Sławomir Wolski

The spin-dependent scattering process in a system of a topological insulator and a quantum dot is studied. The unitary scattering process is viewed as a gate transformation applied to an initial state of two electrons. Due to the randomness imposed through the impurities and alloying-induced effects of band parameters, the formalism of the random unitary gates is implemented. For quantifying entanglement in the system, we explored concurrence and ensemble-averaged Rényi entropy. We found that an applied external magnetic field leads to long-range entanglement on distances much larger than the confinement length. We showed that topological features of itinerant electrons sustain the formation of robust long-distance entanglement, which survives even in the presence of strong disorder.

Currently, medicine uses typical industrial structure techniques, including reverse engineering, data processing, 3D-CAD modeling, 3D printing, and coordinate measurement techniques. Taking this into account, one can notice the applications of procedures used in the aviation or automotive industries based on the structure of Industry 4.0 in the planning of operations and the production of medical models with high geometric accuracy. The procedure presented in the publication shortens the processing time of tomographic data and increases the reconstruction accuracy within the hip and knee joints. The procedure allows for the partial removal of metallic artifacts from the diagnostic image. Additionally, numerical models of anatomical structures, implants, and bone cement were developed in more detail by averaging the values of local segmentation thresholds. Before the model manufacturing process, additional tests of the PLA material were conducted in terms of its strength and thermal properties. Their goal was to select the appropriate type of PLA material for manufacturing models of anatomical structures. The numerical models were divided into parts before being manufactured using the Fused Filament Fabrication technique. The use of the modifier made it possible to change the density, type of filling, number of counters, and the type of supporting structure. These treatments allowed us to reduce costs and production time and increase the accuracy of the printout. The accuracy of the manufactured model geometry was verified using the MCA-II measuring arm with the MMDx100 laser head and surface roughness using a 3D Talyscan 150 profilometer. Using the procedure, a decrease in geometric deviations and amplitude parameters of the surface roughness were noticed. The models based on the presented approach allowed for detailed and meticulous treatment planning.

The application of anatomical models and surgical templates in maxillofacial surgery allows, among other benefits, the increase of precision and the shortening of the operation time. Insufficiently precise anastomosis of the broken parts of the mandible may adversely affect the functioning of this organ. Applying the modern mechanical engineering methods, including computer-aided design methods (CAD), reverse engineering (RE), and rapid prototyping (RP), a procedure used to shorten the data processing time and increase the accuracy of modelling anatomical structures and the surgical templates with the use of 3D printing techniques was developed. The basis for developing and testing this procedure was the medical imaging data DICOM of patients treated at the Maxillofacial Surgery Clinic of the Fryderyk Chopin Provincial Clinical Hospital in Rzeszów. The patients were operated on because of malignant tumours of the floor of the oral cavity and the necrosis of the mandibular corpus, requiring an extensive resection of the soft tissues and resection of the mandible. Familiarity with and the implementation of the developed procedure allowed doctors to plan the operation precisely and prepare the surgical templates and tools in terms of the expected accuracy of the procedures. The models obtained based on this procedure shortened the operation time and increased the accuracy of performance, which accelerated the patient’s rehabilitation in the further course of events.

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions.

We consider 2- and 3-dimensional cubic monocrystalline and polycrystalline materials. Expressions for Young's and shear moduli and Poisson's ratio are expressed in terms of eigenvalues of the stiffness tensor. Such a form is well suited for studying properties of these mechanical characteristics on sides of the stability triangles. For crystalline high-symmetry directions lines of vanishing Poisson's ratio are found. These lines demarcate regions of the stability triangle into areas of various auxeticity properties. The simplest model of polycrystalline 2D and 3D cubic materials is considered. In polycrystalline phases the region of complete auxetics is larger than for monocrystalline materials.