The Lax formulation of the hyper-Hermiticity condition in four dimensions is used to derive a potential that generalises Plebanski’s second heavenly equation for hyper-Kahler 4-manifolds. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a com...

In this paper we present a novel method for motion generation task specification for spherical mechanisms. This is accomplished with a new methodology for determining the optimal design sphere and the orientations on this design sphere for a finite set of desired spatial positions. In addition, we include a modification to the method which enables the designer to require that one of the n desir...

We study complexified Harmonic Oscillator with a position dependent mass, referred to in the present paper as Exotic Oscillator (EO), in arbitrary dimensions. The real space EO has an interesting dynamics: in the equation of motion the full Hamiltonian operator appears in place of the frequency parameter in a Harmonic Oscillator. We reveal some generic features in the constraint structure of th...

Let V be a finite dimensional complex vector space and W ⊆ GL(V ) be a finite complex reflection group. Let V reg be the complement in V of the reflecting hyperplanes. We prove that V reg is a K(π, 1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection group. The complexified real case follows from a theorem of Deligne and, after ...

Real and complex norms of a linear operator acting on a normed complexified space are considered. Bounds on the ratio of these norms are given. The real and complex norms are shown to coincide for four classes of operators: 1. real linear operators from Lp(μ1) to Lq(μ2), 1 ≤ p ≤ q ≤ ∞; 2. real linear operators between inner product spaces; 3. nonnegative linear operators acting between complexi...

Many quaternion numbers associated with Fibonacci and Lucas or even their generalizations have been defined widely discussed so far. In all the studies, coefficients of these quaternions selected from consecutive terms numbers. this study, we define other for usual quaternions. We also present some properties, including Binet's formulas d'Ocagne's identities, types quaternions.

The effect of the built-in supersymmetric quantum mechanical language on the spectrum of the (1+1)-Dirac equation, with position-dependent mass (PDM) and complexified Lorentz scalar interactions, is re-emphasized. The signature of the ”quasi-parity” on the Dirac particles’ spectra is also studied. A ”quasi-free” Dirac particle with PDM via an inversely linear plus linear model, a Dirac particle...

The effect of the built-in supersymmetric quantum mechanical language on the spectrum of the (1+1)-Dirac equation, with position-dependent mass (PDM) and complexified Lorentz scalar interactions, is re-emphasized. The signature of the ”quasi-parity” on the Dirac particles’ spectra is also studied. A Dirac particle with PDM and complexified scalar interactions of the form S (z) = S (x− ib) (an i...