The Gauss map of submanifolds in the Heisenberg group

We obtain criteria for the harmonicity of the Gauss map of submanifolds in the Heisenberg group and consider the examples demonstrating the connection between the harmonicity of this map and the properties of the mean curvature field. Also, we introduce a natural class of cylindrical submanifolds and prove that a constant mean curvature hypersurface with harmonic Gauss map is cylindrical.     

Harmonic almost contact structures via the intrinsic torsion

We proceed further in the study of harmonicity for almost contact metric structures already initiated by Vergara-Díaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric structures. Additionally, we study the harmonicity of such structures as a map into the quotient bundle of the oriented orthonormal frames by the action of the structural group U(n)×1. Finally, by using a Bochner type formula proved by Bör and Hernández Lamoneda, we display some examples which give the absolute minimum for the energy.     

Harmonicity of the inverse of a harmonic diffeomorphism

In this paper we obtain a necessary and sufficient condition for the harmonicity of the inverse of a harmonic diffeomorphism. As an application, we give explicit representations of reversible logharmonic diffeomorphisms. As another application, we connect the harmonicity of the inverse of a hyperbolic harmonic quasiconformal diffeomorphism with the Schoen conjecture.     

Eigenvalues of harmonic almost submersions

Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues yield characterizations of harmonicity, totally geodesic maps and biconformal changes of metric preserving harmonicity. A Schwarz lemma for pseudo harmonic morphisms is proved, using the dilatation of the eigenvalues and, in dimension five, a Bochner technique method, involving the Laplacian of the difference of the eigenvalues, gives conditions forcing pseudo harmonic morphisms to be harmonic morphisms.     

On harmonic and 2-stein spaces of Iwasawa type

We study geometric properties of solvable metric Lie groups S of Iwasawa type; in particular harmonicity and the 2-stein condition. One restriction we obtain is that harmonic spaces of Iwasawa type have algebraic rank one, that is, the commutator subgroup of S has codimension one.We show that among Carnot solvmanifolds the only harmonic spaces are the Damek–Ricci spaces. Moreover, this rigidity result remains valid if harmonicity is replaced by the weaker 2-stein condition. As an application, we show that a harmonic Lie group of Iwasawa type with nonsingular 2-step nilpotent commutator subgroup is, up to scaling, a Damek–Ricci space.     

Generating a reference trajectory with defined kinematics for the IRb-6 manipulator

The fundamental problem in industrial robots control concerns algorithms generating reference trajectories.References [1–4] suggest generating algorithms of a reference trajectory, which are based on an arbitrary discretization of the manipulator's internal coordinates. Each point of discretization in the external space approximating a reference trajectory corresponds to known discretized internal coordinates of the manipulator.In [5–7], iteration methods of determining the internal coordinates corresponding to external coordinates of the reference trajectory point have been suggested. In this method of internal coordinates, determining the point of the reference trajectory is being approached in successive steps of an iterative computation. In [5], a modified iterative method of generation of a straight segment reference trajectory has been presented.Analytic formulae, which are the solution of an inverse problem of manipulator kinematics, enable design of trajectory generating algorithms which compute, in one step only, the internal coordinates of points lying exactly on the reference trajectory, with the accuracy resulting from the computer register length.In this paper, the author has presented an original PLAN2 computer algorithm generating reference trajectories of motion for a task. The kinematics of those trajectories is defined at selected points through which a task is to be passed, the distances between them being optional. The algorithm is based on formulae which are analytic solutions to an inverse problem for an IRb-6 manipulator kinematics.     

Analytic and harmonic maps into a topological space

The relationship between the harmonicity and analyticity of a continuous map from the open unit disc to the underlying space of a real algebra is investigated.     

Supply of harmonic functions of an infinite number of variables. III

We obtain optimal (in a certain sense) harmonicity conditions on functions on a Hilbert space which follow from estimates for sums of independent random variables. Together with the harmonicity conditions obtained earlier, based on estimates of the order of growth for sums of dependent random variables and for sums of orthogonal random variables, they make it possible to consider new classes of harmonic functions of an infinite number of variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 417–423, March, 1992.     

Supply of harmonic functions of an infinite number of variables. II

There are exceptionally many harmonic functions of an infinite number of variables. Using for the estimate of the infinite-dimensional Laplacian introduced by P. Levy, estimates of the germ of sums of orthogonal random variables, there are obtained optimal (in a certain sense) conditions of the harmonicity of the functions in a Hilbert space. Along with harmonicity conditions obtained earlier based on estimates of the germ of sums of dependent random variables, they allow one to encompass the manifold of harmonic functions of an infinite number of variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1687–1693, December, 1990.     

On the interplay between the Hilbert transform and conjugate harmonic functions

As is well‐known, there is a close and well‐defined connection between the notions of Hilbert transform and of conjugate harmonic functions in the context of the complex plane. This holds e.g. in the case of the Hilbert transform on the real line, which is linked to conjugate harmonicity in the upper (or lower) half plane. It also can be rephrased when dealing with the Hilbert transform on the boundary of a simply connected domain related to conjugate harmonics in its interior (or exterior). In this paper, we extend these principles to higher dimensional space, more specifically, in a Clifford analysis setting. We will show that the intimate relation between both concepts remains, however giving rise to a range of possibilities for the definition of either new Hilbert‐like transforms, or specific notions of conjugate harmonicity. Copyright © 2006 John Wiley & Sons, Ltd.     

Exact discrete compactlike breathers in soft φ4 potentials

We demonstrate the existence of exact discrete compact breather solutions in nonlinear Klein–Gordon systems. We show numerically that the breathers stability is related to the lattice boundary conditions, and to the coupling term, and the harmonicity parameter. Finally, we propose an electronic device allowing to verify experimentally these theoretical predications.     

Accelerated reference systems in AdS space

We consider reference systems of uniformly accelerated observers in anti-de Sitter space. We construct coordinate transformations for the transition from an inertial reference system to a uniformly accelerated reference system for all acceleration values, both greater and less than critical. The basis for the construction are the Beltrami coordinates, natural coordinates for describing a uniformly accelerated motion because geodesics in anti-de Sitter space in these coordinates become straight lines, i.e., can be described by linear functions. Because translations of space–time coordinates in anti-de Sitter space are non-Abelian, a nontrivial problem of defining the comoving inertial reference system arises. Constructing the coordinate system of an accelerated observer using this auxiliary comoving inertial reference system requires additional transformations that not only equalize the velocities of the two systems but also equalize the system origins. The presence of a critical acceleration in anti-de Sitter space leads to a difference in explicit expressions in passing to an accelerated coordinate system for accelerations greater and less than critical.     

Harmonic Almost Complex Structures with Respect to General Natural Metrics

We continue here the study initiated in [9] on the harmonicity of certain geometric objects on the total space TM of the tangent bundle of a Riemannian space form (M(c), g). Precisely, in this paper we find all the general natural metrics on TM, with respect to which the canonical almost complex structure J on TM is harmonic. We also study the harmonicity of this tensor field with respect to the natural diagonal metrics. In particular, we obtain that J is harmonic with respect to the Sasaki metric on TM if and only if the base manifold is flat.     

Self-dual connections,hyperbolic metrics and harmonic mappings on Riemann surfaces

The Sampson-Wolf model of Teichmüller space (using harmonic mappings) is shown to be exactly the same as the more recent Hitchin model (utilizing self-dual connections). Indeed, it is noted how the self-duality equations become the harmonicity equations. An interpretation of the modular group action in this model is mentioned.     

Two parameter optimal stopping and bi-Markov processes

Summary In this paper, the optimal stopping problem is solved for particular two-parameter processes, here called bi-Markov processes. A subsequent potential theory is developed with respect to a pair of one-parameter semi-groups. We introduce a new notion of harmonicity for two-variable functions and we interpret it in the framework of the theory of bi-Markov processes.     

Closed forms transverse to singular foliations

We describe a homological criterion for the existence of a closed form transverse to a foliation which is allowed to have certain tame singularities, generalizing results in the nonsingular case by Sullivan. As illustrations of the method, we derive results about the existence of singular symplectic forms and about the characterization of intrinsic harmonicity for forms.     

𝒢-constructibility of planar graphs

In this paper, the concept of the ??-constructibility of graphs is introduced and investigated with particular reference to planar graphs. It is conjectured that the planar graphs are minimally N-constructible, where N is a finite set of graphs and an infinite set ?? is obtained such that the planar graphs are also minimally ??-constructible. Finally, some properties of the set of all N-constructible graphs are discussed and compared with the corresponding properties of planar graphs.     

Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Necessary and sufficient conditions of harmonicity of functions of infinitely many variables (Jacobian case)

A criterion of harmonicity of functions in a Hilbert space is given in the case of weakened mutual dependence of the second derivatives. Ukrainian Scientific and Industrial Wood Working Concern, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6 pp. 785–788, June, 1994     

Criteria for (Sub-)Harmonicity and Continuation of (Sub-)Harmonic Functions

We obtain criteria for harmonicity and subharmonicity of a function in a domain in R ^d , d2, in terms of special Arens–Singer and Jensen measures. We also establish a criterion for (sub-)harmonicity of a -subharmonic function in terms of the associated Riesz charge and special Arens–Singer and Jensen functions. To this end, we use the theorem of this article on continuation of (sub-)harmonic functions to polar sets.