Trends in Differential Geometry, Complex Analysis and Mathematical Physics

The following sections are included:

• PREFACE

• ORGANIZING COMMITTEES

• PRESENTATIONS

• CONTENTS

In this article we propose discrete models of trajectories for Kähler magnetic fields. We consider regular graphs whose edges are colored by 2 colors and compare the asymptotic behaviors of the number of prime cycles on them with those of prime trajectory-cycles on compact quatients of complex hyperbolic spaces.

We consider the question whether with the constraint from the Bethe ansatz solution on the ASEP boundary parameters the matrix product ansazt fails to produce the stationary state. We discuss applicability of the matrix product ansatz from the point of view of the boundary algebra which reveals deep symmetry properties of the ASEP. We argue that the Bethe ansatz integrability constraint is the defining condition for a finite dimensional representation of the boundary algebra when the system reaches a steady state satisfying detailed balance.

These remarks are devoted to a basic properties of the double-complex Laplace operator , where z, w are complex variables and U(z, w) is a double-complex value function. A decomposition into two complex operators is given. An initial problem for the double-complex Laplace operator on the whole space ℂ(1, j) is solved with a D'Alember's type formula. An initial problem on the cartesian product of two unit squares in ℂ for the zero eigenvalue problem for the double-complex Laplace operator is treated with the Fourier method of separating of the variables in double-complex context. The corresponding system of two equations of second order and the corresponding operator of fourth order with real coefficients are obtained.

Generalizations of conjugate connections are studied in this paper. It is known that generalized conjugate connections, and semi-conjugate connections are generalizations of conjugate connections. To clarify relations of these connections, the notion of dual semi-conjugate connections is introduced. Then their relations are elucidated. Local triviality of generalized conjugate connection is also studied.

We consider a family of Herglotz functions F_y (y ϵ ℝ) generated from a Herglotz function F, and obtain their integral representations. We review the theory of value distribution for boundary values of Herglotz functions, and present a result for limiting generalized value distribution for translated Herglotz functions.

In the paper we consider cyclic hyper-number systems which are naturally related to the partial differential equations over the paracomplex algebra with zero divisors. This research is somehow parallel to the notion of the classical hypercomplex systems. At the end we shortly describe the Hankel geometry as a special cyclic geometry. Due to some page restrictions in this volume, we shall develop the analytic and the geometric approaches to this specific kind of cyclic geometry in another article of ours.

This paper deals with plane curves which being integral curves of a certain dynamical system of two degrees of freedom can be thought of as trajectories of a particle of unit mass in a potential field. Two special cases of this system that are integrable by quadratures are identified in another paper of this volume. Typical examples of curves, corresponding to each of these two cases are the so-called Lévy's elasticae and the profile curves of the Delaunay surfaces, respectively. Here their parameterizations are derived in explicit form and illustrated by a number of figures.

This paper aims to relate some properties of photon-like objects, considered as spatially finite time-stable physical entities with dynamical structure, to well defined properties of the corresponding electromagnetic strains defined as Lie derivatives of the Minkowski (pseudo) metric with respect to the eigen vector fields of the Maxwell-Minkowski stress-energy-momentum tensor. First we recall the geometric sense of the concept of strain, then we introduce and discuss the notion for PhLO. We compute then the strains and establish important correspondences with the divergence terms of the energy tensor and the terms determining some internal energy-momentum exchange between the two components F and _*F of a vacuum electromagnetic field. The role of appropriately defined Frobenius curvature is also discussed and emphasized. Finally, equations of motion and interesting PhLO-solutions are given.

The following sections are included:

• Introduction

• Preliminary

• Minimal surfaces in Riemannian flat tori

• References

We analyze the algebraic aspects of solving the multicomponent nonlinear Schrödinger (MNLS) equations related to the symmetric spaces of BD.I-type. This analysis includes: i) the spectral properties of the MNLS equations under the nonvanishing (constant) boundary conditions; ii) the construction of new equations of MNLS type imposing additional reductions; iii) the involutivity of their integrals of motion proven by using the method of the classical R-matrix; iv) brief but explicit description of their hierarchies of Hamiltonian structures.

The following sections are included:

• Introduction

• Preliminaries

• Hypersurfaces in Imℭ

• On G₂ frame field on S² × R⁴

• References

In the following article we present some recently derived results describing how a class of solutions of a family of nonlinear dispersive equations evolve over time. We describe the physical relevance of certain integrable members of this family, including applications in modelling shallow water waves over a flat bed and the propagation of waves in hyperelastic rods. The results contained in this article are of relevance to the soliton solutions of these equations.

Classical Bézier curves (the parabolic case) with the natural barycentric description are considered. The explicit expressions for lenght, curvature, shape etc. are given. It is introduced the notion trajectory of Bézier curve and by means of the projective isotomic conjugation the corresponding envelope conic surfaces in the space are obtained.

The Heisenberg relations are derived in a quite general setting when the field transformations are induced by three representations of a given group. They are considered also in the fibre bundle approach. The results are illustrated in a case of transformations induced by the Poincaré group.

A class of equations describing the geodesic flow for a right-invariant metric on the group of diffeomorphisms of ℝⁿ is reviewed from the viewpoint of their Lie-Poisson structures. A subclass of these equations is analogous to the Euler equations in hydrodynamics (for n = 3), preserving the volume element of the domain of fluid flow. An example in n = 1 dimension is the Camassa-Holm equation, which is a geodesic flow equation on the group of diffeomorphisms, preserving the H¹ metric.

The following sections are included:

• Introduction

• Basic items

• Parallel surface

• Linear Weingarten surface

• References

We study two wave interacting system by the inverse scattering method. We develop the algebraic approach in terms of classical r-matrix and give an interpretation of the Poisson brackets as linear r-matrix algebra. The solutions are expressed in terms of Weierstrass functions and Jacobian functions.

The paper is inspired by an elegant generalization by Z.-D. Zhang (2007) of the exact solution by L. Onsager (1944) to the problem of description of the E. Ising lattices. We start with mentioning the relationship with Bethe-type fractals. We characterize then the differences when taking into account the first, second, and third nearest neighbours of an atom in an alloy. In turn we pass to discussing the Ising-Onsager-Zhang lattices in the context of possible applications of the Jordan-von Neumann-Wigner (1934) approach (Theorem 1). We decide to use some fractals of the algebraic Clifford structure in relation with the quaternionic sequence of Jordan algebras, and this appears to be quite promising (Theorem 2). Finally we outline further perspectives of investigation in this direction.

We characterize Clifford minimal hypersurfaces S^r(nc/r) × S^n-r(nc/(n - r)) with 1 ≦ r ≦ n - 1 in a sphere Sⁿ⁺¹(c) of constant sectional curvature c by observing their geodesics from this ambient sphere.

A type of almost contact hypersurfaces with Norden metric of a Kähler manifold with Norden metric is considered. The curvature tensor and the special sectional curvatures are characterized. The canonical connection on such manifolds is studied and the form of the corresponding Kähler curvature tensor is obtained. Some curvature properties of the manifolds belonging to the widest integrable main class of the considered type of hypersurfaces are given.

In this paper we study submanifolds of almost contact manifolds with Norden metric of codimension two with totally real normal spaces. Examples of such submanifolds as a Lie subgroups are constructed.

This short communication deals with several properties of the double-complex Laplace operator, namely it is constructed a fundamental distribution solution, it is proved the partial C^∞ hypoellipticity and it is proved a theorem on microlocal propagation of singularities of the solutions along the leaves (two dimensional) of the real and imaginary parts of the Hamiltonian vector field of the symbol.

The aim of this paper is to make an overview on three generalizations to higher dimensions of the function's theory of a complex variable. The first one concerns the so-called monogenic functions which were introduced by F. Brackx, R. Delanghe and F. Sommen, the second is the theory of hyper-monogenic functions developed by H. Leutwiler, and the last one studies the theory of holomorphic Cliffordian functions due to G. Laville and I. Ramadanoff. The basic notions in this three theories will be given. Their links and differences will also be commented.

Some classes of exact solutions of eikonal equation has been found by using some decomposition method. There is also prsented its application to quantum mechanics.

We take a convex pentagon R = ABCDE satisfying , and ∠B = ∠E = π/2. By use of 4 pentagons congruent to R we make a fundamental hexagon and give a tessellation of a Euclidean plane, which is called a tessellation of tiling-type 4 by congruent pentagons. This tessellation has 3-valent and 4-valent vertices. We study Dirichlet property for such tessellations of a plane. We also consider similar tessellations by congruent original and reversed pentagons which have fundamental regions made by 4 pieces and study their Dirichlet property.

A four-parametric family of linear connections preserving the almost complex structure is defined on an almost complex manifold with Norden metric. Necessary and sufficient conditions for these connections to be natural are obtained. A two-parametric family of complex connections is studied on a conformal Kähler manifold with Norden metric. The curvature tensors of these connections are proved to coincide.

Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.

We present new examples of multicomponent mKdV type equations associated with symmetric spaces of BD.I series. These are integrable equations obtained by using the method of reduction group. Their 1-soliton solutions are derived as well by applying the dressing procedure with an appropriately chosen dressing factor.

Let the curvature of a plane curve parametrized by arclength s be given as a function of the Cartesian co-ordinates of the points the curve is passing through in the Euclidean plane. Then, the co-ordinates of its position vector are determined by a system of equations arising from the Frenet-Serret relations that can be regarded as a dynamical system of two degrees of freedom determining the motion (trajectories) of a particle of unit mass, s playing the role of time. Here, two classes of integrable systems of the foregoing type are identified. For that purpose, we explore the variational symmetries of a generic system of this kind with respect to Lie groups of point transformations of the involved variables. As a result, a set of sufficient conditions are found which ensure that such a system possesses two functionally independent integrals of motion and, consecutively, is integrable by quadratures. In each such case, we achieve either an explicit parameterization of the corresponding trajectory curves in terms of their curvatures or, at least, a separation of the dependent variables.

The following section are included:

• AUTHOR INDEX