Lviv University

Linear geometry describes geometric properties that depend on the fundamental concept of a line. In this article, we survey the main concepts and results of linear geometry that depend on the flat hulls: flats, exchange, rank, regularity, modularity, and parallelity.The article is the first in a series of reviews of linear geometry, the branch of geometry that studies geometric properties that depend only on lines, scheduled for publication in the journal Matematychni Studii. Linear geometry emerged as a separate branch of geometry in the 20th century. We present the main ideas and results of linear geometry using Bourbaki’s approach to mathematics as a science that studies mathematical structures, in particular, such a fundamental mathematical structure of linear geometry as the structure of a liner.

Abstract Microstructural and microprobe analyses of as-cast and annealed alloys (1,000 °C for 61 h) revealed a new congruently melting Hf 0.45 Ti 0.55 Ni 2 compound in the Hf–Ti–Ni system. The compound forms two-phase equilibria with Hf 7 Ni 10 , Hf 3 Ni 7 , αHfNi 3 and TiNi 3 and exhibits a homogeneity range of 18–24 at.% Ti at 65 ± 0.5 at.% Ni. Its crystal structure was determined by single-crystal and powder X-ray diffraction. Hf 0.45 Ti 0.55 Ni 2 crystallizes in a derivative of the Co 1.75 Ge prototype exhibiting statistical mixture of hafnium and titanium atoms on 2 d site and two 2 a , 2 c sites occupied by nickel atoms: PS hP 6, SG P 6 3 / mmc , a = 4.225(2), c = 5.044(4) Å. The Rietveld-refined composition, Hf 0.482(2) Ti 0.518(2) Ni 2 , is in close agreement with both the single-crystal X-ray diffraction data and the energy-dispersive X-ray analysis results. The homogeneity range of the compound Hf x Ti 1- x Ni 2 has been refined from powder X-ray diffraction and energy-dispersive X-ray data: (0.55 ≤ x ≤ 0.72), variation of the lattice parameters is the following: a = 4.2045(3) – 4.185(1), c = 5.0055(4) – 5.000(1) Å. The microstructure shows a characteristic twin morphology. The microhardness of the compound was measured as 1,160 ± 20 and 1,140 ± 60 H μ for single-phase as cast alloys containing 18 and 24 at.% Ti, respectively. Electronic structure calculations based on the LMTO method for the ordered model HfTiNi 4 ( P 6/ mmm ) indicate a metallic behavior with significant Ti–Ni, Hf–Ni, and Ti–Hf bonding interactions. The analysis of DOS and COHP indicates strong covalent contributions alongside Ni–Ni antibonding states, suggesting a complex hybrid metallic–covalent bonding nature that stabilizes the Hf–Ti–Ni three-dimensional network.

Streptomyces viridosporus (formerly ghanaensis) ATCC14672 is the best-studied producer of moenomycins, a family of phosphoglycolipid natural products directly inhibiting peptidoglycan glycosyltransferases. The uniqueness of moenomycins, in terms of structure and mode of action, and their extreme potency against several Gram-positive pathogens (including vancomycin- and methicillin-resistant cocci), are their advantages as a drug candidate. However, they are not orally bioavailable; in the bloodstream, moenomycins are characterized by an exceedingly long half-life. These shortcomings are thought to be caused by the long C25 lipid chain (moenocinol) of moenomycins. Here we report the generation and initial studies of two ATCC 14672 mutants to access a greater diversity of moenomycins around its lipid chain. The dO5 mutant cannot produce any moenomycins due to the deletion of the moeO5 gene for the first step in moenomycin assembly. The M12 mutant of ATCC 14672 (knockout of the prenyltransferase gene moeN5) accumulates moenomycins with a C15 lipid chain instead of moenocinol. We demonstrate and discuss the potential applications of the dO5 and M12 mutants for the discovery of novel moenomycins and their genetic determinants.

Harvesting naturally dynamic forests causes losses of habitat quality and functional connectivity. Focusing on Sweden as a case study of high-yield rotation forestry, we provide analyses supporting spatial prioritisation of protection, management and restoration of representative functional forest habitat networks. Habitat suitability index modelling of focal birdspecies was used to analyse how forest naturalness, habitat patch size and functional connectivity affect representative forest habitat networks in Sweden's five ecoregions. Habitat modelling for the least demanding bird species showed that of the mountain ecoregion 57-77% was functional, but in the other three boreal ecoregions only 8-9% were functional. For nemoral forests, the proportions of functional habitat networks were < 3%. More demanding species have even less functional habitat. We highlight the importance of the mountain ecoregion for forest biodiversity conservation, and the urgent need for landscape planning of protection, conservation management and nature restoration in Sweden.

The object of research is the thixotropy of meat batters with different ingredient compositions characteristic of sausage products. The research analyzes modern approaches to evaluating the rheological properties of meat emulsion systems and summarizes the factors influencing their structural organization. The regularities of changes in viscous-plastic characteristics of pork and turkey batters under different levels of load are investigated, and the features of structural destruction and recovery are established. A comprehensive assessment of thixotropic properties is based on the analysis of the structural-mechanical characteristics of batter systems during their formation with various ingredients. It has been demonstrated that the thixotropic state depends on the concentration of protein components and the degree of hydration of the protein system, which determines the stability of the sausage matrix during mechanical processing. The shear stress was calculated for time intervals in the range of 1–1000 s. The output parameters of the proposed model were adapted to the properties of soft meat batters: for minced pork, coefficients A = 80 Pa and B = –12 were adopted, and for minced turkey, A = 60 Pa and B = –9. Since the value of B is negative for both types of raw materials, this confirms the presence of pronounced thixotropic properties in the studied systems. Controlling these parameters allows regulation of the thixotropy of the batter to achieve optimal consistency and structural stability of the finished product. Thixotropic properties play a crucial role in technological operations, including grinding, mixing, filling casings, and structure formation during thermal processing. Excessive structural stability complicates processing, while excessive thixotropy reduces the shape retention of products. The determination of rheological parameters allows predicting the behavior of batter systems under production conditions, ensuring stable texture and high product quality. The results obtained have practical value for optimizing technological regimes in sausage production and developing new products with predictable rheological properties.

The Ukrainian educational system has experience of the digital quicker uptake by two significant crises in the country that occurred in the year 2019-2025 as compared to the rest of the world. The study examines how AI and Generative AI GenAI) have been adopted and adapted within the Ukrainian educational context across multiple crisis phases between 2019 and 2025. A longitudinal mixed-methods was used for the study. A total of 200 university instructors in Ukrainian higher education was selected for survey while 20 participants was selected for interview using purposive sampling technique. Qualitative and quantitative data were collected using interviews and structured online questionnaires respectively. The result showed that AI tools in education increased steadily and significantly between 2019 and 2025. AI adoption factors changes from technical (4.32 ± 0.58) to contextual (4.32 ± 0.58) over time. There is a significant difference in AI/GenAI adoption and motivation across the crisis’s phases (p &lt; 0.05) with a partial η² value of 0.162 and 0.125 respectively. AI-LMS platforms, Adaptive Learning Systems and GenAI are the most adaptable to the Ukrainian context. The result showed that language barriers, infrastructure limitations, and resource availability were statistically significant challenges of AI adoption (p&lt;0.0). The study concluded that there is an exponential growth in AI adoption and turning points, however crises and technological breakthroughs jointly redefined digital education. Policies and ethical frameworks of AI-use should be introduced to universities to steer students in the ethical use of AI and academic honesty.

For an entire transcendental function $f$ and for a sequence $(\lambda_n)$ of positive numbers increasing to $+\infty$, a series $A(z)=\sum\limits_{n=1}^{\infty}a_nf(\lambda_n z)$ is said to be regularly convergent in $\{z: |z|&lt;R[A]&lt;+\infty\}$, if $\mathfrak{M}(r,A)=\sum\limits_{n=1}^{\infty} |a_n|M_f(r\lambda_n)&lt;+\infty$ for all $r\in [0, R[A])$, where $R[A]$ is the radius of a regular convergence of $A(z)$ and $M_f(r)=\max\{|f(z)|:|z|=r\}$. We have found conditions on $(\lambda_n)$ and $f$, under which $\ln\mathfrak{M}(r,A)\sim \ln\,\mu(r,A)$ as $r\to R[A]$, where $\mu(r,A)= \max\{|a_n|M_f(r\lambda_n):n\ge 1\}$ is the maximal term of the series. At the end of the paper, an unresolved problem is stated.

The collective work by Svitlana Romaniuk, Larysa Kolibaba, and Oleksandra Antoniv — a new textbook for Polish students «The Ukrainian Language in Diplomacy and Politics», published in Warsaw — has been reviewed. It is emphasized that this educational publication is especially relevant due to its professional orientation: today it is extremely important to have abroad representatives of governmental institutions or diplomatic missions who have a command of the Ukrainian language and therefore deeply understand the political situation, maintain a clear international stance, and actively support our struggle against aggression for the peaceful existence of the state. The structure of the textbook, the content of its thematic units, innovative methodological approaches, and the quality of design are analyzed. Attention is drawn to the variety of authentic texts and their professional orientation. The educational materials aimed at developing language competence at the lexical, morphological, and syntactic levels as well as materials valuable from the perspective of linguistic culture are described. It is noted that the textbook’s professional focus is reflected in the use of a large number of diplomatic and political terms, excerpts from academic texts in the field of international relations, and examples of legislative and diplomatic documents. The inclusion of authentic specialized texts in Polish is considered methodologically appropriate: translation practice deepens language knowledge and strengthens writing skills. It is emphasized that the practical tasks for independent work and mini-projects help enhance the activity-based nature of learning. Importantly, all didactic materials are aligned with the norms of the current Ukrainian Orthography (2019). Given the authors’ extensive experience in teaching Ukrainian as a foreign language in Ukraine and abroad and in preparing numerous educational publications, it can be asserted with confidence that the textbook materials were tested in advance among foreign learners. This has ensured that in its scope, content, and presentation, the new textbook deserves a positive evaluation. The reviewed educational publication will deepen foreigners’ knowledge about our country as a European state and help them master Ukrainian in yet another strategically important sphere.

Conventional electrochemical mechanisms for electrical energy storage face fundamental limitations in achieving ultra-high energy density and high-power output. These constraints arise from the intrinsic nature of the electrochemical processes themselves. Overcoming this challenge requires a paradigm shift—from electrochemical to quantum mechanisms of energy storage. As shown by theoretical models, this concept can be implemented in nanostructured materials consisting of clusters with tunnel-transparent shells. It is possible to build such a structure using supramolecular complexes and clathrate organization of matter. For this purpose, we synthesized a supramolecular clathrate with a hierarchical sub-host&lt;host&lt;guest&gt;&gt; architecture and investigated its conductive and polarization properties using impedance spectroscopy. As shown by the results of the research, in this structure it was possible to combine a high value of the dielectric permittivity with a dielectric loss tangent below unity in the ultra-low-frequency range. This was facilitated by the presumably specific energy structure of the clathrate, as evidenced by the measured spectra of thermally stimulated discharge currents. The ability of the clathrate to accumulate an electric charge is evidenced by the measured hysteresis current-voltage characteristic. The value of the specific capacitance of this clathrate reaches the value that arises from the theoretical model of a quantum supercapacitor.