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Conducting surveys of multi-storey buildings is a laborious task, because large volumes of visual and instrumental research should be carried out. Reduction of labor costs with an increase in the reliability of information about the state of damage and technical condition is an actual scientific and practical task. One of the ways to solve it is to use non-destructive vibration diagnostic methods. The purpose of carrying out diagnostics with the use of vibration based damage detection methods is to search for damages in structural elements that can cause the deviation of the dynamic parameters of a structure from calculated ones. Determination of the dynamic parameters of the structure, in particular natural frequencies and mode shapes of mechanical systems, is one of the most important tasks that allows obtaining integral information about the state of a structure. This article presents the results of calculations for the localization of slabs defects in a multi-storey building with a transverse crack, span L = 4.5 (m), height H = 0.2 (m), with prestressed reinforcement d = 0.05 (m). Vibration based Damage Index method was used to localize the defect. During the study, reliable localization values of the defect area of the slab were obtained, this indicates that the vibration method for determining the damage index with a sufficient degree of accuracy allowed predicting the site of damage to the structure.

Deep learning approaches for multi-source sound localization face significant challenges, particularly the need for extensive training datasets encompassing diverse spatial configurations to achieve robust generalization. This requirement leads to substantial computational demands, which are further exacerbated when localizing overlapping sources in complex acoustic environments with reverberation and noise. In this paper, a new methodology is proposed for simultaneous localization of two overlapping sound sources in the time–frequency domain in a closed, reverberant environment with a spatial resolution of 10° using a small-sized microphone array. The proposed methodology is based on the integration of the sound source separation method with a single-source sound localization model. A hybrid model was proposed to separate the sound source signals received by each microphone in the array. The model was built using a bidirectional long short-term memory (BLSTM) network and trained on a dataset using the ideal binary mask (IBM) as the training target. The modeling results show that the proposed localization methodology is efficient in determining the directions for two overlapping sources simultaneously, with an average localization accuracy of 86.1% for the test dataset containing short-term signals of 500 ms duration with different signal-to-signal ratio values.

This paper investigates D-finite discrete generating series and their sections. The concept of D-finiteness is extended to multidimensional discrete generating series and its equivalence to p-recursive sequences is rigorously established. It is further shown that sections of the D-finite series preserve D-finiteness, and that their generating functions satisfy systems of linear difference equations with polynomial coefficients. In the two-dimensional case, explicit difference relations are derived that connect section values with boundary data, while in higher dimensions general constructive methods are developed for obtaining such relations, including cases with variable coefficients. Several worked examples illustrate how the theory applies to solving difference equations and analyzing multidimensional recurrent sequences. The results provide a unified framework linking generating functions and recurrence relations, with applications in combinatorics, number theory, symbolic summation, and the theory of discrete recursive filters in signal processing.

We extend existing functional relationships for the discrete generating series associated with a single-variable linear polynomial coefficient difference equation to the multivariable case.

The current engineering theories on bending vibrations and the stability of beam structures are based on solving eigenvalue problems through similarly formulated differential equations. Solving the eigenvalue problem for engineering calculations is particularly laborious, especially for non-classical supports, where factors like the stiffness of supports, axial forces, or temperature must be considered. In this case, the solution can be obtained only by numerical methods using specially created programs, which makes it difficult to select supports for a given planar beam structure in engineering practice. This work utilizes established solutions from eigenvalue problems in the theory of vibrations and stability of beams, incorporating factors such as axial forces, temperature, and support stiffness. This combined solution is applicable to beam structures of any type and cross-section, as it is determined solely by the selected support conditions (stiffness) and loading (axial force, temperature). Approximation of eigenvalue problem solutions through continuous functions allows the readers to use them for the analytical solution of the design problem of choosing a support system to ensure the frequency of vibrations and stability of the planar beam structure. At the same time, the known solutions given in the reference books on bending vibrations and stability become their particular solutions. This approach is applicable to solving problems of vibrations and loss of stability of various types (torsional, longitudinal, etc.), and is also applicable in other disciplines where solving problems for eigenvalues is required.

We consider a variant of the Cauchy problem for a multidimensional difference equation with constant coefficients, which connected with a lattice path problem in enumerative combinatorial analysis. We obtained a formula in which generating function of the solution to the Cauchy problem is expressed in terms of generating functions of the Cauchy data and a formula expressing solution to the Cauchy problem through its fundamental solution and Cauchy data

The article presents a comprehensive method for diagnosing underground structures using the example of an underground pedestrian crossing located in Rostov-on-Don. The problem of assessing the condition of buildings and structures is very relevant at all stages of the life cycle. There is a special need for continuous monitoring of bearing structures for many buildings with critical applications especially if reliability of which determines the life and health of people. This work considered complex method included geodetic research, geological study of the state of sub-soil, analysis of state of steel and reinforced concrete structures, vibro diagnostics of load-bearing structures under dynamic technogenic impacts and an assessment of the mechanical characteristics of steel and reinforced concrete structures in order to develop a conclusion on the overall state of the transition and the possibility of its further operation. Analysis of the response from arbitrary non-stationary effects on structural elements is carried out on the base of the calculated spectrum of reference impacts.

An effective method of the layered structures mechanical properties determination is presented. The method is based on the non-stationary analysis of the structure response that is acted by impact load on the surface of multi-layered structure. The assessment of the design parameters using a genetic algorithm is performed. During the solving procedure there are several methods for breeding and selection processes are used, that demonstrated different results. The most appropriate parameters for described algorithms are selected and analysed.

We define a set of polynomial difference operators which allows us to solve the summation problem and describe the space of polynomial solutions for these operators in equations with the polynomial right-hand side. The criterion describing these polynomial difference operators was obtained. The theorem describing the space of polynomial solutions for the operators was proved.

We synthesized star-shaped pentagonal microcrystals of boron carbide with an extremely low carbon content (~5%), from m-carborane under high pressure (7 GPa) and high temperature (1370–1670 K). These crystals have five-fold symmetry and grow in the shape of stars. A 5-fold symmetry in large micron-sized crystals is extremely rare making this a striking observation.