We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of pseudo-triangulations which was useful for implicit solution of the carpenter’s rule problem and proved later to give a nice tool for graph embeddings. On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for con...

Let $$\Omega _+$$ be either the open unit disc or upper half plane right plane. In this paper, we compute norm of basic operator $$A_\alpha =\Pi _\Theta T_{b_\alpha }|_{\mathcal {H}(\Theta )}$$ in vector valued model space $$\mathcal )=H^m_2 \ominus \Theta H^m_2$$ associated with an $$m\times m$$ matrix inner function $$\Theta $$ and show that is attained. Here $$\Pi denotes orthogonal projecti...

In this paper, we continue the study of contractive conditions for mappings in complete partial metric spaces. Concretely, we present fixed point results for weakly contractive and weakly Kannan mappings in such a way that the classical metric counterpart results are retrieved as a particular case. Special attention to the cyclical case is paid. Moreover, the well-posedness of the fixed point p...

For a cardinal of the form κ = בκ, Shelah’s logic L 1 has characterisation as maximal above $${ \cup _{\lambda < \kappa }}{L_{\lambda ,\omega }}$$ satisfying strengthening undefinability well-order. Karp’s chain [20] is known to satisfy well-order and interpolation. We prove that if singular countable cofinality, . Moreover, we show strong limit cardinals $$L_{ , }^c{ }}L_{\lambda ,\lambda }^c$...

We find concrete necessary and sufficient conditions for the existence of contractive completions of Hankel partial contractions of size 4×4 in the extremal case. Along the way we introduce a new approach that allows us to solve, algorithmically, the contractive completion problem for 4 × 4 Hankel matrices. As an application, we obtain a concrete example of a partially contractive 4× 4 Hankel m...

BACKGROUND Fuchs' uveitis (FU) is occasionarlly complicated with heavy vitreous opacity. We have performed vitrectomy procedures to remove vitreous opacity in affected patients as part of differential diagnosis for primary vitreoretinal lymphoma (PVRL). CASE PRESENTATION We retrospectively reviewed the clinical records of five patients who first visited the Uveitis Clinic of the University of...

Barnsley (2006) introduced the notion of a fractal top, which is an addressing function for the set attractor of an Iterated Function System (IFS). A fractal top is analogous to a set attractor as it is the fixed point of a contractive transformation. However, the definition of IFS is extended so that it works on the colour component as well as the spatial part of a picture. They can be used to...

Let Ω be a bounded C∞-smoothly bounded domain in C. For such a domain we define a new notion between strict pseudo-convexity and pseudo-convexity: the size of the set W of weakly pseudo-convex points on ∂Ω is small with respect to Minkowski dimension: near each point in the boundary ∂Ω, there is at least one complex tangent direction in which the slices of W has a upper Minkowski dimension stri...