Given an oriented immersed hypersurface in hyperbolic space H n + 1 $\mathbb {H}^{n+1}$ , its Gauss map is defined with values the of geodesics which endowed a natural para-Kähler structure. In this paper, we address question whether immersion G $G$ universal cover $n$ -manifold M $M$ equivariant for some group representation π ( ) $\pi _1(M)$ Isom $\mathrm{Isom}(\mathbb {H}^{n+1})$ . We fully ...

A random system with complete connections associated with a piecewise fractional linear map with explicit invariant measure is defined and its ergodic behaviour is investigated. This allows us to obtain a variant of Gauss-Kuzmin type problem for the above linear map.

In this paper, we present the theoretical studies of a refractive index map to implement a Gauss to a J(0)-Bessel-Gauss convertor. We theoretically demonstrate the viability of a device that could be fabricated on a Si/Si(1-y)O(y)/Si(1-x-y)Ge(x)C(y) platform or by photo-refractive media. The proposed device is 200 μm in length and 25 μm in width, and its refractive index varies in controllable ...

In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern–Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely rela...

Many applications, such as contour machining, rapid prototyping, and reverse engineering by laser scanner or coordinate measuring machine, involve sampling of free-from surfaces along section cuts by a family of parallel planes with equidistant spacing ∆ and common normal N. To ensure that such planar sections provide faithful descriptions of the shape of a surface, it is desirable to choose th...

In contour machining of free{form surfaces with a ball{end cutter, the tool contacts the surface along loci that correspond to sections by a set of parallel planes, with equidistant spacing and common normal N. For a given tool radius, the greatest scallop height between successive tool paths is suppressed by choosing an orientation for the contouring planes that will maximize, over the entire ...

We show that the Lyapunov exponent (LE) of periodic orbits with Lebesgue measure zero from the Gauss map can be used to determine the main qualitative behavior of the LE of a Hamiltonian system. The Hamiltonian system is a one-dimensional box with two particles interacting via a Yukawa potential and does not possess Kolmogorov-Arnold-Moser (KAM) curves. In our case the Gauss map is applied to t...