Proceedings of the YSU A: Physical and Mathematical Sciences
Proceedings of the YSU A: Physical and Mathematical Sciences aims to publish original research papers and survey articles in all areas of physics, mathematics and informatics. Proc. YSU A: Phys. Math. Sci. accepts also review articles, short communications, conference proceedings, algorithms, Ph.D and doctoral thesis’s and other items with a detailed exposition of results, proofs, experiments and examples. One of purposes is to reflect the progress of the research in all areas of physics, mathematics and informatics in Armenia and, by providing an international forum, to stimulate its further developments.
ABOUT THE MAXIMUM SUM OF CASUAL NUMBER OF INDEPENDANT CASUAL QUANTITY
Let ${\xi_n}$ be a sequence of independent random variables $S_n=\xi_1+\dots+\xi_n, ~\bar{S_n}=\max\limits_{1\leq k\leq n}|S_k|, ~n\geq1.$ It is shown that the correct variation of the function $N_n(t)=\sum\limits_{k=1}^n P(|\xi_k|\geq t)$ at infinity, where $P$ is the probability sign, implies the limit relations $\lim\limits_{t\rightarrow+\infty}\frac{P(\bar{S_n}\geq t)}{N_n(t)}= \lim\limits_{t\rightarrow+\infty}\frac{P(|S_n|\geq t)}{N_n(t)}=1.$ Let $v>0$ be a function independent of ${\xi_n}$ is an integer random variable with finite mathematical expectation, $$ S_v=\xi_1+\dots+\xi_v, ~\bar{S_v}=\max\limits_{1\leq n\leq v}|S_n|.$$ Consider the following model. Let ${\delta_n}$ be a sequence of positive numbers, $\alfa\geq0, ~L(t) $ vary slowly at infinity, $\lim\limits_{t\rightarrow+\infty}\big(P(|\xi_n|\geq t) \ t^{-\alfa}L(t)\big)= \delta_n$ uniformly on $n\geq1.$ Denote $c_n=(v=n), A_n=\sum\limits_{k=1}^n \delta_n,~ n\geq1,~A=\sum\limits_{n\geq 1}c_n \cdot A_n$. It has been proven that within the framework of the model $\lim\limits_{t\rightarrow +\infty}\frac{P(\bar{S_v}\geq t)}{ t^{-\alfa}L(t)}= \lim\limits_{t\rightarrow+\infty}\frac{P(|S_v|\geq t)}{ t^{-\alfa}L(t)}=A.$ Under the above conditions with existence of asymmetry in $P(\xi_n <x),~ n\geq1,$, asymmetry in $P(S_v<x)$ was found.
Copyright (c) 2000 Proceedings of the YSU https://creativecommons.org/licenses/by-nc/4.0/
Let the sequences \big \{beta_k\big \} ^\infty_{n=1}$ be fixed such that $\lim\limits_{k\rightarrow\infty}(M_{2k}-M_{2k-1})=+\infty, \beta_k>0, $\lim\limits_{k\rightarrow\infty} \beta_k=0.$ In this paper, we prove that there exists a function $f_0(x) \in L^1_{[0,1]}$ such that the Fourier series of the function $f_0(x)$ with respect to the subsystem $ \big \{W_{n_k}(x) $ \big \}_{k=1}^\infty=\big \{W_m(x):M_{2s-1}\leq m\leq M_{2s}, s= 1,2,... \big \}$ diverges in the $L^1_{[0,1]}$ metric, and the Fourier-Walsh coefficients satisfy the condition $\sum\limits_{k=1}^\infty |a_{n_k}|\beta_{n_k}<\infty.$
ESTIMATES FOR EIGENVALUES AND COMPLETENESS OF EIGENFUNCTIONS OF NON SELFADJOINT SEMIELLIPTIC DIFLERENTIAL OPERATORS
This paper is a continuation of [1] for a certain class of non-self-adjoint semi-elliptic operators. In Section 1, an estimate for the eigenvalues of a class of non-self-adjoint semi-elliptic operators in a bounded domain is proved. In Section 2, the completeness of the eigenfunctions in L2(Ω) is demonstrated; more precisely, it is proved that sp’(T)= L2(Ω), where sp’(T) denotes the closed subspace covered by the eigenvectors of T corresponding to the non-zero eigenvalues.
This article considers a two-input productivity function that relates income, capital, and labor and satisfies certain initial conditions. For the intensive form of the productivity function, the existence of an asymptote is investigated.
ABOUT IRRADIATION ABSORPTION ANOMALIES EFFECTS IN THE MEDIA WITH HELICAL PERIODICAL STRUCTURE
This paper examines the transmission and reflection of light under normal incidence on a layer of a medium with a spiral periodic structure, the axis of which is perpendicular to the boundary surfaces. The influence of changes in various medium parameters (layer thickness, real and imaginary parts of the average permittivity, and dielectric anisotropy) on previously discovered effects is studied. The characteristics of anomalously strong absorption of radiation in periodic media near the diffraction reflection region are investigated. The physical mechanisms underlying the observed patterns are discussed.
THE PULSARS RADIO ILLUMINATION AND MAGNETIC MOMENTS OF NEUTRON STARS
Using the formulas of the theory of radio emission of pulsars given in the review [1], based on the available observational data, the radiances of pulsars and the magnetic moments of 491 neutron stars were calculated.
ABOUT ONE PROBLEM OF OPTIMAL STABILIZATION OF FIXED PLATE
We study the problem of optimally stabilizing the vibrations of a rectangular plate, one side of which is rigidly fixed, while the others are free. The plate is stabilized using a control force applied to its upper surface. The problem is solved using beam functions. An optimal control action is obtained that minimizes the objective functional, which represents the total energy.
SOLUTION OF MIXED PROBLEM OF ELASTICITY THEORY FOR CYLINDRICAL ORTHOTROPIC PLATE IN THE FORM OF RECTANGULAR
This paper presents a solution to a plane elasticity problem for a cylindrically orthotropic rectangular plate with displacements specified on one edge and stresses on the other three. A small physical parameter is introduced, and the stress function is considered in the problem as a power of this parameter. In recurrent boundary value problems, the stress function is represented as a Fourier series. Infinite systems of linear algebraic equations are obtained to determine the expansion coefficients. The solution to the problem for specific parameter values is reduced to numerical results, allowing one to study the nature of the stress distribution near the embedment.
ON APPROXIMATION OF THE SOLUTIONS OF BOUNDARY-VALUE PROBLEMS IN UNBOUNDED DOMAINS
In this paper, it is proved that the solution of the boundary value problem in an unbounded domain $\Omega$ can be obtained as the limit as $r \rightarrow \infty$ of the solution ur of the boundary value problem in a bounded domain $\Omega \cap \Omega B_ {r, \mu$, where $B_ {r, \mu}=\left\{x; \big |x \big |_\mu < r\right\}$ is a generalized ball. A similar result is obtained for solutions of problems in a half-space.
HYPERIDENTITIES OF LEFT AND RIGHT DISTRIBUTIVITIES IN VARIETIES OF SEMI­GROUPS
An equality in a semigroup variety is called a superidentity if, substituting any term of the variety into the functional symbols of this equality, we obtain identities. Accordingly, an equality in a semigroup variety is called a presuperidentity if, substituting any term of the variety, excluding projection terms, we obtain identities into the functional symbols of this equality. In this paper, satisfiability criteria for superidentities and presuperidentities of left and right distributivities of various ranks were found.
ON THE SPECTRAL THEOREM FOR A BANACH REPRESENTATION OF A COMPACT GROUP GENERATED BY DELSARTE SHIFTS
This paper is devoted to the spectral theorem for a Banach representation of a compact group G in a Banach space X generated by Delsarte shifts.
ATTRACTORS OF THE DEGENERATING EVOLUTIONARY EQUATIONS
This article proves the existence of semigroup attractors generated by mixed problems for degenerate second-order evolution equations, where degeneration can occur either on the entire boundary or on a part of it. The compactness of the attractors in the space L2 is established, and a Lyapunov function is constructed to elucidate their structure.
ON A SYSTEM OF FUNCTIONS GENERATING CLASSICAL ORTHOGONAL SYSTEMS OF ALGEBRAIC POLYNOMIALS
In contrast to the two well-known methods for introducing classical orthogonal systems of algebraic polynomials (see [1--5], respectively), this paper proposes a completely different method for introducing these same polynomials. At the same time, integral representations for them that are different from the known ones are also given. The latter, in comparison with the Rodrigues formulas known in the literature, are simple and, what is more important, the integrands are not multivalued, but single-valued. In this paper it is proved that the system of functions $$I^{(\alpha,\beta)}_{n,n_1,a}=\frac{{\ae}_{n,n_1,a}}{2\pi_i}\int\limits_C\frac{(1-\frac{x}{a})^{n_1+\zeta}(\frac{x}{a})^{-\zeta}}{\Gamma(n_1+\zeta+\alpha+1)\Gamma(-\zeta+\beta+1)}\cdot\frac{d\zeta}{\prot\limits^m_{v=0}(\zeta+v)},$$ where $n_1\geq n\geq 0$ are integers, $\alpha>-1, \beta>-1, a>0$ are arbitrary numbers, $x\in(0,a)$, the simple contour $C$ encloses neighborhoods of points $0,-1,-2,...,-n, {\ae}{n,n_1,a}=\Gamma(n_1+n+\alpha+\beta=2),$ generates the above-mentioned orthogonal polynomials, namely: for $n_1=n, a=1$ we obtain the Jacobi polynomials, and for $n_1=a\rightarrow\infty$ -- the Laguerre polynomials $L^{(\beta)}_n(x),$ multiplied by $\exp^{-x}$. At the endpoints $[0,a] \mathrm{Y}\limits^{(\alpha, \beta)_{n,n_1,a}(x)} $ is understood in the sense of $x\rightarrow0+, x\rightarrow a–$.
LOADS TRANSFER FROM THE SYSTEMS OF FINITE NUMBER FINITE-LENGTH STRINGERS TO AN INFINITE SHEET THROUGH ADHESIVE LAYERS
The article considers the problem for an elastic infinite plate (sheet), which along of two parallel lines of its upper surface is strengthened by systems of finite number finite-lenght stringers having different elastic properties. The interaction between infinite sheet and stringers take place through thin, uniform, elastic adhesive layers having other physical-mechanical properties and geometric configuration. The stringers are deformed under the action of horizontal concentrated forces, which are applied at one end points of stringers. The problem of determining unknown contact forces acting between infinite sheet and stringers is reduced to the system of Fredholm integral equations of second kind with respect to arbitrary finite number of unknown functions, which are specified along of two parallel lines on different finite intervals. Further, are determined of the change regions of the problem characteristic parameters, for which this system of integral equations allows the exact solution and which can be solved by the method of successive approximations. Some particular cases are considered and the character and behavior of unknown shear contact forces near the end points of the stringers are investigated. For these cases numerical results depending on the multiparameters of the problem are investigated in the previous article (A.V. Kerobyan, K.P. Sahakyan, Proc. YSU. Phys. Math. Sci. 57 (3) (2023), 86--100).
Copyright (c) 2025 Proceedings of the YSU https://creativecommons.org/licenses/by-nc/4.0/
STUDY OF $\mathrm{PbWO}_4$ CRYSTAL FOR ePIC EmCal PROTOTYPE
The study presents the characterization of $\mathrm{PbWO}_4$ crystals intended for the prototype of the EmCal electromagnetic calorimeter of the ePIC detector, which is being constructed at the Electron-Ion Collider in the Brookhaven National Laboratory at the USA. Measurements were performed on 20 crystals produced by "Crytur" company, each of which was then thoroughly examined under a microscope. Transversal transparency measurements were made at the center of the crystals, as well as at several fixed points equidistant from the center to the right and left sides to study uniformity. The average transparency of the crystals is 21.3%, 65.6%, and 71.7% for wavelengths of 360 nm, 440 nm, and 600 nm, respectively. The transmittance measurements repeated 10 times in the center of each crystal show that accuracy of our measurement is better than 10%. The light yield of $\mathrm{PbWO}_4$ was estimated to be an average 16 pe/MeV. The optical characteristics of Crytur crystals meet the requirements of the EIC electromagnetic calorimeter. After performing all the necessary measurements, 16 crystals in good condition were selected for the calorimeter prototype. A 4$\times$4 prototype of EmCal was designed, constructed and tested with cosmic muons.
ON A RESULT CONCERNING ALGEBRAIC CURVES PASSING THROUGH n-INDEPENDENT NODES
Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e. each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,n-3)+3= (n+1)+n+\cdots+5+3.$ In this paper we prove that there are at most three linearly independent curves of degree less than or equal to $n-1$ that pass through all the nodes of $\mathcal X.$ We provide a characterization of the case when there are exactly three such curves. Namely, we prove that then the set $\mathcal X$ has a very special construction: either all its nodes belong to a curve of degree $n-2,$ or all its nodes but three belong to a (maximal) curve of degree $n-3.$ This result complements a result established recently by H. Kloyan, D. Voskanyan, and H. Hakopian. Note that the proofs of the two results are completely different.
Copyright (c) 2022 Proceedings of the YSU https://creativecommons.org/licenses/by-nc/4.0/
FINITE-DIFFERENCE STOCHASTIC SCHEMES FOR MINIMIZING A STRONGLY QUASICONVEX NON-DIFFERENTIABLE FUNCTION ON ℝn. THE NURMINSKII METHOD
In this work, stochastic approximation methods based on finite-differences are investigated for the problem of minimizing quasiconvex functions. The main result of this work is the derivation of convergence rate estimates for stochastic finite-difference methods in the case of quasiconvex functions. The obtained results significantly extend the applicability of stochastic finite-difference methods to nonsmooth quasiconvex optimization problems and provide a rigorous justification for their use in black-box settings where the oracle returns only function values.
Copyright (c) 2026 Proceedings of the YSU https://creativecommons.org/licenses/by-nc/4.0/
ADHESIVE CONTACT PROBLEM FOR AN INFINITE SHEET STRENGTHENED BY THREE PARALLEL FINITE-LENGTH STRINGERS
The article considers the problem for an elastic infinite sheet, which along of three parallel lines of its upper surface is strengthened by three parallel finite-length stringers having different elastic properties. The interaction between infinite sheet and three parallel stringers takes place through thin elastic adhesive layers having other physical-mechanical properties and geometric configuration. The stringers are deformed under the action of horizontal concentrated forces, which are applied at one end points of the stringers. The problem of determining the law of distribution of unknown contact forces acting between infinite sheet and stringers is reduced to the system of Fredholm integral equations of second kind with respect to three unknown functions, which are specified on three parallel finite intervals. Further, are determined of the change regions of the problem characteristic parameters, for which this system of integral equations allows the exact solutions and which can be solved by the method of successive approximations. Some special cases are considered and the character and behavior of unknown shear forces near the end points of the stringers are investigated. For these cases numerical results depending on the multiparameters of the problem are investigated in the article by A.V. Kerobyan and K.P. Sahakyan (Proc. YSU. Phys. Math. Sci., 57 (3) (2023), 86-100).
HERMITE MULTIVARIATE INTERPOLATION FORMULA
We present a new formula for the Hermite multivariate interpolation problem in the framework of the Chung--Yao approach.
KNOWLEDGE-DISTILLED VARIATIONAL BAYESIAN FRAMEWORK FOR EFFICIENT LARGE-SCALE IMAGE DEHAZING
Real-time processing of large-scale image databases with state-of-the-art dehazing methods presents a significant computational challenge: methods that achieve superior generalization typically require substantial inference time, limiting their deployment in real-time applications. High-performing methods employ complex multi-scale processing and deep architectures, typically achievin less than 5 FPS on high-resolution images. Building on the preliminary multi-scale variational Bayesian framework [1, 2], which achieves strong synthetic-to-real generalization, this paper proposes knowledge distillation to transfer the generalization capabilities of high-performance models to a lightweight Vision Transformer-based student network. The student leverages patch-based processing and reduced architectural complexity to achieve over $150\times$ speedup, while maintaining competitive performance through a theoretically-grounded distillation framework integrated into the variational Bayesian objective. Additionally, the atmospheric scattering model is extended to estimate space-variant atmospheric light, improving performance on varying haze regions. Trained solely on synthetic Haze4K data, the proposed method stays competitive on synthetic-to-real generalization and downstream object detection (on the augmented KITTI dataset) tasks, while achieving superior inference speed for large-scale real-world applications.
OPERATIONAL ENVIRONMENT FOR DYNAMIC LOADING AND MANAGEMENT OF MODULAR APPLICATIONS IN EMBEDDED SYSTEMS
Ensuring modularity under strict resource constraints is a critical challenge in the design of embedded systems. This paper presents "Part Only RTOS", an operational environment developed for devices based on 8-bit ATmega2560 microcontrollers. The core feature of the proposed system is the ability to dynamically load and execute application modules (in .bin format) from an SD card without requiring full hardware reprogramming. The developed three-tier architecture, comprising the Launcher, Bootloader, and Kernel, ensures reliable software validation via unique pattern matching and efficient memory utilisation. Experimental results demonstrate that the system kernel occupies less than $2.5~KB$ of Flash memory, and the loading time for a 10 KB module is approximately 132 ms. This solution enables the creation of flexible and high-performance modular embedded systems on low-power microcontroller platforms.
ON SUM EDGE-COLORINGS OF SOME REGULAR GRAPHS
A sum edge-coloring of a graph is an assignment of positive integers to the edges of the graph, so that adjacent edges correspond to different numbers (colors) and the sum of the numbers on all the edges is minimum possible. This minimum possible sum is called the edge-chromatic sum of the graph, and the minimal number of colors needed for a sum edge-coloring is called the edge-strength of the graph. In this paper, we give the exact values of the edge-chromatic sums and edge-strengths for cycle powers and generalized cycles.
NEAR-INTERVAL EDGE-COLORINGS OF COMPLETE BIPARTITE GRAPHS
A proper edge-coloring of a graph $G$ is a mapping $\alpha:E(G)\longrightarrow \mathbb{N}$ such that $\alpha(e)\not=\alpha(e')$ for every pair of adjacent edges $e$ and $e'$ in $G$. If $\alpha $ is a proper edge-coloring of $G$ and $v\in V(G)$, then $S_{G}\left(v,\alpha \right)$ denotes the set of colors appearing on edges incident to $v$. A proper edge-coloring $\alpha$ of a graph $G$ with colors $1,\ldots,t$ is called a near-interval $t$-coloring if all colors are used, and for each vertex $v\in V(G)$, $S_G(v,\alpha)$ is an interval of integers with no more than one gap. If a graph $G$ has such a coloring, the minimum number of colors in a near-interval coloring of a graph $G$ is denoted by $w^{1}(G)$. It is known that all complete bipartite graphs admit near-interval colorings. In this paper, we determine or bound the parameter $w^{1}(K_{m,n})$ ($m,n\in\mathbb{N}$) for complete bipartite graphs.
ON DEFICIENCY OF COMPLETE 3-PARTITE AND 4-PARTITE GRAPHS
A proper t-edge-coloring of a graph G is a mapping $\alpha: E(G)\rightarrow \{1,\ldots,t\}$ such that all colors are used, and $\alpha(e)\neq \alpha(e^{\prime})$ for every pair of adjacent edges $e,e^{\prime}\in E(G)$. If $\alpha $ is a proper edge-coloring of a graph G and $v\in V(G)$, then the spectrum of a vertex v, denoted by $S\left(v,\alpha \right)$, is the set of all colors appearing on edges incident to v. The deficiency of $\alpha$ at vertex $v\in V(G)$, denoted by $\mathrm{def}(v,\alpha)$, is the minimum number of integers that must be added to $S\left(v,\alpha \right)$ to form an interval, and the deficiency $\mathrm{def}\left(G,\alpha\right)$ of a proper edge-coloring $\alpha$ of G is defined as the sum $\displaystyle\sum_{v\in V(G)}\mathrm{def}(v,\alpha)$. The deficiency of a graph G, denoted by $\mathrm{def}(G)$, is defined as follows: $\mathrm{def}(G)=\min_{\alpha}\mathrm{def}\left(G,\alpha\right)$, where the minimum is taken over all possible proper edge-colorings of G. In 2019, Davtyan, Minasyan, and Petrosyan provided an upper bound on the deficiency of complete multipartite graphs. In this paper, we improve this bound for complete tripartite and some complete 4-partite graphs. We also confirm the conjecture that states the deficiency of a graph is bounded by the number of vertices of the graph for all tripartite graphs containing up to 10 vertices.
ON SUM EDGE-COLORINGS OF COMPLETE TRIPARTITE GRAPHS
A proper edge-coloring of a graph is called a sum edge-coloring if it minimizes the total sum of colors on all the edges of the graph. The aforementioned minimal sum is called the edge-chromatic sum of the graph, and the minimal number of colors needed for a sum edge-coloring is called the edge-strength of the graph. In this paper, upper bounds on the values of the edge-chromatic sums of some complete tripartite graphs are given, while for some other complete tripartite graphs, the exact values of both parameters are obtained.
ON THE CALCULATION OF THE COEFFICIENTS OF CUBIC SPLINES ON A SET OF EQUIDISTANT KNOTS
As is known, the coefficients of the interpolation cubic spline are found by solving a tridiagonal system of linear algebraic equations of a special type. To solve the system, a well-known numerical algorithm is usually used. In this paper, an alternative method for finding the coefficients of a natural cubic spline on a uniform set of knots is proposed. The method is based on the analytical inversion of the tridiagonal matrix, which made it possible to obtain closed-form expressions for the coefficients. This approach allows us both identify the analytical dependence of the spline coefficients on its values at the knots and obtain simple formulas for calculating these coefficients, by passing the solution of the system.
THE ANALYSIS OF VIBRATIONS OF CURVILINEAR TETRAGON WITH RECTILINEAR ANISOTROPY
The behaviour of frequencies of vibrations of curvilinear tetragon with a rectilinear anisotropy is investigated.
Copyright (c) 2001 Proceedings of the YSU https://creativecommons.org/licenses/by-nc/4.0/
ON THE REPRESENTATION OF FUNCTIONS BY SERIES OFLEGANDRE POLYNOMIALS IN WEIGHTED $ L_\mu ^q[-1, 1]$ SPACES
In this paper we construct the weighted $L_\mu ^q[-1, 1], q \in[1, 4/3]\cup [4, +\infty],$ and Legander series $\sum \limits_{k=1} ^\infty c_pP_k(x)$, which is universal in $L_\mu ^q [-1, 1]$.
EINSTEIN REPRESENTATION OF TENSOR-SCALAR THEORIES OF GRAVITATION
In this paper considerations on conformal transformations of tensor-scalar theory are represented. It is shown that under certain conditions conformal transformed theory of Jordan–Brans–Dicke is equivalent to Einstein theory with a minimally bounded scalar field source.
ON VIBRATIONS OF ELECTROCONDUCTIVE PLATE IN LONGITUDINAL MAGNETIC FIELD
The problem of vibrations of transversal-isotropic electroconductive plate in view of transversal shear strains and induced electromagnetic field at presence of constant longitudinal magnetic field is investigated.