Bukhara State University, Bukhara State University named after Fayzulla Khodjayev, is a higher educational institution in Bukhara, Uzbekistan that trains scientific and pedagogical personnel.
Numerical integration of definite integrals is important in fundamental and applied sciences. The error of approximate integral calculations depends on the initial data and specific requirements, which leads to the imposition of various conditions on the obtained calculations. In this paper, we consider the problem of constructing an optimal quadrature formula for the approximate calculation of Fourier integrals using the φ -function method. The error of the quadrature formula is estimated from above using the integral of the square of the function φ from the Hilbert space. Next, a function φ is chosen for which the integral of the square of the function on this interval takes the smallest value. The coefficients of the optimal quadrature formula are calculated using the obtained φ function. The optimal quadrature formula is exact for the functions e^σ x and e^ - σ x , where σ is a nonzero real parameter.
Developing environmentally friendly and highly effective adhesive compositions is a pressing challenge in modern materials science. This study synthesized and comprehensively characterized a new adhesive based on oxalic acid, formaldehyde, and phthalimide. These compounds were chosen as starting materials due to their expected adhesive properties and relative environmental friendliness. The resulting sample was analyzed using infrared (IR) and ultraviolet-visible (UV-Vis) spectroscopy to identify characteristic functional groups and intermolecular interactions. Additionally, a quantum-chemical analysis using density functional theory (DFT) was performed to evaluate the electronic structure, chemical bonding characteristics, and molecular stability. IR spectroscopy confirmed the presence of absorption bands corresponding to esterification and amide bonds, while UV-Vis spectroscopy data indicated the formation of conjugated systems that provide structural strength. Theoretical calculations are consistent with experimental observations and confirm the favorable electronic properties of the molecule. A combined experimental and theoretical approach provided a comprehensive understanding of the molecular architecture and potential adhesive properties. Thus, the adhesive based on oxalic acid, formaldehyde, and phthalimide exhibits good adhesion strength and is environmentally friendly to synthesize, making it promising for a wide range of industrial and sustainable applications.
The article discusses inverse problems for the fractional diffusion equation with the Hilfer operator in time. The direct problem is the initial-boundary value problem for this equation with Cauchy-type initial data and Dirichlet boundary conditions. The first inverse problem, which involves determining a time-dependent coefficient, is reduced to an equivalent Volterra-type integral equation. The existence and uniqueness of the solution are proven using the contraction mapping principle. The second inverse problem involves determining a function dependent on the spatial variable on the right-hand side of the equation. This problem is studied using the Fourier method and the properties of the Mittag-Leffler function. The solution is constructed in the form of a series based on eigenfunctions.
In this paper, a family of block operator matrices 𝒜_h(K), K ∈( - π/h;π/h]^3 , associated with the Hamiltonian of a system with a non-conserved number of particles not exceeding three on a non-integer lattice (hℤ)^3 with step h > 0 , is considered. It is established that the operator 𝒜_h(0), 0: = (0,0,0), has a finite number of negative eigenvalues if the corresponding generalized Friedrichs model has a zero eigenvalue. It is shown that the operator 𝒜_h(0) possesses an infinite number of negative eigenvalues accumulating at zero (the Efimov effect) if the generalized Friedrichs model has a zero-energy resonance. An asymptotic formula is obtained for the number N_h(z) of eigenvalues of the operator 𝒜_h(0) lying below z, z ⩽ 0 as the spectral parameter z → - 0.
To synthesize a bifunctional catalyst on an Al2O3 support, the initially used adsorbent Al2O3 was regenerated at 600 degrees C for 360 minutes in the absence of atmospheric oxygen. A process model for synthesizing bifunctional catalysts on regenerated Al2O3 support was developed using Ni, Co, Mo, and P precursors. According to the elemental composition analysis of the regenerated adsorbent by XRF (X-ray fluorescence) method, it was determined to consist of 81.5 % Al2O3 . Using the regenerated Al2O3 as the support, (NH4)6Mo7O24 & centerdot;4H2O, Ni A process model was developed to synthesize a bifunctional catalyst using salts such as Ni(NO3)(2)& centerdot;6H(2)O, Co(NO3)2 & centerdot;6H(2)O, and H3PO4 acid, and based on this model, three types of catalyst samples were synthesized. To determine the morphological changes and textural properties of the surface of the obtained catalyst samples, analyses were performed using SEM (Scanning Electron Microscope), TEM (Transmission Electron Microscope), and BET (Brunauer-Emmett-Teller) model-based methods. These analyses were applied to evaluate morphological changes and textural characteristics of the prepared catalyst samples.
