Three-pencil lattices on triangulations

In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S ₃ on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.      

Membrane switch hypothesis. 1. Cell density influences lateral domain structure of tumor cell membranes

The domain structure of human cancer cells membranes was investigated by electron paramagnetic resonance (EPR) in different phases of cell growth, and the results were compared to those obtained for nonmalignant cells. On the basis of computer simulation of the EPR spectra using a newly developed GHOST condensation routine it was suggested that plasma membranes of cancer cells have less lateral lipid domain types at confluent conditions than in the exponential growing phase, while in nonmalignant cells the domain structure does not change significantly during cell growth. In accordance to our experimental data we propose a membrane switch hypothesis: disappearance of certain membrane domain types might act as a switch promoting the clustering of membrane constituents into the active units in a common lipid membrane domain and thus influencing the physiology of cells.     

Lattices on simplicial partitions

In this paper, (d+1)-pencil lattices on simplicial partitions in R^d are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of this fact leads to an efficient computer algorithm for the design of a lattice.     

The structure of irreducible matrix groups with submultiplicative spectrum

We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p ²× p ² matrices if p is an odd prime.     

Approximation of circular arcs by parametric polynomial curves

In this paper the approximation of circular arcs by parametric polynomial curves is studied. If the angular length of the circular arc is h, a parametric polynomial curve of arbitrary degree \(n \in {\mathbb{N}}\) , which interpolates given arc at a particular point, can be constructed with radial distance bounded by h ²ⁿ . This is a generalization of the result obtained by Lyche and Mørken for odd n.     

Interpolation by $G^2$ Quintic Pythagorean-Hodograph Curves

In this paper, the $G^2$ interpolation by Pythagorean-hodograph (PH) quintic curves in $\mathbb{R}^d$, $d ≥2$, is considered. The obtained results turn out as a useful tool in practical applications. Independently of the dimension $d$, they supply a $G^2$ quintic PH spline that locally interpolates two points, two tangent directions and two curvature vectors at these points. The interpolation problem considered is reduced to a system of two polynomial equations involving only tangent lengths of the interpolating curve as unknowns. Although several solutions might exist, the way to obtain the most promising one is suggested based on a thorough asymptotic analysis of the smooth data case. The numerical algorithm traces this solution from a particular set of data to the general case by a homotopy continuation method. Numerical examples confirm the efficiency of the proposed method.     

Symbolic dynamics in investigation of quaternionic Julia sets

We focus our attention on the dynamics of the simplest quaternionic quadratic function f_Q(X) = X² + Q. The discussion can be reduced to a complex parameter Q and a three dimensional subspace. The images of quaternionic Julia sets suggest a natural decomposition. We find that it can be derived from a certain symbolic dynamics giving rise to fractal fibrations. The starting point are the equators and their preimages. If the parameter Q is real, fibrations are trivial, obtained by rotation of the complex Julia set. Repeating itineraries, on the other hand, define curves connecting periodic points.     

Integrated endoscopy system for simultaneous imaging and spectroscopy for early lung cancer detection

An integrated endoscopy system for simultaneous imaging and spectroscopy was developed to facilitate more accurate and convenient detection of early lung cancers. A specially designed three-CCD camera in combination with a dedicated light source permits capture of both white-light color images and tissue autofluorescence images without the need to switch between two different cameras. A mirror with an optical fiber at its center, placed at an interim imaging plane inside the camera unit, facilitates simultaneous imaging and spectroscopy measurements in either white-light reflectance mode or fluorescence mode. The system has been successfully tested in a clinic, demonstrating a practical approach to improve both diagnostic sensitivity and specificity at the same time.