Issue No. 2 (2025)

This introductory  review is devoted to the newest section of the theory of symmetries -- the theory of quantum groups.
The principles of the theory of quantum groups are reviewed from the point of view of the possibility of their use for deformations of symmetries in physics models. The R-matrix approach to the theory of quantum groups is discussed in detail and is taken as the basis of the quantization of classical Lie groups, as well as some Lie supergroups. We start by laying out the foundations of non-commutative and non-cocommutative Hopf algebras. Much attention has been paid to Hecke and Birman-Murakami-Wenzl (BMW) R-matrices and related quantum matrix algebras. Noncommutative differential geometry on quantum groups of special types is discussed. Trigonometric solutions of the Yang-Baxter equations associated with the quantum groups GL_q(N), SO_q(N), Sp_q(2n) and supergroups GL_q(N|M), Osp_q(N|2m), as well as their rational (Yangian) limits, are presented. Rational R-matrices for exceptional Lie algebras and elliptic solutions of the Yang-Baxter equation are also considered. The basic concepts of the group algebra of the braid group and its finite dimensional quotients (such as Hecke and BMW algebras) are outlined. A sketch of the representation theories of the Hecke and BMW algebras is given, including methods for finding idempotents  (quantum Young projectors) and their quantum dimensions. Applications of the theory of quantum groups and Yang-Baxter equations in various areas of theoretical physics are briefly discussed.

This is a modified version of the review paper published in 2004 as a preprint  of the Max-Planck-Institut für Mathematik in Bonn.

The non-Markovian two-dimensional dynamics of charge carriers in a dissipative non-magnetic medium is studied. The possibility of observing a new classical Hall-type effect in the absence of a magnetic field is predicted.

The amyloid-beta peptide (Aβ peptide) is proposed to play a central role in the onset of Alzheimer’s disease (AD). The pathology is associated with the fast accumulation of neurotoxic amyloid aggregates in brain tissues, though the fundamentals of the disease’s progression remain unsolved. It is noted that the preclinical stage of AD may play a crucial role in its further irreversible development. Namely, interactions between lipid membranes and Aβ-peptide molecules incorporated therein at relatively low concentrations should be under a close attention. In this review, we discuss recent works devoted to studying the lipid peptide interactions with a specific focus on the lipid membrane reorganizations caused by Aβ (25–35) peptide in the preclinical AD mimicking conditions. The interactions observed are believed to be important in understanding the mechanisms of the Aβ-peptide destructive effects on lipid membranes and the corresponding onset of the disease. The methods of applied nuclear physics have proven remarkably relevant in such research. The scattering methods provided instrumental information on a level of supramolecular assemblies, while spectrometry allowed obtaining information on the molecular level. Finally, molecular dynamics simulations provided details unachievable by experimental approaches, though the validation role of the latter cannot be undermined. Altogether, the recent advances in research results prove these complementary approaches the most appropriate for tackling the complex issues of biomembrane interactions.

Accurate identification of disease and correct treatment policy can save and increase yield. Different deep learning methods have emerged as an effective solution to this problem. Still, the challenges posed by limited datasets and the similarities in disease symptoms make traditional methods, such as transfer learning from models pre-trained on large-scale datasets like ImageNet, less effective. In this study, a self-collected dataset from the DoctorP project, consisting of 46 distinct classes and 2615 images, was utilized. DoctorP is a multifunctional platform for plant disease detection oriented on agricultural and ornamental crop. The platform has different interfaces like mobile applications for iOS and Android, a Telegram bot, and an API for external services. Users and services send photos of the diseased plants in to the platform and can get prediction and treatment recommendation for their case. The platform supports a wide range of disease classification models. MobileNet_v2 and a Triplet loss function were previously used to create models. Extensive increase in the number of disease classes forces new experiment with architectures and training approaches. In the current research, an effective solution based on ConvNeXt architecture and Large Margin Cosine Loss is proposed to classify 46 different plant diseases. The training is executed in limited training dataset conditions. The number of images per class ranges from a minimum of 30 to a maximum of 130. The accuracy and F1-score of the suggested architecture equal to 88.35% and 0.9 that is much better than pure transfer learning or old approach based on Triplet loss. New improved pipeline has been successfully implemented in the DoctorP platform, enhancing its ability to diagnose plant diseases with greater accuracy and reliability.