UDC 510 We consider some issues related to the 2-orthogonal polynomials (2-OP). The answers to these issues can be regarded a supplement to the article [K. Douak, P. Maroni, On a new class of 2-orthogonal polynomials, I: The recurrence relations and some properties, Integral Transform and Special Functions (2020)]. The conditions imposed on the parameters of two original recurrence relations (the first of these conditions is for the 2-OP, while the second condition is for their normalized derivatives) and guaranteeing the ``"classical" nature of the 2-OP in Hahn's sense are clarified. It is constructively proved that these recurrence relations do not cover all possible "classical" 2-OPs. An example of ``"classical" 2-OP generated by the generating function constructed by using a Bessel function of the first kind of order zero is presented. These OPs are unique because their properties are similar to the classical OPs. In particular, this concerns the fact that their zeros are real and their location. The analysis of the available literature and our own numerous numerical-analytic experiments reveals the absence of other examples of the ``"classical" 2-OPs with similar properties.

This work focuses on obtaining approximate analytical solutions for a fractional-order smoking epidemic model formulated with the Caputo derivative. The model divides the population into five compartments—potential smokers, current smokers, occasional smokers, permanent smokers, and temporary quitters—allowing the fractional framework to capture the long-term memory effects in smoking dynamics. Transition rates between these compartments are described by a set of parameters that reflect realistic behavioral changes. To solve the non linear fractional differential system efficiently, we propose a new hybrid computational strategy that combines the Junaid integral transform TN G with the Adomian Decomposition Method (ADM) and an Artificial Neural Network (ANN). The combined TN G–ADM stage ensures rapid convergence and accurate series solutions, while the ANN improves predictive performance by learning directly from the system dynamics. Numerical simulations validate the effectiveness and computational efficiency of the proposed approach, demonstrating its suitability for modelling memory-dependent epidemiological processes.

One of the basic tasks of communication structures is to safely carry loads and then transfer them to the ground. In the case of railway surfaces, in addition to static loads, dynamic loads are of particular importance. In the discussed case, this involves adopting an appropriate model to describe the behavior of geosynthetics in railway engineering structures. When analyzing the cooperation of geosynthetics with elements of an engineering structure (e.g. subgrade, subsoil, pipeline), we can treat them, for example, as elastic membranes or coatings placed on various types of substrate. Modeling a real system means idealizing it in terms of taking into account those features of the medium and its elements that seem most important from the point of view of the analyzed problem. A physical model will be built, followed by a mathematical model using non-classical operational calculus. The mentioned operational calculus is not based, as is the case with the classical operator calculus, on integral transformations (Laplace, Laplace-Carson, Fourier, Mellin, Hankel, etc.). Therefore, it is freed from many limiting assumptions required in the case of classical operational calculus. The non-classical operational calculus is based on three arbitrary linear operations S, T, s and two arbitrary linear spaces L0, L1. This allows us to create continuous and discrete models and analyze them in a uniform way. In the mathematical description of the behavior of geosynthetics in railway engineering structures in the continuous domain, the Laplace operator △ = ∇2 and the d'Alembert operator □ are used. The d'Alembert operator is treated as an operation S, for which operations T, s are selected and the location of points P from the geosynthetic at time t is determined. Assuming S as an operation defined in the space of multi-point sequences allows us to analyze the formulated problem in the discrete domain. Such approach to the analyzed problem allows us to determine the location of P points from the geosynthetic at time t using analytical, numerical or hybrid methods.

Hypergeometric functions are divided into complete and confluent functions. For the first time, Srivastava and Karlsson described a method for constructing a set of all possible triple Gaussian hypergeometric series and compiled a table showing definitions and areas of convergence for 205 different complete series (Srivastava-Karlsson List) depending on three variables. Several authors subsequently obtained various integral representations and transformations for the functions proposed by Srivastava and Karlsson. In this work we compile integral representations and transformation formulas for all confluent forms of one complete hypergeometric function in three variables from the Srivastava-Karlsson List.To prove integral representations for 8 confluent hypergeometric functions of three variables, properties of beta and gamma functions are used. The Boltz formula allows us to derive transformation formulas for the confluent functions under consideration.

Analysis of binding kinetics and mass transport in SPR-based biosensor using the Generalized Integral Transform Technique and the Markov Chain Monte Carlo Method.

The transition to an environmentally oriented AIC requires significant investments aimed at implementing innovative solutions, modernizing production processes, developing organic farming, implementing bioenergy projects, rational water use systems, and effective waste management. The article investigates the current state and dynamics of investments in environmentally oriented projects within Ukraine’s agro-industrial complex (hereafter – AIC). It analyzes the volume of investments from 2019 to 2024, the structure of funding sources, and trends in the attraction of both private and institutional capital. Investment activity trends are visualized through dynamic graphs, and key factors influencing investor decisions in the field of sustainable agriculture are outlined. Particular attention is paid to the impact of government support programs, international technical assistance, and donor funds on the development of organic farming, rational agriculture technologies, bioenergy, greenhouse gas reduction, and climate change adaptation. A comparative analysis of investment activity by types of economic activity was conducted, which made it possible to identify industries with the highest and lowest levels of capital attraction. The main factors influencing the investment attractiveness of individual economic sectors were identified, including institutional conditions, profitability levels, government support, and infrastructure availability. Trends in the development of investment flows were examined in the context of current economic challenges. The impact of macroeconomic factors on the dynamics of investment activity across various industries was analyzed. Recommendations are provided to stimulate green investment, including improving the legislative framework, developing public-private partnership mechanisms, encouraging the involvement of financial institutions in sustainable projects, and expanding access to the preferential lending. The research results can be used to shape an effective national and regional policies in the field of sustainable agricultural production in the context of European integration transformations.

Micropropagation is key for the sustainable conservation of critically endangered plants and often enhances metabolite production. The biochemical fidelity of these regenerated plants is assessed through metabolite profiling. Here, we developed an efficient in vitro regeneration protocol for Nardostachys jatamansi, achieving the highest callus induction rate (93.33% ± 4.08%) using 1.5 mg/L TDZ and 1 mg/L NAA. Combining 1.5 mg/L TDZ and 0.1 mg/L NAA yielded a regeneration efficiency of 68% ± 4.89%, with the highest shoot number (6.4 ± 0.50) under conditions of 15°C ± 2°C. GC-MS analysis revealed elevated primary and secondary metabolites in the regenerated leaves compared to those in the mother plant. In addition, an Agrobacterium tumefaciens-mediated transformation protocol was optimised for subcellular localisation, transient gene expression and virus-induced gene silencing. Results revealed,in comparision to EHA105 (30% ± 5.77%), GV3101 (43.33% ± 3.33%) showed higher transformation efficiency under similar conditions. The highest transformation efficiency for transient expression was achieved with an OD₆₀₀ = 0.6, a 4-h infection, and a 3-day co-cultivation, confirmed through RT-PCR and western blotting. Virus-induced gene silencing using pTRV2:NjPDS effectively silenced the NjPDS gene. This integrated platform, combining micropropagation, transformation and metabolite profiling, provides a scalable tool for enhancing bioactive compounds and genetics in N. jatamansi.

We study an integral transform—here called the Master Integral Transform—in which the kernel is an arbitrary entire function of finite order. When the nonzero Taylor coefficients of the kernel have positive Beurling–Malliavin density, we prove completeness and global injectivity in a Cauchy-weighted Hilbert space, and we furnish explicit Mellin–Fourier inversion formulae with exponentially decaying integrands. Classical Fourier, Laplace, and Mellin transforms appear only as strict special cases. Beyond these, we establish structural properties (multiplier/composition law, dilation covariance, parameter regularity) and present applications not captured by fixed-kernel frameworks, including inverse-kernel identification and hybrid boundary value models, e.g., the Poisson–Airy pair produces a closed-form transformed Green’s function and a solvable variable-coefficient PDE, illustrating capabilities unavailable to fixed-kernel frameworks.

This article investigates combining traditional methods for solving differential equations with Aboodh integral transformation and the Bernstein method. This approach offers a new perspective in solving various differential and integral equations. The solution approach obtained from the combination of Bernstein polynomials and integral transformations stands out in terms of theoretical and practical implications. This method can be applied to various examples of differential equations, and its practical applicability can be evaluated through studies conducted using programming languages. The error between the exact solution and the proposed method’s approximate solution has been calculated. We compare the numerical results of the Aboodh-Bernstein, Laplace-Bernstein, and Sumudu-Bernstein methods.

The article reveals the theoretical and methodological foundations of improving the mechanisms of public regulation of economic development in the context of European integration transformations, which provide for a combination of institutional, legal and economic instruments. It is established that the effectiveness of public administration is determined by the ability to ensure effective management of social processes, to achieve compliance between the goals of management and the actual results of activity. It is stated that in the conditions of European integration transformations, the requirements for regulatory compatibility with the EU acquis, digitalization of public services and institutional capacity of the state are increasing. It is determined that modern integration processes directly stimulate the introduction of innovative management solutions and technological modernization. The peculiarities of the Ukrainian experience of adaptation of regulatory mechanisms to European standards in the context of post-war recovery and external risks are revealed. It is found that the European Commission's Ukraine Report 2024 focuses on digitalizing procedures, improving financial management, and strengthening anti-corruption institutions. It was emphasized that the introduction of industrial parks and special economic zones are key mechanism for stimulating investment and regional development. It is emphasized that at the regulatory level, there is a desire to improve the legal regime of industrial parks, unify the conditions of state support and increase their investment attractiveness. The importance of the National Economic Strategy until 2030 and government plans as tools of strategic planning in the field of European integration reforms is revealed. It was emphasized that the implementation of the Export Strategy of Ukraine until 2030 allows stimulating the development of an export-oriented economy and integration into global production chains. It was emphasized that the Communication Strategy on European Integration of Ukraine for the period until 2026 is aimed at forming public support and consolidating resources for the implementation of reforms. It is stated that the communication dimension of European integration directly affects the perception of reforms and increases their effectiveness.

In this paper, we highlight on application of the Kushare Integral Transform to solve Fractional Newton’s Law of Cooling problems.

ABSTRACT This paper introduces two new mathematical functions, the sine Krätzel function and the cosine Krätzel function, which extend the classical Krätzel function into the trigonometric domain. These functions are rigorously defined through integral representations and their fundamental properties, such as absolute convergence, generating functions, and derivative formulas, are investigated in detail. The integral transforms of these functions, including Mellin, Fourier, and Laplace transforms, are derived, highlighting their analytical flexibility. Furthermore, the study explores the applications of these functions in wave optics, specifically in the context of Fresnel and Fraunhofer diffraction patterns, demonstrating their utility in understanding light diffraction phenomena. Numerical examples and graphical visualizations are provided to illustrate the influence of key parameters on diffraction patterns, emphasizing the potential of these functions in applied mathematical and physical research.

This work deals with the physical system governed by a Hamiltonian operator, in two-dimensional space, of spinless charged particles subject to a perpendicular magnetic field B, coupled with a harmonic potential in the context of nonextensive statistical thermodynamics. The thermodynamics of such a quantum gas system is elaborated in the framework of Tsallis statistics by obtaining the q versions of the partition function, magnetization, and susceptibility after performing the Hilhorst integral transformation. The results are discussed in the q → 1 limit.

We prove a Hardy-type inequalities in Dunkl setting, integrated with an HI-potential. Our approach utilizes the h-harmonic expansion of functions and integrating techniques such as integral transformations, spherical coordinate formulas, and separation of variables, we derive the main result presented in Theorem 1. These outcomes build upon and extend the foundational work of Ghoussoub and Moradifam (2013), which addressed Hardy-type inequalities involving the Laplace operator and the Lebesgue measure in conjunction with an HI-potential. Consequently, our findings advance the generalization of Hardy inequality within broader context of Dunkl theory. Moreover, this research carries substantial implications for analyzing differential equations and partial differential equations that exhibit singularities, thereby providing enhanced understanding of the qualitative properties and behaviors of solutions in these equation classes. This extension not only refines existing inequalities but also opens avenues for applications in mathematical physics and functional analysis.

The legal regulation of sustainable rural development has a long history. This comprehensive analysis of legislation regulating the public relations in rural areas views it as part of development of domestic science, technology, economy, and society. A retrospective analysis of the Soviet period and the transitional period revealed the main stages and approaches to sustainable rural development. A set of formal-legal, comparative-legal, systemic, historical-legal, sociological, and statistical methods made it possible to identify the goals for improving the regulatory framework for sustainable rural development, e.g., an integrated human-centered approach, high-quality resource management, integral digital transformation of local governments, digital literacy of rural residents, independent information technology resources for the analysis of urban-rural migration, organized rural tourism, green economy based on socio-ecological subsystems, green legislation, etc.

This study presents a comparative analysis of the conceptual foundations for understanding basic concepts of early childhood education in Ukrainian and German scientific and pedagogical discourse. The research addresses the urgent need for terminological harmonization in the context of Ukraine’s European integration processes and the necessity to align national educational concepts with European standards and approaches. The study develops an original methodology for comparative analysis of terminological apparatus through the prism of sociocultural and historical-pedagogical contexts. For the first time, a structural-functional classification of basic concepts of early childhood education has been implemented, dividing them into two groups: system-structural concepts that characterize the architecture of the early childhood education system and its institutional components, and dynamic-processual concepts that characterize functional and transformational aspects of the system. The research substantiates an evolutionary paradigm of conceptual approaches to interpreting early childhood education, demonstrating transformation from preparatory models to personality-oriented paradigms in both educational systems over the period from 1990 to 2025. Specific German terms without Ukrainian analogues have been identified, including “Frühkindliche Bildung”, “Erziehungspartnerschaft”, “Bildungsdokumentation”, “Kindertagespflege”, “Familienzentrum”, and “Fachberatung”, which reflect national pedagogical traditions, sociocultural features of educational systems, and different philosophical approaches to early childhood development. The study determines conceptual differences in understanding the quality of early childhood education, educational environment, and development trends as results of cultural traditions and socio-pedagogical contexts influence. A system of author’s definitions of key research concepts has been created, forming theoretical and methodological foundations for comparative studies in early childhood education. The practical significance lies in forming a terminological basis for harmonizing national understanding of early childhood education with European approaches in the context of Ukraine’s European integration transformations and creating conceptual prerequisites for effective implementation of progressive international experience.

This comprehensive study investigates the pivotal role of vocational English instruction in advancing China’s “Vocational Education Going Global” strategy within the “Integration of Industry and Education” paradigm. Through multidimensional analysis of transnational initiatives, pedagogical innovations, and policy frameworks, the research elucidates how industry-aligned English pedagogy enhances global workforce competitiveness, facilitates cross-border educational collaboration, and supports the international expansion of Chinese enterprises. Critical challenges are examined, including curriculum-industry misalignment, teacher capacity gaps, and intercultural communication barriers. The paper proposes an integrated transformation framework centered on collaborative curriculum design, hybrid teacher development ecosystems, and culturally responsive pedagogical approaches. These insights contribute actionable pathways for positioning vocational English education as strategic infrastructure in China’s global skills leadership agenda.

The article presents the method for statistical leaning of nonlinear model parameters, its principle and efficiency of application. It proposes a solution to the shortcomings of using the differential spectra balance approach through the integral form of least squares in the scheme of differential non-Taylor transformations. The object of the study is the process of statistical learning of nonlinear model parameters. The purpose of this paper is to formulate the statistical learning method for evaluation of nonlinear model parameters using LSM in integral form and differential non-Taylor transformations. It is relevant in many areas of modern activity and is necessary for applying the statistical training methodology to a more complex time series, as well as increasing the accuracy of the received expectations. To achieve this goal the statistical learning methodology was proposed, which is based on the creation of process model in integral form of least squares with simplification using non-Taylor transformations. It differs from existing approaches by incorporating all the available differential discretes in the created model, which allows for better predictability and circumvents the problem of unequal count of discretes inside the models, which allows for better application of the method for different model forms. The algorithm for the process was formed, using which it can be applied different models. In the paper several experiments were conducted to verify the efficacy of the proposed method in different situations. These experiments use generated datasets that are polluted with stochastic errors to better simulate real data. The results of modeling are shown, and the statistical characteristics of the obtained expectation are compared with the results of the application of the statistical training methodology using the differential spectra balance. During the study, features of the technique were found that must be considered for its application to data with a high number of stochastic deviations. Based on the obtained metrics, a conclusion is made about the effectiveness of using the technique relative to time series reflecting processes of different nature.

Mobile organisms integrate multimodal self-motion signals - including motor commands, vestibular inputs, optic flow and proprioceptive feedback - to accurately perceive their heading and speed of traversal. These instantaneous cues are processed, via continuous temporal integration and progressive spatial transformations, to facilitate path-integration-based navigation. Recent cutting-edge neurophysiological recordings in animal models have revealed several ubiquitous cross-modal algorithms that contribute to this processing: vestibular-visual convergence to enhance self-motion perception, predictive coding integration to enable optimal dynamic state estimates, landmark-referenced error correction‌ to mitigate path-integration drift and facilitate cognitive spatial map construction, and egocentric-to-allocentric conversion‌ via integration with proprioceptive cues from the eyes, head, body or limbs. Thus, multisensory coding plays an important role in self-motion perception and self-localization during navigational behaviour.

To carry out radiation sterilization, one needs to determine the permissible irradiation modes, which is carried out using computer dosimetry methods. Nowadays, the choice of optimal irradiation modes can be based on the models of the depth-dose curve at different incidence angles of the electron beam on the layer of matter. In the present paper, the distribution of transferred energy in the volume of the target initiated by the normal incidence of a point beam of radiation on the surface of a semi-infinite medium (Dose-Map object) is used to develop such models. Semi-empirical models of the Dose-Map object are designed based on two assumptions. One is that the target has axial symmetry relative to the direction of the radiation particle incidence on the target. The second is that the dose spatial distribution is uniform or normal (Gaussian distribution) in the cross-sections of the Dose-Map object at all depths. For a two-parameter approximation of the Dose-Map object, three-dimensional geometric figures are suggested, which surfaces are formed by rotating the plots of power functions around the abscissa axis. Semi-empirical models are developed based on the assumption that the parameters of the Dose-Map object in its eigen coordinate system do not change when the beam incidence angle changes. Expressions are obtained for calculating the depth-dose curves from radiation incident on the target at an angle θ in the form of an integral transformation of the depth-dose curve for normal incidence of the radiation beam on the target. Software has been developed for calculating depth-dose curves in a semi-infinite medium under uniform irradiation by an electron beam. The implemented algorithms for calculating the depth-dose curves from an electron beam incident on the target at the angle θ are tested. Satisfactory agreement is established between the results obtained using the developed semi-empirical models and the results of Monte Carlo modeling of the depth-dose curves at different incidence angles of the electron beam on the target. Good agreement is established between the results obtained using the semi-empirical model "Cone" and the results obtained using the developed two-parameter semi-empirical models SEM2U and SEM2N. The capabilities of the developed two-parameter models for a more complete description of the technological characteristics of the radiation sterilization process are investigated using the numerical methods. Examples are provided where the developed two-parameter models allow for the simultaneous description of two technological characteristics of the two-sided irradiation process: the optimal target thickness and the dose uniformity ratio (DUR) in the target. Consistent data on these characteristics allow choosing optimal modes of electron beam irradiation during radiation sterilization in a reasonable manner. The possibilities of using the approach suggested in the present paper for developing a set of semi-empirical models of computer dosimetry of irradiation processes in radiation technologies are noted.