This research proposes a novel approach to the Word Sense Disambiguation (WSD) task in the Georgian language, based on supervised fine-tuning of a pre-trained Large Language Model (LLM) on a dataset formed by filtering the Georgian Common Crawls corpus. The dataset is used to train a classifier for words with multiple senses. Additionally, we present experimental results of using LSTM for WSD. Accurately disambiguating homonyms is crucial in natural language processing. Georgian, an agglutinative language belonging to the Kartvelian language family, presents unique challenges in this context. The aim of this paper is to highlight the specific problems concerning homonym disambiguation in the Georgian language and to present our approach to solving them. The techniques discussed in the article achieve 95% accuracy for predicting lexical meanings of homonyms using a hand-classified dataset of over 7500 sentences.

Recently it has been found that quantum chromodynamics (QCD) phase diagram possesses a duality between chiral symmetry breaking and pion condensation. For the first time this was revealed in the QCD motivated toy model. Then it was demonstrated in effective models as well and new additional dualities being found. We briefly recap the main features of this story and then discuss its applications as a tool to explore the QCD phase structure. The most appealing application is the possibility of getting the results on the QCD phase diagram at large baryon density. Taking the idea from large $1/N_{c}$ universalities it was argued that the scenario of circumventing the sign problem with the help of dualities seems plausible. It is also discussed that there is a persistent problem about whether there should be catalysis or anti-catalysis of chiral symmetry breaking by chiral imbalance. One can probably say that the issue is settled after lattice results (first principle approach), where the catalysis was observed. But they used an unphysically large pion mass so it is still interesting to get additional indications that this is the case. It is shown just by the duality property that there exists catalysis of chiral symmetry breaking. So, having in mind our results and the earlier lattice simulations, one can probably claim that this issue is settled. It is demonstrated that the duality can be used to obtain new results. As an example, it is showcased how the phase structure of dense quark matter with chiral imbalance (with possibility of inhomogeneous phases) can be obtained from the knowledge of a QCD phase diagram with isopin asymmetry.

Since the pioneering work Lohani et. al., Phys. Rev. X 9, 041063 (2019), it became clear that quantum skyrmions have highly unusual properties as compared to the classical skyrmions and, due to their quantumness, cannot be described by continuous magnetic textures akin to the classical skyrmions. Competing nearest-neighbor and next-nearest-neighbor ferromagnetic and antiferromagnetic interactions in triangular spin-frustrated magnets lead to the formation of quantum skyrmion states. In frustrated magnets, skyrmions are characterized by the helical degree of freedom, which can store quantum information. In the limit of a weak electric field, the system can be described as a two-level system, i.e., a skyrmion qubit. Here, we propose a more general formulation of the problem and obtain general analytic solution of the model previously introduced in Psaroudaki et. al., Phys. Rev. Lett. 127, 067201 (2021). Our solution is valid not only for small barrier but for the arbitrary electric field. In the case of a significant barrier, we prove that the system's state is not a Skyrmion qubit as it was thought before, but a Skyrmion qudit. We constructed the density matrix of the Skyrmion qudit and studied its evolution in time. The obtained results suggest that the proposed model can be exploited further to meet the needs of quantum information theory and quantum skyrmionics. We showed that the $l_1$ norm of coherence of the skyrmion quantum qudit is a thousand times larger than the coherence of the skyrmion quantum qubit. The obtained result opens new perspectives for quantum skyrmion-based resource theory.

This research proposes a novel approach to the Word Sense Disambiguation (WSD) task in the Georgian language, based on supervised fine-tuning of a pre-trained Large Language Model (LLM) on a dataset formed by filtering the Georgian Common Crawls corpus. The dataset is used to train a classifier for words with multiple senses. Additionally, we present experimental results of using LSTM for WSD. Accurately disambiguating homonyms is crucial in natural language processing. Georgian, an agglutinative language belonging to the Kartvelian language family, presents unique challenges in this context. The aim of this paper is to highlight the specific problems concerning homonym disambiguation in the Georgian language and to present our approach to solving them. The techniques discussed in the article achieve 95% accuracy for predicting lexical meanings of homonyms using a hand-classified dataset of over 7500 sentences.

This paper describes the application of the method of probabilistic solutions (MPS) to numerically solve the Dirichlet generalized and classical harmonic problems for irregular n sided pyramidal domains. Here, generalized means that the boundary function has a finite number of first kind discontinuity curves, with the pyramid edges acting as these curves. The pyramid base is a convex polygon, and its vertex projection lies within the base. The proposed algorithm for solving boundary problems numerically includes the following steps: a) applying MPS, which relies on computer modeling of the Wiener process; b) determining the intersection point between the simulated Wiener process path and the pyramid surface; c) developing a code for numerical implementation and verifying the accuracy of the results; d) calculating the desired function value at any chosen point. Two examples are provided for illustration, and the results of the numerical experiments are presented and discussed.

We investigate a recent semantics for intermediate (and modal) logics in terms of polyhedra. The main result is a finite axiomatisation of the intermediate logic of the class of all polytopes -- i.e., compact convex polyhedra -- denoted PL. This logic is defined in terms of the Jankov-Fine formulas of two simple frames. Soundness of this axiomatisation requires extracting the geometric constraints imposed on polyhedra by the two formulas, and then using substantial classical results from polyhedral geometry to show that convex polyhedra satisfy those constraints. To establish completeness of the axiomatisation, we first define the notion of the geometric realisation of a frame into a polyhedron. We then show that any PL frame is a p-morphic image of one which has a special form: it is a 'sawed tree'. Any sawed tree has a geometric realisation into a convex polyhedron, which completes the proof.