J. Phys. A: Math & Gen. 34, 9425-9461 (2001).

[5] V. S. Gerdjikov. Inverse Problems 2, n. 1, 51–74, (1986); V. S. Gerdjikov. Theor. Math. Phys. 92, 374–386 (1992).

[6] A. V. Mikhailov, M. A. Olshanetzky, A. M. Perelomov. Comm. Math. Phys. 79, 473–490, 1981; D. Olive, N. Turok. Nucl. Phys. B B265 [FS15] 469–484 (1986).

[7] V. S. Gerdjikov. Z_{N}–reductions and new integrable versions of derivative nonlinear Schrödinger equations. In Nonlinear evolution equations: integrability and spectral methods, Ed. A. P. Fordy, A. Degasperis, M. Lakshmanan, Manchester University Press, (1981), p. 367–379.

The Caudrey-Beals-Coifman Systems and the Gauge Group Action

Georgi Grahovski Institute for Nuclear Research and Nuclear Energy 72 Tzarigradsko chaussee , 1784 Sofia, Bulgaria and Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, 2 avenue A. Chauvin, F-95302 Cergy-Pontoise Cedex, France e-mail: grah@inrne.bas.bg

The fundamental properties of the generalized Zakharov–Shabat systems with complexvalued Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) [1, 2] and their gauge equivalent are considered. This class contains such physically important equations like the N-wave equations and their gauge equivalent [3], the complex Toda chain, etc.

In the present talk the following topics will be discussed: the properties of their fundamental analytical solutions (FAS) associated to the corresponding Lax operator; the minimal set of scattering data [4]; the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures.

References

[1] R. Beals, R. Coifman, Commun. Pure and Appl. Math. 37 (1984), 39. R. Beals, R. R. Coifman. Inverse Problems 3 (1987), 577. P. J. Caudrey. Physica D 6 (1982), 51.

[2] V S. Gerdjikov, A B. Yanovski, J. Math. Phys. 35 (1994), 3687.

[3] V. S. Gerdjikov, G. G. Grahovski, N. A. Kostov, Eur. Phys. J. B 29 (2002), 243.

[4] G. G. Grahovski, In: ”Geomerty, Integrability and Quantization III”, Eds: I. Mladenov and G. Naber, Coral Press, Sofia (2002), p. 262.

Lie groups as four-dimensional Riemannian product manifolds.

Dobrinka K. Gribacheva - Shtarbeva Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv 4003, Bulgaria e-mail: dobrinka@pu.acad.bg

The basic class of the integrable Riemmanian almost product manifolds with Norden metric is considered. Three examples of 4-dimensional Riemmanian product manifolds are constructed by means of Lie groups and Lie algebras. The form of the curvature tensor for each of the examples is obtained.

On almost Hermitian structures of 6-dimensional submanifolds in the octonions

Hideya Hashimoto Meijo University, Tempaku, Nagoya, Japan e-mail: hhashi@ccmfs.meijo-u.ac.jp

On the real hypersurfaces of Hermitian manifolds

Milen Hristov University of Veliko Tarnovo, 2 T. Tarnovski str., 5000 Veliko Tarnovo, Bulgaria e-mail: milenjh@yahoo.com

There are sixteen classes of real hypersurfaces of Kähler manifold, generated by the four basic classes W_{1}, W_{2}, W_{4} and W_{6} in the classification scheme for almost contact metric manifolds [1]. In [2] these classes are described in terms of the hypersurface second fundamental form. In the partial case when the ambient manifold is complex space form the identities for the Riemannian and for the Hermitian-like curvatures are introduced [3, 5]. The paper deals with the classification problem for the real hypersurfaces of Hermitian manifolds by using the ideas and results in [4, 6].

References

[1] V. Alexiev, G. Ganchev. On the classification of almost contact metric manifolds, Math. and Educ. in Math., Proc. of the XV Spring Conf. of UBM. Sunny Beach, (1986) 155-161.

[2] G. Ganchev, M. Hristov, Real hypersurfaces of a Kähler manifold (the sixteen classes), Perspectives of Complex Analysis, Differential Geometry and Mathematical Physics, Proc. 5th Int. Workshop on Complex Structures and Vector Fields, Varna, 2000, Eds S. Dimiev and K. Sekigawa, World Scientific, Singapore, (2001) 147-158.

[3] G. Ganchev, M. Hristov, Riemannian curvatures of the four basic classes of real hypersurfaces of a complex space form, Third Int. Conf. on Geometry, Integrability and Quantization, June 14-23, 2001, Varna, Bulgaria, Ivailo M. Mladenov and Gregory L. Naber, Editors, Coral Press, Sofia 2001, pp238-248.

[4] G. Ganchev, V. Mihova, Kähler manifolds of quasi-constant holomorphic sectional curvatures, arXiv:math.DG/0505671 v1 31 May 2005.

[5] Hristov, Milen J. Hermitian-like curvatures of the four basic classes of real hypersurfaces of a complex space form. Trends in complex analysis, differential geometry and mathematical physics (St. Konstantin, 2003), 63–75, World Sci. Publishing, River Edge, NJ, 2003.

[6] L.Ornea, L.Vanheche, Harmonicity and minimality of vector fields and distributions on locally conformally Kähler and hyperkähler manifolds, Bull.Belg.Math.Soc. Simon Stevin 12, no.4(2005), 543 – 555.

Dislocations in Complex Materials

Isaac Iroegbu Ibenme Cairo wireless computer engineerng institute, Cairo, Egipt e-mail: prof ut jacob@earthling.net

Deformation of metals and alloys by dislocations gliding between well-separated slip planes is a well-understood process, but most crystal structures do not possess such simple geometric arrangements. Examples are the Laves phases, the most common class of intermetallic compounds and exist with ordered cubic, hexagonal, and rhombohedral structures. These compounds are usually brittle at low temperatures, and transformation from one structure to another is slow. On the basis of geometric and energetic considerations, a dislocation-based mechanism consisting of two shears in different directions on adjacent atomic planes has been used to explain both deformation and phase transformations in this class of materials. We report direct observations made by Z-contrast atomic resolution microscopy of stacking faults and dislocation cores in the Laves phase Cr2Hf. These results show that this complex dislocation scheme does indeed operate in this material. Knowledge gained of the dislocation core structure will enable improved understanding of deformation mechanisms and phase transformation kinetics in this and other complex structures.

Centroidal Voronoi Tessellation Based Algorithms for Vector Fields Visualization and Segmentation

Ikechukwu Clement Ijeh Cairo wireless computer engineering institute, Cairo, Egipt e-mail: prof ut jacob@earthling.net

A new method for the simplification and the visualization of vector fields is presented based on the notion of Centroidal Voronoi tessellations (CVT’s). A CVT is a special Voronoi tessellation for which the generators of the Voronoi regions in the tessellation are also the centers of mass (or means) with respect to a prescribed density. A distance function in both the spatial and vector spaces is introduced to measure the similarity of the spatially distributed vector fields. Based on such a distance, vector fields are naturally clustered and their simplified representations are obtained. Our method combines simple geometric intuitions with the rigorously established optimality properties of the CVTs. It is simple to describe, easy to understand and implement. Numerical examples are also provided to illustrate the effectiveness and competitiveness of the CVT-based vector simplification and visualization methodology.

Normal frames and linear transports along paths in line bundles. Applications to classical electrodynamics

Bozhidar Z. Iliev Institute for Nuclear Research and Nuclear Energy 72 blvd.Tzarigradsko chaussee, 1784 Sofia, Bulgaria e-mail: bozho@inrne.bas.bg

The definitions and some basic properties of the linear transports along paths in vector bundles and the normal frames for them are recalled. The formalism is specified on line bundles and applied to a geometrical description of the classical electrodynamics. The inertial frames for this theory are discussed.

Anisotropic Depth Migration on Complex-Structure Land Data

Oladippo Daniel Ipadeola Cairo wireless computer engineering institute, Cairo, Egipt e-mail: prof ut jacob@earthling.net

Anisotropic depth migration (ADM) in thrust-belt environments has become more commonplace over the past four years. As our understanding of the problems with seismic imaging of structures below dipping clastic strata increases, so does our ability to build anisotropic velocity models and correct for imaging and position errors cause by seismic anisotropy. The theoretical background and numerical modelling of seismic propagation effects of waves through media that exhibits tilted transverse isotropy (TTI) illustrate problems with anisotropy. Results of ADM on three complex-structure datasets illustrate the effectiveness of the solution.

Global existence of the solution to the Hartree equation.

\underline{Angel Ivanov}, George Venkov Technical University, 8 Kl. Ohridski blvd., 1000 Sofia, Bulgaria

We consider nonlinear Schrodinger equation with nonlocal interaction and prove that if the solution has finite H^{1/2}− norm at t = 0, then it exists globally in time. Moreover, its H^{1/2}− norm remains bounded for all times.

Camassa-Holm equation as a geodesic flow of the H^{1} right-invariant metric

Rossen Ivanov Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria and School of Mathematics, Trinity College Dublin, Dublin 2, Ireland e-mail: ivanovr@tcd.ie

The fundamental role played by the Lie groups in mechanics, and especially by the dual space of the Lie algebra of the group and the coadjoint action will be illustrated through the Camassa-Holm equation (CH). In 1998 Misiolek observed that the CH is a geodesic flow equation on the group of diffeomorphisms, preserving the so-called H^{1} metric. This example is analogous to the situations in hydrodynamics, where the infinitedimensional groups of diffeomorphisms preserve the volume element of the domain of fluid flow and to the theory of rigid body with a fixed point, where the Hamiltonian is actually a left-invariant metric on SO(3). The momentum map and some explicit parametrizations of the diffeomorphism group, related to recently obtained solutions for the CH equation will be presented.

Quadratic forms on the double-complex algebras

Branimir Kiradjiev University of forestry 10 Kliment Okhridsky Blvd., 1756 Sofia, Bulgaria e-mail: Kiradjiev2003@yahoo.com

Quadratic forms over double-complex algebras appeared naturally. For simplicity here we consider such forms of two double-complex variables α, β ∈ C(1, j), i.e. α = z + jw, z, w, u, v ∈ ℂ, j^{2} = i, i ∈ ℂ.

In general a quadratic form Q((α, β)) over C(1, j) × C(1, j) is defined by a bi-linear symetric functional B(x, y) with complex coefficients (defined over the underlying vector space with base (1, 1), (1, j), (j, 1), (j, j), such that Q = Q((α, β)) = B((α, β), (α, β)). In terms of z, w, u, v we have Q = a(z + jw)^{2} + 2b(z + jw)(u + jv) + c(u + jv)^{2}, a, b, c ∈ ℂ. So Q = a(z^{2} + iw^{2} + 2jzw) + 2b(zu + iwv + j(zv + wu)) + c(u^{2} + iv^{2} + 2juv) = a(z^{2} + iw^{2}) + 2b(zu + iwv) + c(u^{2} + iv^{2}) + j(2azw + b(zv + wu) + 2cuv). In fact we obtain two quadratic form over ℂ^{4}. The classic theorem of Lagrange is established in Q-orthogonal basises.

The quadratic form Q is a double-complex holomorphic form with respect to the double-complex variable z + jw iff it satisfies the Cauchy-Riemann type equations introduced by S. Dimiev for C(1, j):

\frac{∂Q_{0}}{∂z} = \frac{∂Q_{1}}{∂w} \frac{∂Q_{0}}{∂w} = j \frac{∂Q_{1}}{∂z} ,

where Q_{0} = a(z^{2} +iw^{2})+2b(zu+iwv)+c(u^{2} +iv^{2}), and Q_{1} = 2azw+2b(zv+wu)+2cuv. So the equalities 2az + 2bu = 2az + 2bu and 2iaw + i2bv = i(2aw + 2bv) show that Q = Q_{0} + jQ_{1} is always holomorphic.

Examples: the quadratic form a(z + jw)^{2} is holomorphic over C(1, j), a ∈ ℂ. As ∂(az^{2} + aiw^{2})/∂z = ∂(2azw)/∂w (or 2az = 2az) and ∂(az^{2} + aiw^{2})/∂w = i∂(2azw)/∂z (or 2aiw = i2aw). The same is true for c(u + iv)^{2}. But the (C × C)-holomorphic form Q = z^{2} − iw^{2} (Q_{0} = z^{2} − iw^{2}, Q_{1} = 0) (which is important for the definition of the isotropic cone of C(1, j)) is not C(1, j)-holomorphic.

Another negative example is given by the quadratic form (α^{∗})^{2} = (z − jw)^{2}. It is not double-complex holomorphic as it does not satisfy the above mentioned Cauchy-Riemann system. The same is true for the quadratic forms of the type aαα^{∗} + bββ^{∗} etc.

Effects of finite arithmetic - myths and realities.

\underline{Mihail Konstantinov}^{1}, Petko Petkov^{2}

1University of Architecture, Civil Engeneering and Geodesy 1, Christo Smirnenski blvd. 1046 Sofia, Bulgaria 2Department of Systems and Control, Technical University of Sofia, 1756 Sofia, Bulgaria, e-mails: mmk_{−}fte@uacg.bg, php@tu-sofia.bg

In this paper we analyze the main effects of using finite arithmetic (FA) in the solution of computational problems. As a particular model of FA we use the standard floatingpoint binary arithmetic satisfying the IEEE Standard. In particular we describe the influence of the three main groups of constants which govern the effects in using FA:

1. The rounding unit and the range of FA.

2. The relative condition number of the (regular) computational problem.

3. The constants in the definition of numerical stability of the algorithm used to solve the problem.

We give a simple formula which shows how these constants define an upper bound on the relative accuracy of the solution of a regular computational problem in FA.

Next we discuss some myths and misconceptions which are wide spread among users of computer systems for mathematical computations.

Thus the aim of the paper is to help lecturers in informatics and mathematics as well as students and young specialists in science and engineering to avoid bad computational practices.

Symmetries of the Manev Problem and its Real Hamiltonian Form

\underline{Assen V. Kyuldjiev}^{1}, Vladimir S. Gerdjikov^{1}, Giuseppe Marmo^{2}, Gaetano

Vilasi^{3} 1Institute for Nuclear Research and Nuclear Energy 1784 Sofia, Bulgaria 2Dipartimento di Scienze Fisiche, Università Federico II di Napoli Via Cintia, 80126 Napoli, Italy 3Dipartimento di Fisica “E.R. Caianiello”, Universita di Salerno and Istituto Nazionale di Fisica Nucleare, 84100 Salerno, Italy e-mails: kyuljiev@inrne.bas.bg, marmo@na.infn.it, vilasi@sa.infn.it

The Manev model and its real form dynamics are known to possess Ermanno-Bernoulli type invariants similar to the Laplace-Runge-Lenz vector of the Kepler model. Using these additional invariants, we demonstrate here that both Manev model and its real Hamiltonian form have exactly the same 𝔰𝔬(3) or 𝔰𝔬(2, 1) symmetry algebra in addition to the angular momentum algebra. Thus Kepler and Manev models are shown to have identical symmetry algebras and hence sharing more features than previously expected.

The Manakov model as two moving interacting curves

\underline{Nikolay A. Kostov}^{1}, Rossen Dandoloff^{2}, Vladimir Gerdjikov^{3}, Georgi Grahovski^{2,3} 1 Institute of Electronics, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria 2 Université de Cergy-Pontoise, 2 avenue, A. Chauvin, F-95302, Cergy-Pontoise Cedex, France 3Institute for Nuclear Research and Nuclear Energy, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria e-mails: nakostov@ie.bas.bg, rossen.dandoloff@u-cergy.fr, gerjikov@inrne.bas.bg, grah@inrne.bas.bg

The two time-dependent Schrödinger equations in a potential V (s, u), u denoting time, can be interpreted geometrically as a moving interacting curves whose Fermi-Walker phase density is given by −(∂V/∂s). The Manakov model appears as two moving interacting curves using extended da Rios system and two Hasimoto transformations.

Clusters and fractals in mathematical physics

Julian Ławrynowicz , Institute of Physics, University of Łódź Pomorska 149/153, PL-90-236 Łódź, Poland Institute of Mathematics, Polish Academy of Sciences Łódź Branch, Banacha 22, PL-90-238 Łódź, Poland e-mail: jlawryno@uni.lodz.pl

It is well known that starting with real structure, the Cayley–Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 2^{7}. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p → p + 2 → p + 4 → …, they have constructed 2^{4}-dimensional “bipetals” for p = 9 and 2^{7}-dimensional “bisepals” for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the “pistil” and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the “stamens”. As a consequence of an effective, explicit determination of the periods and expressing them in terms of complex and quaternionic structures, the following applications are outlined: fractals vs. biholomorphic invariants, importance of fractals when studying the entropy-depending structures, hyperholomorphy and harmonicity, and applications to nonlinear parabolic equations. The lecture is based on a research joint with Stefano Marchiafava (Roma) and Małgorzata Nowak-Kȩpczyk (Radom, Poland).

Characterization of parallel isometric immersions of space forms into space forms in the class of isotropic immersions

\underline{Sadahiro Maeda}, S. Udagawa Department of Mathematics, Shimane University, Matsue 690-8504, Japan e-mail: smaeda@riko.shimane-u.ac.jp

We characterize parallel isometric immersions of space forms into space forms in the class of isotropic submanifolds M’s under conditions that the mean curvature vector of M is parallel and the sectional curvature K of M^{n} satisfies some inequality.

On quasi-Kähler manifolds with Norden metric

\underline{Mancho Manev}, Kostadin Gribachev, Dimitar Mekerov Department of Geometry, Faculty of Mathematics and Informatics, University of Plovdiv, 236 Bulgaria blvd., 4003 Plovdiv, Bulgaria e-mails: mmanev@yahoo.com, costas@pu.acad.bg, mircho@pu.acad.bg

The basic class of the non-integrable almost complex manifolds with Norden metric is considered. Its curvature properties are studied. A 4-parametric family of 4-dimensional quasi-Kähler manifolds with Norden metric is constructed by means of a Lie group with a non-Abelian almost complex structure. This family is characterized geometrically. Necessary and sufficient conditions for the special case of isotropic Kähler manifolds are given.

Distributional boundary values of the functions in the Djarbjashian classes

\underline{Vesna Manova-Erakovik}, Nikola Pandeski Institute of Mathematics, Faculty of Natural Sciences and Mathematics St. Cyril and Methodius, University of Skopje, Macedonia e-mail: vesname@iunona.pmf.ukim.edu.mk In the paper by M.Djarbashian [1] the author introduced, on the unit disk Δ = {z, |z| < 1}, the Djarbjashian spaces denoted by H^{p}(α), −1 < α < ∞, 0 < p < ∞, i.e.

f ∈ H^{p}(α) ↔ \frac{1 + α}{π} ∫_{0}^{1} ∫_{0}^{2π}(1 − r^{2})^{α} |f(re^{iθ})|^{p} rdr dθ < ∞,

with the norm

||f||_{p}^{α} = (\frac{1 + α}{α} ∫_{Δ}(1 − |z|^{2})^{α}|f(z)^{p}|dx dy)^{1/p} , z = x + iy.

In our paper we found the distributional boundary values of the functions in the Djarbjashian spaces.

References

[1] M.Djarbashian. ”O kanonichnom predstavleni meromorfnyh v edinichnom kruge funkcii, DAN Arm. SSR, 3 No 1, 3-9, (1945).

Homogenization Of Nonlinear Parabolic Operators Of High Order

Marin Marinov New Bulgarian University, 21 Montevideo Str., 1618 Sofia, Bulgaria e-mail: mlmarinov@nbu.bg

In this work the problem of homogenization of nonlinear parabolic operators of the type

P_{ε}(u) = \frac{∂u}{∂t} − ∑_{|α|≤m}(−1)^{|α|}D^{α}a_{ε}(ε^{−μ}t, ε^{−r}x, ∂u)

is solved, where the functions a(τ, y, ζ) are periodic in (τ, y). Depending on the relation between μ and r, the behavior of P_{ε}(u) when ε → 0 is examined. A homogenization operator P (u) is found and the corresponding formulas for calculation of the effective coefficients are proved.

Application of Unscented and Extended Kalman Filtering for Estimating Quaternion Motion

Vanya Markova Institute of Control and System Research PO Box 565, 139 Russkii blvd, 4000 Plovdiv e-mail: markovavanya@yahoo.com

The unscented Kalman filter is a superior alternative to the extended Kalman filter for a variety of estimation and control problems. However, its effectiveness for improving direction of motion tracking for virtual reality applications in the presence of noisy data has been unexplored. In this paper, we present an empirical study comparing the performance of unscented and extended Kalman filtering for improving model for forecasting. Specifically, we examine wind orientation motion, represented with quaternions, which are critical for correct viewing perspectives in virtual reality. Our experimental results and analysis indicate that unscented Kalman filtering performs equivalently with extended Kalman filtering. However, the additional computational overhead of the unscented Kalman filter and quasi-linear nature of the quaternion dynamics lead to the conclusion that the extended Kalman filter is a better choice for estimating quaternion motion in virtual reality applications.

Almost Kähler-Einstein structures on 8-dimensional Walker manifolds ρ. Counterexamples to the Goldberg conjecture of neutral version

Yasuo Matsushita Section of Mathematics, School of Engineering, University of Shiga Prefecture, Hikone 522-8533, Japan e-mail: yasuo@mech.usp.ac.jp

There is a famous conjecture, by Goldberg [6], which states that the almost complex structure of a compact almost Kähler-Einstein Riemannian manifold is integrable. Sekigawa proved that the Goldberg conjecture is true if the scalar curvature is non-negative [11], [12]. Nurowski and Przanowski [10] have shown that the conjecture is false when one omits the assumption of compactness and one of us (Haze) has demonstrated the same for the neutral-signature analogue of Goldberg’s conjecture by constructing a neutral Ricci-flat metric on an open subset of ℝ^{4} which is almost-Kähler with respect to a non-integrable almost complex structure. (The assertion in [10] that Sekigawa has established Goldberg’s conjecture in four dimensions is erroneous, [13].)

In [8], Haze’s metric is shown to be an instance of a Walker metric. By a Walker manifold, we mean a pseudo-Riemannian n-manifold which admits a field of parallel null r-planes, with r ≤ \frac{n}{2}. The corresponding metric is called a Walker metric [14]. When n = 2k and r = k, such a metric is of neutral signature. Walker metrics have proven useful in various circumstances, e.g., in [5]. I have recently begun a study of neutral Walker metrics in four dimensions [7], [8]. See also joint works with Davidov and Muškarov, and with my colleague in Spain as follows: [1], [2], [3] and [4].

By considering 8-dimensional Walker manifolds, we are able to exhibit almost Kähler-Einstein neutral structures which are not Kähler, initially on ℝ^{8} and then on an 8-torus. Thus, the neutral-signature version of Goldberg’s conjecture fails. Whether our result has any bearing on the original conjecture of Goldberg for Riemannian manifolds remains unclear, although our example, being Ricci flat, indicates that Sekigawa’s result [11], [12] fails for neutral signature. Our almost Kähler-Einstein neutral structure on an 8dimensional torus is the first counterexample for a compact manifold.

The counterexamples will be published in Monatshefte für Mathematik [9].

References

[1] M. Chaichi, E. García-Río and Y. Matsushita, Curvature properties of fourdimensional Walker metrics, Classical Quantum Gravity 22 (2005), 559-577.

[2] J. Davidov, J. C. Díaz-Ramos, E. García-Río, Y. Matsushita, O. Muškarov and R. Vázquez-Lorenzo, Almost Kähler Walker four-manifolds, Preprint June 2006.

[3] J. Davidov and O. Muškarov, Self-dual Walker metrics with a two-step nilpotent Ricci operator, J. Geom. Phys. in press.

[4] E. García-Río, S. Haze, N. Katayama and Yasuo Matsushita, Symplectic, Hermitian and Kähler structures on Walker 4-manifolds, Preprint May 2006.

[5] R. Ghanam and G. Thompson, The holonomy Lie algebras of neutral metrics in dimension four, J. Math. Phys. 42 (2001), 2266–2284.

[6] S. I. Goldberg, Integrability of almost Kähler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96–100 .

[7] Y. Matsushita, Four-dimensional Walker metrics and symplectic structures, J. Geom. Phys. 52 (2004), 89–99.

[8] Y. Matsushita, Walker 4-manifolds with proper almost complex structures, J. Geom. Phys. 55 (2005), 385-398.

[9] Y. Matsushita, S. Haze and P. R. Law, Almost Kähler-Einstein structures on 8dimensional Walker manifolds, Monatsh. Math., to appear.

[10] P. Nurowski and M. Przanowski, A four-dimensional example of a Ricci flat metric admitting almost-Kähler non-Kähler structure, Classical Quantum Gravity 16 (1999), no. 3, L9–L13.

[11] K. Sekigawa, On some 4-dimensional compact Einstein almost Kähler manifolds, Math. Ann. 271 (1985) 333–337.

[12] K. Sekigawa, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan 36 (1987) 677–684.

[13] K. Sekigawa, private communications December 2004.

[14] A. G. Walker, Canonical form for a riemannian space with a parallel field of null planes, Q. J. Math. Oxford (2) 1 (1950), 69–79.

Information geometry and affine differential geometry

Hiroshi Matsuzoe Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan e-mail: matsuzoe@nitech.ac.jp

Affine differential geometry studies hypersurfaces immersed into an affine space, and information geometry studies geometric structures of sets of probability distributions. A key fact is that duality of affine connections arises naturally, then these geometries have common geometric ideas.

Let us focus to elucidate the relation between Bayesian statistics and affine differential geometry. Bayesian statistics is a statistical theory which uses previous knowledge for inference, and the knowledge is expressed by a prior probability distribution. Hence existence of prior distributions and their choice are important problems. From the viewpoint of differential geometry, prior distributions can be regarded as volume elements on statistical manifolds, and equiaffine structures work essentially for the existence of natural prior distributions.

In this talk, we give a geometric interpretation of Bayesian statistics. Moreover, we will give sufficient conditions for a statistical submanifold to have an equiaffine structure, as a result for differential geometry.

On the Geometric Structure of Hypersurfaces of Conullity Two in Euclidean Space

Velichka Milousheva Institute of Mathematics and Informatics, Acad. G. Bonchev Str. bl. 8, 1113 Sofia, Bulgaria e-mail: milousheva@vsu.bg

We introduce the notion of a semi-developable surface of codimension two in Euclidean space as a generalization of the notion developable surface of codimension two and give a characterization of the developable and semi-developable surfaces of codimension two in terms of their second fundamental forms. We prove that each hypersurface of conullity two in Euclidean space is locally a foliation of developable or semi-developable surfaces of codimension two.

We give a geometric description of the class 𝔎_{0} of hypersurfaces of conullity two with involutive geometric two-dimensional distributions proving that the integral surfaces of these distributions are surfaces with flat normal connection, which are not developable and conversely, that any two-dimensional surface with flat normal connection, which is not developable, generates a hypersurface of conullity two from the class 𝔎_{0}. In this way the hypersurfaces of conullity two from the class 𝔎_{0} are in one-to-one correspondence with the two-dimensional surfaces with flat normal connection, which are not developable.

We characterize the hypersurfaces of conullity two also as envelopes of two-parameter families of hyperplanes, proving that a hypersurface in Euclidean space is locally a hypersurface of conullity two if and only if it is the envelope of a two-parameter family of hyperplanes. This geometric characterization allows us to obtain a parametrization of each hypersurface of conullity two by a pair of a unit vector function l(u, v) and a scalar function r(u, v). We obtain a characterization in terms of a system of partial differential equations for the geometric functions l(u, v) and r(u, v) of two main classes of hypersurfaces of conullity two: the class of minimal hypersurfaces of conullity two and the class of umbilical hypersurfaces of conullity two.

On some non-integrable almost contact manifolds with Norden metric of dimension 5

Galia Nakova University of Veliko Tarnovo, 2 T. Tarnovski str., 5000 Veliko Tarnovo, Bulgaria e-mail: gnakova@yahoo.com

A Lie group as a 5-dimensional almost contact manifold with Norden metric is considered. By using the complexification of the real Lie algebra this manifold is constructed. The obtained manifold is non-integrable and belongs to one of the basic classes. Curvature properties of the constructed manifold are given.

On the definition of the Kobayashi-Buseman pseudometric

\underline{Nikolay Nikolov}^{1}, Peter Pflug^{2} 1Institute of Mathematics and Informatics Acad. G. Bonchev 8, 1113 Sofia, Bulgaria 2Carl von Ossietzky Universität Oldenburg Institut für Mathematik, Fakultät V D-26111 Oldenburg, Germany e-mails: nik@math.bas.bg, pflug@mathematik.uni-oldenburg.de

We prove that the (2n − 1)-th Kobayashi pseudometric of any domain D ⊂ ℂ^{n} coincides with the Kobayashi–Buseman pseudometric of D, and that 2n−1 is the optimal number, in general.

Non-oscillating trajectories of vector fields and Zariski’s local uniformization.

Peace Udom Nwaobilor Cairo wireless computer engineering institute Cairo, Egipt e-mail: prof ut jacob@earthling.net

A transcendent non-oscillating trajectory of an analytic germ of real vector field induces a structure of Hardy field for the meromorphic functions. It has a natural valuation associated to it. The study of this valuation allows to get a reduction of singularities of the vector field following the strict transform of the trajectory. More generally, for a holomorphic complex vector field, the above results can be generalized for a given valuation of the field of the meromorphic functions. We obtain in this way a local uniformization in the sense of Zariski, that should be globalized in dimension three, following the classical results of Zariski. The key of these results is a construction (due to J. Cano and Grigoriev-Singer) based on the Newton Polygon of a differential operator, that assures finiteness results on the valuation allowing the local uniformization

On the hypoellipticity of complex valued vector fields

Petar R. Popivanov Institute of mathematics and informatics, acad. G. Bontchev str., 1113 Sofia, Bulgaria e-mail: angp@math.bas.bg

This talk deals with the (micro)local C^{∞} and Gevrey hypoellipticity and subellipticity of some systems of complex valued vector fields. In the case of second-order overdetermined systems of PDE generated by the same vector fields we prove C^{∞} hypoellipticity results which do not depend on the lower order terms. The sharp loss of regularity in the Sobolev spaces in the latter case is equal to 1, while in the first one it runs from 1/2 to 1.

Some critical almost Kähler structures

Takashi Oguro^{1}, \underline{Kouei Sekigawa}^{2} 1Department of Mathematical Sciences, School of Science and Engineering, Tokyo Denki University, Saitama, 350-0394, Japan 2Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan e-mail: sekigawa@sc.niigata-u.ac.jp

Let M be a compact orientable manifold of dimension m. We denote by ℳ (M) the set of all Riemannian metrics on M and 𝒩 (M) the set of all Riemannian metrics of the same volume element. It is well-known that a Riemannian metric g ∈ 𝒩 (M) is a critical point of the functional 𝒜 on 𝒩 (M) defined by

𝒜 (g) = ∫_{M} τ dv_{g}

if and only if g is an Einstein metric, where τ is the scalar curvature of g and dv_{g} is the volume element of the Riemannian metric g.

Now, let M be a compact manifold of dimension m = 2n admitting an almost complex structure. We denote by 𝒜ℋ (M) the set of all almost Hermitian structures and 𝒜ℋ (M, Ω) the set of all almost Hermitian structures with the same Kähler form Ω. An almost Hermitian manifold M = (M, J, g) with the closed Kähler form Ω (dΩ = 0) is called an almost Kähler manifold. Let M = (M, J, g) be a compact almost Kähler manifold and Ω the corresponding Kähler form. Then, we may note that any almost Hermitian structure (J, g) ∈ 𝒜ℋ (M, Ω) is an almost Kähler structure on M. In this case, we denote 𝒜ℋ (M, Ω) by 𝒜𝒦 (M, Ω). In [1], Blair and Ianus studied critical points of the functional ℱ on 𝒜𝒦 (M, Ω) defined by

ℱ (J, g) = ∫_{M}(τ^{∗} − τ) dv_{g},

where τ^{∗} is the ∗-scalar curvature of (M, J, g), and proved that (J, g) is a critical point of ℱ on 𝒜𝒦 (M, Ω) if and only if the Ricci tensor ρ is J-invariant.

We denote by 𝒜𝒦 (M, [Ω]) the set of all almost Kähler structures on M with the same Kähler class [Ω] in the de Rham cohomology group. In [3], [4], Koda studied critical points of the functional ℱ_{λ,μ} on 𝒜ℋ (M, Ω) and 𝒜𝒦 (M, [Ω]) defined by

ℱ_{λ,μ}(J, g) = ∫_{M}(λτ + μτ^{∗}) dv_{g}, (λ, μ) ∈ ℝ^{2}\(0, 0).

On one hand, it is known that the following integral formula (∗) holds on a compact almost Kähler manifold M = (M, J, g) ([6]):

∫_{M} (f_{1} − \frac{1}{2}f_{2} + f_{3} − 2f_{4}) dv_{g} = 0,  (∗)

where f_{1}, f_{2}, f_{3}, f_{4} are certain functions on M, which will be introduced in the talk. The above integral formula (∗) plays an important role in the arguments on the Goldberg conjecture.

In this talk, we shall discuss the first variation of the above integral formula (∗) on 𝒜𝒦 (M, [Ω]) and introduce the corresponding Euler-Lagrange type equations. Further, we shall also discuss another first variational problem on 𝒜𝒦 (M, [Ω]) for a 4-dimensional compact almost Kähler manifold M = (M, J, g) which is based on the Wu’s theorem.

References

[1] D.E. Blair and S. Ianus, Critical associated metrics on symplectic manifolds, Contemporary Math., 51 (1986), 23–29.

[2] S.I. Goldberg, Integrability of almost Kähler manifolds, Proc. Amer. Math. Soc., 21 (1969), 96–100.

[3] T. Koda, Critical almost Hermitian structures, Indian J. Pure Appl. Math., 26 (1995), 679–690.

[4] T. Koda, Almost Kähler structures with a fixed Kähler class, Math. J. Toyama Univ., 27 (2004), 125–131.

[5] T. Oguro and K. Sekigawa, The first variation of certain integral formula on compact almost Kähler manifolds, preprint.

[6] K. Sekigawa, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan, 39 (1987), 677–684.

[7] S. Tanno, Deformations of Riemannian metrics on 3-dimensional manifolds, Tôhoku Math. J., 27 (1975), 437–444.

Differential forms of double-complex variables

Petar Stoev University of Architecture, Civil Engeneering and Geodesy 1, Christo Smirnenski blvd. 1046 Sofia, Bulgaria

Differential 1-forms over the double-complex algebra C(1, j) are considered

ω = ϕ(α)dα + ψ(α)dα^{∗}

where α ∈ C(1, j), α = z + jw, α^{∗} = z − jw, j^{2} = i, i ∈ ℂ, z, w ∈ ℂ (ℂ is the field of complex numbers). By definition

\frac{∂}{∂α} := \frac{1}{2} (\frac{∂}{∂z} − ji\frac{∂}{∂w} ) , \frac{∂}{∂α^{∗}} := \frac{1}{2} (\frac{∂}{∂z} + ji\frac{∂}{∂w} ) ,

It is proved that df = ∂f/∂αdα + ∂f/∂α^{∗}dα^{∗}, or shortly df = ∂f + ∂^{∗}f. In the case ∂f/∂α^{∗} = 0, the function f(α) is a holomorphic function of the double-complex variable α. We study the equation ∂∂^{∗}f = 0 of the complex harmonic functions.

Characterization of Totally Umbilic Immersions by Curves of Order 2

Tadashi Sugiyama Division of Mathematics and Mathematical Science, Department of Computer Science and Engineering, Nagoya Institute of Tecnhnology, Gokisocho Showaku Nagoya 466-8555 Japan e-mail: tadashi@zelus.ics.nitech.ac.jp

In my talk we study submanifolds by use of extrinsic shapes of some curves having points of order 2. Here a smooth curve γ parameterized by its arclength is said to be of order 2 at γ(t_{0}) if

i) ∇γ̇ γ̇(t_{0}) ≠ 0,

ii) ∇γ̇(\frac{1}{∥∇γ̇ γ̇∥}∇γ̇ γ̇)(t_{0}) is parallel to γ̇(t_{0}).

We give a condition that they are totally umbilic, which gives an extension of Nomizu-Yano’s result on a characterization of extrinsic sphere. We also study Kähler and quaternionic Kähler submanifolds from this point of view and characterize totally geodesic submanifolds in these classes.

Lie groups as several complex manifolds with Norden metric

Marta Teofilova Department of Geometry, Faculty of Mathematics and Informatics, University of Plovdiv, 4003 Plovdiv, Bulgaria e-mail: mar@gbg.bg

The basic classes of the integrable almost complex manifolds with Norden metric are considered. An example of a 4-dimensional conformal Kähler manifold with Norden metric and a 4-dimensional special complex manifold with Norden metric are constructed by means of Lie groups and Lie algebras. Both manifolds are characterized geometrically. The form of the curvature tensor for one of the examples is obtained. The conditions for these manifolds to be isotropic Kählerian are given.

Bäcklund transformations and Riemann-Hilbert problem for N-wave equations with additional symmetries

Vladimir Gerdjikov, \underline{Tihomir Valchev} Institute for Nuclear Research and Nuclear Energy 72 Tzarigradsko chaussee , 1784 Sofia, Bulgaria e-mails: gerjikov@inrne.bas.bg, valtchev@inrne.bas.bg

We obtain soliton solutions of N-wave equations with reductions imposed on them. The method we make use of is a special type of Bäcklund transformation, namely the dressing method. There is an equivalence between the inverse scattering problem and Riemann-Hilbert problem. This way we construct singular solutions for the Riemann-Hilbert problem associated with the N-wave equation.

Phase transitions in binary alloy thin films

Ilona Zasada Solid State Physics Department, University of Łódź Pomorska 149/153, PL-90-236 Łódź, Poland e-mails: izasada@mvii.uni.lodz.pl

The theory, of order-disorder phenomena in binary alloy thin films of AB3 type is considered in the context of order parameters of different physical nature. The main idea of the present description was applied at first by Valenta and Sukiennicki in the case of permalloy films for their order-disorder characterisation and consists in the model of pair interactions between nearest neighbour atoms belonging to the sublattices in the form of the monoatomic layers parallel to the surface. Such description is based on the thermodynamic approach to small particles when they should be treated as thermodynamically inhomogeneous systems. The thin film is considered as the thermodynamically inhomogeneous system constructed of the homogeneous subsystems in the form of monoatomic layers. Different order parameters, as lattice long- and short-range order, magnetic order and crystallinity parameters, are introduced and considered in the context of their mutual interdependence.

LIST of PARTICIPANTS

Toshiaki ADACHI Department of Mathematics, Nagoya Institute of Technology Nagoya 466-8555, Japan e-mail: adachi@nitech.ac.jp

Vestislav Apostolov UQAM Université du Québec Montréal, Case postale 8888, succursale Centre-ville Montréal (Québec), H3C 3P8 Canada e-mail: apostolo@math.uqam.ca

Lilia N. Apostolova Institute of mathematics and informatics, acad. G. Bontchev str., 1113 Sofia, Bulgaria e-mail: liliana@math.bas.bg

Vladimir Balan University Politehnica of Bucharest Faculty of Applied Sciences, Department of Mathematics I, RO-060042, Bucharest, Romania. e-mail: vbalan@mathem.pub.ro

Rossen Dandoloff LPTM, Université de Cergy-Pontoise, F-95302 Cergy-Pontoise Cedex, France e-mail: rossen.dandoloff@u-cergy.fr

Stancho Dimiev Institute of mathematics and informatics, acad. G. Bontchev str., 1113 Sofia, Bulgaria e-mail: sdimiev@math.bas.bg

Stoil Donev Institute for Nuclear Research and Nuclear Energy, 72 blvd.Tzarigradsko chaussee 1784 Sofia, Bulgaria e-mails: sdonev@inrne.bas.bg

Norio Ejiri Department of Mathematics, Meijo University, Tempaku, Nagoya, 468-8502 Japan e-mail: ejiri@ccmfs.meijo-u.ac.jp

Petia Furlinska Plovdiv, Bulgaria e-mail: petiaf@abv.bg

Vladimir Georgiev Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy e-mail: georgiev@dm.unipi.it

Vladimir Gerdjikov Institute for Nuclear Research and Nuclear Energy, 72 Tzarigradsko chaussee , 1784 Sofia, Bulgaria e-mail: gerjikov@inrne.bas.bg

Georgi Grahovski Institute for Nuclear Research and Nuclear Energy, 72 Tzarigradsko chaussee, 1784 Sofia, Bulgaria e-mail: grah@inrne.bas.bg

Kostadin Gribachev Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv 4003, Bulgaria

Dobrinka K. Gribacheva-Shtarbeva Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv 4003, Bulgaria e-mail: dobrinka@pu.acad.bg

Hideya Hashimoto Meijo University, Tempaku, Nagoya, Japan e-mail: hhashi@ccmfs.meijo-u.ac.jp

Milen Hristov University of Veliko Tarnovo, 2 T. Tarnovski str., 5000 Veliko Tarnovo, Bulgaria e-mail: milenjh@yahoo.com

Isaac Iroegbu Ibenme Cairo wireless computer engineerng institute, Cairo, Egipt e-mail: prof ut jacob@earthling.net

Ikechukwu Clement Ijeh Cairo wireless computer engineering institute Cairo, Egipt e-mail: prof ut jacob@earthling.net

Bozhidar Z. Iliev Institute for Nuclear Research and Nuclear Energy 72 blvd.Tzarigradsko chaussee, 1784 Sofia, Bulgaria e-mail: bozho@inrne.bas.bg

Oladippo Daniel Ipadeola Cairo wireless computer engineering institute Cairo, Egipt e-mail: prof ut jacob@earthling.net

Angel Ivanov Technical University, 8 Kl. Ohridski blvd. 1000 Sofia, Bulgaria

Rossen Ivanov Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria and School of Mathematics, Trinity College Dublin, Dublin 2, Ireland e-mail: ivanovr@tcd.ie

Branimir Kiradjiev University of forestry 10 Kliment Okhridsky Blvd. 1756 Sofia, Bulgaria e-mail: Kiradjiev2003@yahoo.com

Mihail Konstantinov University of Architecture, Civil Engeneering and Geodesy 1, Christo Smirnenski blvd., 1046 Sofia, Bulgaria e-mails: mmk_{−}fte@uacg.bg

Assen V. Kyuldjiev Institute for Nuclear Research and Nuclear Energy, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria e-mails: kyuljiev@inrne.bas.bg

Nikolay A. Kostov Institute of Electronics, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria e-mails: nakostov@ie.bas.bg

Julian Ławrynowicz Institute of Physics, University of Łódź Pomorska 149/153, PL-90-236 Łódź, Poland Institute of Mathematics, Polish Academy of Sciences Łódź Branch, Banacha 22, PL-90-238 Łódź, Poland e-mail: jlawryno@uni.lodz.pl

Sadahiro Maeda Department of Mathematics, Shimane University, Matsue 690-8504, Japan e-mail: smaeda@riko.shimane-u.ac.jp

Mancho Manev Department of Geometry, Faculty of Mathematics and Informatics, University of Plovdiv, 236 Bulgaria blvd., 4003 Plovdiv, Bulgaria e-mail: mmanev@yahoo.com

Vesna Manova-Erakovik Institute of Mathematics, Faculty of Natural Sciences and Mathematics St. Cyril and Methodius, University of Skopje, Macedonia e-mail: vesname@iunona.pmf.ukim.edu.mk

Marin Marinov New Bulgarian University, 21 Montevideo Str., 1618 Sofia, Bulgaria e-mail: mlmarinov@nbu.bg

Vanya Markova Institute of Control and System Research PO Box 565, 139 Russkii blvd, 4000 Plovdiv, Bulgaria e-mail: markovavanya@yahoo.com

Yasuo Matsushita Section of Mathematics, School of Engineering, University of Shiga Prefecture, Hikone 522-8533, Japan e-mail: yasuo@mech.usp.ac.jp

Hiroshi Matsuzoe Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan e-mail: matsuzoe@nitech.ac.jp

Dimitar Mekerov Department of Geometry, Faculty of Mathematics and Informatics, University of Plovdiv, 236 Bulgaria blvd., 4003 Plovdiv, Bulgaria e-mail: mircho@pu.acad.bg

Velichka Milousheva Institute of Mathematics and Informatics, Acad. G. Bonchev Str. bl. 8, 1113 Sofia, Bulgaria e-mail: milousheva@vsu.bg

Galia Nakova University of Veliko Tarnovo, 2 T. Tarnovski str., 5000 Veliko Tarnovo, Bulgaria e-mail: gnakova@yahoo.com

Nikolay Nikolov Institute of Mathematics and Informatics Acad. G. Bonchev 8, 1113 Sofia, Bulgaria e-mails: nik@math.bas.bg

Peace Udom Nwaobilor Cairo wireless computer engineering institute Cairo, Egipt e-mail: prof ut jacob@earthling.net

Petko Petkov Department of Systems and Control, Technical University of Sofia, 1756 Sofia, Bulgaria, e-mails: php@tu-sofia.bg

Petar R. Popivanov Institute of mathematics and informatics, acad. G. Bontchev str., 1113 Sofia, Bulgaria e-mail: angp@math.bas.bg

Kouei Sekigawa Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan e-mail: sekigawa@sc.niigata-u.ac.jp

Slavka Slavova Dobrich Pedagogical College block 12 Dobrotica 9300 Dobrich, Bulgaria e-mail: slavova@dobrich.net

Petar Stoev University of Architecture, Civil Engeneering and Geodesy 1, Christo Smirnenski blvd. 1046 Sofia, Bulgaria

Tadashi Sugiyama Division of Mathematics and Mathematical Science, Department of Computer Science and Engineering, Nagoya Institute of Tecnhnology, Gokisocho Showaku Nagoya 466-8555 Japan e-mail: tadashi@zelus.ics.nitech.ac.jp

Marta Teofilova Department of Geometry, Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv 4003, Bulgaria e-mail: mar@gbg.bg

S. Udagawa Department of Mathematics, Shimane University, Matsue 690-8504, Japan

Tihomir Valchev Institute for Nuclear Research and Nuclear Energy 72 Tzarigradsko chaussee, 1784 Sofia, Bulgaria e-mail: valtchev@inrne.bas.bg

George Venkov Technical University, 8 Kl. Ohridski blvd. 1000 Sofia, Bulgaria

Ilona Zasada Solid State Physics Department, University of Łódź Pomorska 149/153, PL-90-236 Łódź, Poland e-mails: izasada@mvii.uni.lodz.pl