AN INTRODUCTION TO THE INDEPENDENT TOPOLOGICAL STRUCTURE GENERATED BY FUZZY SOFT γ\gamma-OPEN SETS

In this study, we propose a new generalized fuzzy soft open set, namely the fuzzy soft $\gamma$-open set. Notably, the newly defined fuzzy soft $\gamma$-open set is a special type of fuzzy soft-pre-open set. Additionally, a diagram (Fig. 3) is used to show how fuzzy soft $\gamma$-open sets are related to various existing stronger and weaker forms of fuzzy soft-open sets. The main focus of this paper is on the autonomous topological structure produced by fuzzy soft $\gamma$-open sets. Furthermore, we introduce the concepts of fuzzy soft $\gamma$-interior and $\gamma$-closure operators, which provide another way to define fuzzy soft $\gamma$-topology. Finally, we introduce and explore fuzzy soft $\gamma$-continuity as the application of the defined notions in this regard

ON OBSERVABILITY CONTROL FOR DIFFERENTIAL EQUATIONS

We consider a controlled linear differential equation with constraints as in the author's previous paper. The controller's goal is to displace an initial state of $x_0$ to a specified final state $x_T$. An observer, unaware of the system's state vector, attempts to determine $x_T$ by analyzing the vector $y(t)$, which is linked to $x(t)$. Using $y(t)$, the observer constructs a set of possible values for $x_T$. When specific constraints are used for the controls (or disturbances, from the observer's opinion), this set becomes an ellipsoid, characterized by a set of differential equations. The controller, in turn, aims to achieve its own objectives while simultaneously generating the most challenging signals for the observer. Unlike the previous article of the author not scalar, but two-criterion control observation problem is considered here. It is solved in functional spaces in two ways, without passing to sampling of a system. The solution boils down to determination of finite-dimensional parameters of optimal control from the system of linear algebraic equations. As the third option the problem can be solved also by sampling, but then the solution turns out piecewise-constant. We explore an example to illustrate these concepts

EQUILIBRIUM TRAJECTORIES FOR CONTROL SYSTEMS WITH HETEROGENEOUS DYNAMICS

The paper considers the construction of equilibrium in bimatrix games with heterogeneous dynamics of players' interaction. Heterogeneity of dynamics is connected with difference in maximal rates of the participants. In such a formulation, the switching curves of players' controls are represented by fractional rational functions and are constructed on the basis of N.N. Krasovskii's guaranteed strategies using elements of L.S. Pontryagin's maximum principle. Equilibrium trajectories are generated within the framework of the concept of the dynamic Nash equilibrium introduced by A.F. Kleimenov and are obtained by pasting together the characteristics of the Hamilton-Jacobi equations expressed as exponential functions. The sensitivity analysis is carried out for the shapes of control switching curves with respect to the proportions of players' maximal rates. The comparative analysis is implemented for the values of players' payoffs calculated on equilibrium trajectories of the dynamic game

IMPROVED BRANCH-AND-PRICE ALGORITHM FOR THE EFFICIENT 2-TERMINAL RELIABILITY PROBLEM

The Efficient 2-Terminal Reliability Problem is a nonlinear optimization problem aimed at designing a minimal-cost network within the reliability guaranties. Recent research has provided Branch-and-Price (BnP) solution based on probability relaxation, the Dantzig–Wolfe decomposition, followed by the column generation technique, and Branch-and-Bound scheme. Unfortunately, the performance of this algorithm deteriorates in cases of high-density graphs and stringent unreliability thresholds. By extending our recent approach, we introduce an improved BnP algorithm supplemented with novel valid inequalities, more efficient nonlinear integer pricing problem solver, primal heuristics, and branching strategies. Evaluation results on benchmarking instances demonstrate significant performance advantage of the proposed method

THE IMPACT OF TOXICANTS IN THE MARINE THREE ECOLOGICAL FOOD-CHAIN ENVIRONMENT: A MATHEMATICAL APPROACH

To explore the impact of toxicants on a marine ecological food chain system consisting of three species, this work develops and analyzes a non-linear mathematical model. The model consists of five state variables: phytoplankton, zooplankton, fish, environmental toxicant, and organismal toxicant. We have incorporated the Monod-Haldane functional response as a predation function for each species. Using the Jacobian matrix, the stability analysis was conducted, and necessary constraints were obtained for the system's local and global stability. Hopf bifurcation analysis was performed for carrying capacity ($K$) and the rate of decrease in the growth rate of phytoplankton due to the presence of toxicants ($r_1$). Also, phase portraits are presented for different parameters of the model. In addition, numerical simulations are executed using MATLAB to prove theoretical findings and explore the impact of parameter variation on ecological species behavior

APPROXIMATION OF ONE CLASS OF SMOOTH FUNCTIONS BY ANOTHER CLASS OF SMOOTHER FUNCTIONS ON THE AXIS

This paper investigates the problem of best and best linear approximation in the space of functions on the real axis with bounded Fourier transform. The study focuses on approximating the class $B_1^{n-k}(1)$ of functions whose derivatives of order $n-k-1$ have variation bounded by 1 by the class $B_{2}^n(N)$ of functions whose $n$th-order derivative ($0 \le k 0$. This problem is related to Stechkin's problem and the corresponding sharp Kolmogorov inequality, both previously studied by the author. Stechkin's problem concerns the best approximation in the uniform norm on the real axis of $k$th-order differentiation operators by bounded linear operators from $L_2$ to $C$, considered on the class of functions whose Fourier transform of the $n$th-order derivative ($0 \le k < n$) is summable

A TWO-STAGE METHOD FOR SOLVING A NONLINEAR ILL-POSED OPERATOR EQUATION AND ITS APPLICATION TO THE INVERSE PROBLEM OF THERMAL SOUNDING OF THE ATMOSPHERE

The inverse problem of reconstructing the vertical profiles of CO$_2$ in the atmosphere by IR spectra of the solar light transmission is investigated. To solve this problem, we propose a two-stage method. At the first stage, we use the modified Tikhonov method. At the second stage, to approximate a solution of the regularized equation, we apply a nonlinear analogue of the modified steepest descent method. The convergence theorem is formulated and the results of numerical experiments for retrieving the concentration of carbon dioxide in the atmosphere from measured spectra are discussed

STABILITY OF GENERAL QUADRATIC EULER–LAGRANGE FUNCTIONAL EQUATIONS IN MODULAR SPACES: A FIXED POINT APPROACH

In this paper, we establish a result on the  Hyers–Ulam–Rassias stability of the Euler–Lagrange functional equation. The work presented here is in the framework of modular spaces. We obtain our results by applying a fixed point theorem. Moreover, we do not use the $\Delta_\alpha$-condition of modular spaces in the proofs of our theorems, which introduces additional complications in establishing stability. We also provide some corollaries and an illustrative example. Apart from its main objective of obtaining a stability result, the present paper also demonstrates how fixed point methods are applicable in modular spaces

ASYMPTOTIC BEHAVIOR OF REACHABLE SETS WITH LpL_p-BOUNDED CONTROLS

The paper studies the reachable sets of control systems over a fixed time interval, subject to control constraints defined as a ball in the $L_p$ space for $p \geq 1$. The dependence of reachable sets on the parameter $p$ is investigated. For affine-control nonlinear systems, it is established that these sets are continuous in the Hausdorff metric for all $p$, including $p=1$ and $p=\infty$. In the case of linear systems, estimates for the Hausdorff distance between the sets are derived, and their asymptotic behavior as $p\to 1$ and $p\to \infty$ is analyzed. For $p = 1$, the reachable set, up to closure, coincides with the reachable set of the system with impulse control under a constraint on the magnitude of the impulse. The case $p = \infty$ corresponds to geometric (instantaneous) constraints on the control

ENUMERATION INTERSECTION ARRAYS OF SHILLA GRAPHS WITH B = 6

Let $\Gamma$ be a distance-regular graph of diameter $3$, and let $\theta_1$ be its second eigenvalue. The graph $\Gamma$ is called a Shilla graph if $\theta_1=a_3$. In this case, $\theta_1={(a_1+\sqrt{a_1^2+4k})}/{2}$, and $a=a_3$ divides $k$ We set $b=b(\Gamma)=k/a$. J.H. Koolen and J. Park found the intersection arrays of Shilla graphs with $b\le 3$. J. Cai, I.N. Belousov, and A.A. Makhnev enumerated the intersection arrays of Shilla graphs with $b=4$. H. Li, I.N. Belousov, and A.A. Makhnev found the intersection arrays of Shilla graphs with $b=5$. In this paper, we enumerate the intersection arrays of Shilla graphs with $b=6$