Watched guards in art galleries

The art gallery problem asks how many guards are sufficient to see every point of the interior of a polygon. A set of guards is called watched if each guard itself is seen by at least one of its colleagues. In 1994, Hernández-Pe?alver wrote that watched guards always suffice to guard any polygon with n vertices. However in 2001, Michael and Pinciu, and independently Żyliński, presented a class of polygons that required more than watched guards – which disproved the Hernández-Pe?alver’s result – and they established a new tight bound for watched guards: . Combinatorial bounds for watched guards in orthogonal polygons were independently given by Hernández-Pe?alver , and by Michael and Pinciu, who proved the -bound to be tight. In this paper, tight bounds for polygons of miscellaneous shapes are presented: watched guards for monotone and spiral polygons, and vertex watched guards for star-shaped polygons.     

Theoretical interpretation of the zero-field splitting parameters for Fe3+ ions in wide-band gap semiconductor TlGaSe2 single crystal

Superposition model (SPM) calculations are carried out to provide theoretical interpretation of the zero-field splitting (ZFS) parameters and investigate the local environment around the Fe³⁺ centers in a TlGaSe₂ single crystal. Experimental electron magnetic resonance (EMR) data are analyzed and compared with the ZFS parameter values predicted by SPM based on the orthorhombic approximation of structural data. The results provide an adequate interpretation of the ZFS parameters obtained by fitting EMR spectra and indicate that Fe³⁺ ions substitute for the Ga³⁺ ions in TlGaSe₂ crystal.     

Optical birefringence and molecular orientation of electrospun polycaprolactone fibers by polarizing-interference microscopy

The potential of polarizing-interference Pluta microscope for determination of optical birefringence of individual nanofibers formed by electrospinning was shown. This technique can be applied for measurements of fiber birefringence, practically at diameter above 300 nm. The molecular orientation of individual polycaprolactone (PCL) nanofibers was determined from birefringence assuming the same orientation of both phases, crystal and amorphous. The molecular orientation was determined using DSC crystallinity, crystal intrinsic birefringence calculated for the first time for PCL from bond polarizabilities as well as estimated value of amorphous intrinsic birefringence. Our results indicate that the birefringence and thus molecular orientation are strongly inhomogeneous along the nanofibers, reflecting a complex nature of forces acting during electrospinning process. The average molecular orientation is weak if any, being dependent together with fiber thickness and crystallinity on electrospinning parameters, like applied voltage, concentration and type of solvent. The obtained results indicate that the average molecular orientation displays similar dependence on applied voltage as fiber diameter. Relatively low melting temperature of electrospun nanofibers suggests low crystal size and/or high concentration of defects in crystals. This observation corresponds with low crystallinity and molecular orientation, indicating together relatively low degree of crystal ordering due to high rate of cooling and solvent evaporation during electrospinning, limiting thus crystallization process.     

Polynomial approximation of rational Bézier curves with constraints

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.     

A tabu search algorithm for the minmax regret minimum spanning tree problem with interval data

This paper deals with the strongly NP-hard minmax regret version of the minimum spanning tree problem with interval costs. The best known exact algorithms solve the problem in reasonable time for rather small graphs. In this paper an algorithm based on the idea of tabu search is constructed. Some properties of the local minima are shown. Exhaustive computational tests for various classes of graphs are performed. The obtained results suggest that the proposed tabu search algorithm quickly outputs optimal solutions for the smaller instances, previously discussed in the existing literature. Furthermore, some arguments that this algorithm performs well also for larger instances are provided.     

General theorem for the existence of iterative roots of homeomorphisms with periodic points

We apply Zdun’s factorization theorem (see Zdun (2008) [3]) to give the conditions for the existence and the form of continuous and orientation-preserving iterative roots of homeomorphisms of the circle with a rational rotation number. Our theorem generalizes the previous results given by Jarczyk (2003) in [2], Zdun (2008) in [3] and Solarz (2003, 2009) in [4] and [5].     

Towards a q-analogue of the Kibble–Slepian formula in 3 dimensions

We study a generalization of the Kibble–Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q-Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3-dimensional Normal distribution. We show that this is not the case. Nevertheless we indicate other applications of so-generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam–Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey–Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2-dimensional Poisson–Mehler formula. We also pose several open questions.     

Effects of current localization in high-power microgate bipolar switches with nonideal interconnection between controlled cells

The influence of nonideal interconnection between controlled cells in bipolar microgate switches on current localization at the turn-off stage is analyzed. To estimate the resistance of the distributed electrodes, the entire device is represented as two parallel subsystems of controlled cells interconnected via the effective resistance of the gate circuit. Different scenarios of the turn-off process at nonideal couplings between the cells are numerically simulated for three turn-off circuit regimes by the example of an integrated thyristor with external field control. The turn-off scenarios are studied versus the effective resistance of the gate-off circuit and the ratio between the working surface areas of the cellular subsystems. Limitations on the ultimate switched current are compared. For small-scale inhomogeneities in the series resistance of the gate circuit, the maximum turn-off current in the cascode mode and using a negative voltage source remains an order of magnitude higher than that under the emitter short-circuit conditions.