Isoptic surfaces of polyhedra

The theory of the isoptic curves is widely studied in the Euclidean plane \bE^2 (see \cite{CMM91} and \cite{Wi} and the references given there). The analogous question was investigated by the authors in the hyperbolic \bH^2 and elliptic \cE^2 planes (see \cite{CsSz1}, \cite{CsSz2}, \cite{CsSz5}), but in the higher dimensional spaces there are only a few result in this topic. In \cite{CsSz4} we gave a natural extension of the notion of the isoptic curves to the $n$-dimensional Euclidean space \bE^n $(n\ge 3)$ which are called isoptic hypersurfaces. Now we develope an algorithm to determine the isoptic surface \mathcal{H}_{\cP} of a $3$-dimensional polytop $\mathcal{P}$. We will determine the isoptic surfaces for Platonic solids and for some semi-regular Archimedean polytopes and visualize them with Wolfram Mathematica

Isoptic curves of conic sections in constant curvature geometries

In this paper we consider the isoptic curves on the 2-dimensional geometries of constant curvature \bE^2,~\bH^2,~\cE^2. The topic is widely investigated in the Euclidean plane \bE^2 see for example \cite{CMM91} and \cite{Wi} and the references given there, but in the hyperbolic and elliptic plane there are few results in this topic (see \cite{CsSz1} and \cite{CsSz2}). In this paper we give a review on the preliminary results of the isoptics of Euclidean and hyperbolic curves and develop a procedure to study the isoptic curves in the hyperbolic and elliptic plane geometries and apply it for some geometric objects e.g. proper conic sections. We use for the computations the classical models which are based on the projectiv interpretation of the hyperbolic and elliptic geometry and in this manner the isoptic curves can be visualized on the Euclidean screen of computer

A bounded jump for the bounded Turing degrees

We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=_[bT] 0^[nb] iff X is omega^n-c.e. iff X <=_1 0^[nb], extending the classical result that X <=_[bT] 0' iff X is omega-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that 0^b <=_[bT] X <=_[bT] 0^[2b], there is a Y <=_[bT] 0^b such that Y^b =_[bT] X.Comment: 22 pages. Minor changes for publicatio

THE ROLE OF HEALTH BEHAVIOUR AND SENSE OF COHERENCE IN FORMING THE QUALITY OF LIFE Qualitative and Quantitative Analysis of Subjective Well-Being, Health Behaviour and Sense of Coherence Among High School Students

17