Scattering functions of knotted ring polymers

We discuss the scattering function of a Gaussian random polygon with N nodes under a given topological constraint through simulation. We obtain the Kratky plot of a Gaussian polygon of N=200 having a fixed knot for some different knots such as the trivial, trefoil and figure-eight knots. We find that some characteristic properties of the different Kratky plots are consistent with the distinct values of the mean square radius of gyration for Gaussian polygons with the different knots.Comment: 4pages, 3figures, 3table

On the Limits of Analogy Between Self-Avoidance and Topology-Driven Swelling of Polymer Loops

The work addresses the analogy between trivial knotting and excluded volume in looped polymer chains of moderate length, $N<N_0$, where the effects of knotting are small. A simple expression for the swelling seen in trivially knotted loops is described and shown to agree with simulation data. Contrast between this expression and the well known expression for excluded volume polymers leads to a graphical mapping of excluded volume to trivial knots, which may be useful for understanding where the analogy between the two physical forms is valid. The work also includes description of a new method for the computational generation of polymer loops via conditional probability. Although computationally intensive, this method generates loops without statistical bias, and thus is preferable to other loop generation routines in the region $N<N_0$.Comment: 10 pages, 5 figures, supplementary tex file and datafil

Abundance of unknots in various models of polymer loops

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of $N$ segments follows a decaying exponential form, $\sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of $N_0$ for a variety of polymer models. Among models examined, $N_0$ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.Comment: 13 pages, 6 color figure

Digital Technologies in Teacher Training: New Experience

In the field of training higher educational institution teachers, an era of significant change is beginning. For long, this area has remained the most conservative and was not noticeably subject to changes in society. However, the situation has changed dramatically. First of all, this is due to the rapid development of other areas of human activity, and in particular digital technologies. It is already difficult to imagine a modern society without such technologies, and the main task of a higher education teacher is precisely to train highly qualified specialists with new competencies to solve the challenges facing humanity. Strengthening the explicit link between the needs of society and the objectives of higher education is one of the prerequisites for significant changes in the field of education. The concept of training a teacher of higher education for the future today is largely determined by the development trends of digital technologies. The absence of objective reasons to assume the cessation of the use and development of digital technologies allows us to make an assumption that virtual educational worlds will appear in the foreseeable future, which will become the same familiar environment for acquiring the necessary competencies as higher schools existing in the physical world. This largely determines the relevance of the chosen research topic and determines the need for a thorough analysis of the educational technologies used in the training of teachers. The purpose of this work is, using a combination of analysis, synthesis, idealization and analogy, to generalize at the national and international levels the existing ideas about effective approaches to teaching a higher education teacher, including continuous improvement of competence in his professional field, and modern practice of using digital technologies, and also highlight the most interesting new learning experiences. The necessity of applying the so-called “integrated approach to teaching a higher education teacher” by the authors is proved. The main advantages and disadvantages of using digital technologies are highlighted. Reasonable conclusions are made on the existence of an ideal recipe for combining accumulated educational practices in the framework of traditional education and digital technology

Optimal speed of temperature change of a crystal in a pyroelectric X-ray radiation source

In this work we present the results of the experimental studies of the dependence of the X-ray radiation on the temperature change speed of the lithium tantalate monocrystal in the pyroelectric source of the X-ray radiation. We have found an optimized linear temperature speed change pattern for the pyroelectric crystal designed for generation of the X-ray radiation with enhanced capabilities. The studies that were made open an opportunity to determine the most convenient conditions for generation of X-ray radiation with pyroelectric effect and for development of high-capacity pyroelectric X-ray radiation sources