In this paper we investigate measure-theoretic properties of the class of all weakly mixing transformations on a finite measure space which preserve measurability. The main result in this paper is the following theorem: If $\phi $ is a weakly mixing transformation on a finite measure space $( S, \mathcal A , \mu )$ with the property that $\phi (\mathcal A ) \subseteq \mathcal A ,$ then for every $A, B $ in $\mathcal A$ there is a subset $J(A,B)$ of the set of non-negative integers of density zero such that $\lim _{m \to \infty ,m \notin J(A,B)} \mu (A \cap \phi ^m(B)) = (\mu (A) / \mu (S))\lim _{n \to \infty } \mu \,(\phi^n(B)).$ Furthermore, we show that for most useful measure spaces we can strengthen the preceding statement to obtain a set of density zero that works for all pairs of measurable sets $A$ and $ B.$ As corollaries we obtain a number of inclusion theorems. The results presented here extend the well-known classical results (for invertible weakly mixing transformations), results of R. E. Rice [17] (for strongly mixing), a result of C. Sempi [19] (for weakly mixing) and previous results of the author [8, 10] (for weakly mixing and ergodicity).   2000 Mathematics Subject Classification. Primary: 28D05, 37A25; Secondary: 37A05, 47A35

We consider the following system of rational difference equations in the plane: $$\left\{\begin{aligned}%{rcl}x_{n+1} &= \frac{\alpha_1}{A_1+B_1 x_n+ C_1y_n} \\[0.2cm]y_{n+1} &= \frac{\alpha_2}{A_2+B_2 x_n+ C_2y_n}\end{aligned}\right. \, , \quad n=0,1,2,\ldots $$ where the parameters $\alpha_1, \alpha_2, A_1, A_2, B_1, B_2, C_1, C_2$ are positive numbers and initial conditions $x_0$ and $y_0$ are nonnegative numbers. We prove that the unique positive equilibrium of this system is globally asymptotically stable. Also, we determine the rate of convergence of a solution that converges to the equilibrium $E=(\bar{x},\bar{y})$ of this systems.   2000 Mathematics Subject Classification. 39A10, 39A11, 39A20

We investigate the global character of the difference equation of the form $$ x_{n+1} = f(x_n, x_{n-1},\ldots, x_{n-k+1}), \quadn=0,1, \ldots $$ with several equilibrium points, where $f$ is increasing in all its variables. We show that a considerable number of well known difference equations can be embeded into this equation through the iteration process. We also show that a negative feedback condition can be used to determine a part of the basin of attraction of different equilibrium points, and that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points.   2000 Mathematics Subject Classification. 39A10, 39A11

We investigate the period-two trichotomies of solutions of the equation $$x_{n+1} = f(x_{n}, x_{n-1},x_{n-2}), \quad n=0, 1, \ldots $$ where the function $f$ satisfies certain monotonicity conditions. We give fairly general conditions for period-two trichotomies to occur and illustrate the results with numerous examples.   1991 Mathematics Subject Classification. 39A10, 39A11

Motivated with Hille's first exponential formula for $C_{0}$ semigroups, we prove a formula for $n-$times integrated semigroups. At first we prove a formula for twice integrated semigroup, and, later, we generalize this formula for $n-$times integrated semigroups.   2000 Mathematics Subject Classification. 47D60, 47D62

Enterprise resource planning (ERP) is widely used to boost the total market power of businesses. The wrong selection is one of the key reasons why ERP installations fail. Due to the complexity of the business environment and the range of ERP systems, choosing an ERP system is a complex and time-consuming procedure. ERP alternatives may be assessed using several criteria, so the ERP selection process may be considered a multi-criteria decision-making (MCDM) problem. In this study, the rough best worst method (BWM) was used to determine criteria weights, while the newly developed rough integrated simple weighted sum product (WISP) was used to rank ERP alternatives. Results suggest that the SFT-4 coded software is regarded as the best option, followed by SFT-5, SFT-6, SFT-2, SFT-3, and SFT-1. Results of the newly developed rough WISP method are compared to those of existing rough techniques in the sensitivity analysis. The differences between them have been found to be negligible. The outcomes show how effectively developed rough BWM and WISP integrated method performs in terms of ERP selection with usability, accuracy, ease of use, and consistency. This study will help decision-makers in a context where ERP is implemented choose the best ERP software for different sectors.

In this paper we analyze the values and the properties of the function $S(n,l):=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(n-k)^{l}\ \(n,l\in \mathbb{N\cup }\left\{ 0\right\} ),$ for $n<l.$ At first, we obtain two recurrence relations. Namely, we prove that forevery $n\in \mathbb{N\cup } \left\{ 0\right\} $ and every $l\in \mathbb{N}$ such that $l>n,$ we have\begin{equation*}S(n+1,l)=\sum_{k=1}^{l-n}\binom{l}{k}S(n,l-k),\end{equation*}and also, for every $n\in \mathbb{N\cup }\left\{ 0\right\} $and every $l\in \mathbb{N}$, we have\begin{equation*}S(n+1,l)=(n+1)S(n,l-1)+(n+1)S(n+1,l-1).\end{equation*}Further, we conclude that for every $n\geq 2$ and every $l\geq n$ the following representation formula holds\begin{multline*}S(n,l) =\sum\limits_{k_{1}=1}^{l-(n-1)}\binom{l}{k_{1}}\sum\limits_{k_{2}=1}^{l-k_{1}-(n-2)}\binom{l-k_{1}}{k_{2}}\\\cdot\sum\limits_{k_{3}=1}^{l-k_{1}-k_{2}-(n-3)}\binom{l-k_{1}-k_{2}}{k_{3}}\dots\sum\limits_{k_{n-1}=1}^{l-\sum\limits_{i=1}^{n-2}k_{i}-1}\binom{l-\sum\limits_{i=1}^{n-2}k_{i}}{k_{n-1}}.\end{multline*}We obtain an explicit formula for the calculation $S(n,l),$ especially for $ l=n+1,\dots,n+5,$ and later we give a general result.   2000 Mathematics Subject Classification. 40B05, 11Y55, 05A10

In this paper we describe five methods for the calculation of the moments\begin{equation*}\label{Momenti}\mathbb{M}_{n,m}=\int_{0}^{m}\varphi_{m}(t)t^{n}dt,n\in\mathbb{N}_{0},\end{equation*}where weight function $\varphi_{m}(\cdot)$ is the cardinal B-spline of order $m,m\in\mathbb{N}.$   2000 Mathematics Subject Classification. 65D07, 41A15

Single axis trackers have their application in the efficiency improvement process of photovoltaic systems. In spite of large investment costs, their application is necessary in the areas with a low amount of available solar irradiation. This is applicable only if it is required to exploit all available energy sources at any cost. With optimization and improvement of these systems it is possible to increase their efficiency, reduce energy usage for their movement and improve system reliability. Motion parameters are significant for the optimization of the system. It is necessary to set and monitor the right parameters as a prerequisite for system optimization. The process of calibration of single axis tracker prototype is shown in this paper. It is shown that proper calibration and system adjustment could improve efficiency of the system.

This paper introduces a novel chaotic strategy for controlling the acceleration coefficients within the particle swarm optimization (PSO) algorithm. The PSO algorithm with chaotic exponential-based acceleration coefficients is developed to enhance the exploration of the search space and avoid premature convergence, a common issue associated with the standard PSO algorithm. The PSO algorithm with chaotic exponential-based acceleration coefficients is applied to multilevel image thresholding. The attained experimental results demonstrate that the PSO algorithm utilizing the chaotic strategy to control the cognitive and social acceleration coefficients can be successfully utilized for the selection of image thresholds across a variety of images.

This paper deals with the development and testing of an intelligent vibration damping system that can real-time analyze the vibration of the system, of which it is an integral part, and acting on vibration decreasing. Damping system is based on the pendulum tuned mass damper. The system can determine combinations of changes in mechanical properties and introduce additional disturbances to achieve optimal reduction of induced vibrations. The system's performances were evaluated through experiments conducted on an experimental model. The experiment involved variations in pendulum to observe the system's response for initial perturbance. The embedded system was implemented using an Arduino controller and accompanying components such as an actuator, data acquisition device, accelerometer, and PC. The paper also discusses a proposed adaptive pendulum mass damper with real-time tuning capability for controlling structural vibrations. Combining experimental and simulation results, the paper demonstrates the effectiveness of the developed system and the adaptive pendulum mass damper in actively damping vibrations in structures.

The political systems of Bosnia and Herzegovina, Cyprus, and Lebanon are defined by ethnic or religious divides that have historical roots in their Ottoman and European colonial past. This paper aims to investigate the impact of these historical legacies on popular representation and institutional structures. The Ottomans and European colonizers created institutions that allowed subjects to elect their representatives, shaping these countries' initial experiences with modernity. This paper employs historical institutionalism as a theoretical approach, and path dependency as a methodology to explore how historical legacies have shaped the political institutions and power‐sharing arrangements in Bosnia and Herzegovina, Cyprus, and Lebanon.

Energy production is one of the most crucial scientific, technological, economic, and political challenges in today's world. In response to the increasing demands of the European Union for reducing CO2 emissions and mitigating climate change, countries are obligated to align their national energy plans with clearly defined objectives, aiming for climate neutrality by 2050. This paper explores the location potential and optimal implementation of biomass power plants in Bosnia and Herzegovina. Biomass power plants, particularly those utilizing wood and plant biomass, are a promising solution for simultaneously reducing pollution, addressing unnecessary waste, and improving the energy efficiency of the systems where they are installed. The paper concludes that the potential of wood and plant biomass in Bosnia and Herzegovina remains underutilized. From an eco-perspective, the paper will also examine the reuse of large degraded areas in mines, where the fast-growing energy crops like Miscanthus could be planted.

The Teacher Efficacy for Inclusive Practices (TEIP) scale is a widely used instrument for assessing teachers' effectiveness in implementing inclusive practices. The TEIP has not been validated in Bosnia and Herzegovina (BIH). The goal of the present study was to conduct a confirmatory factor analysis (CFA) of the TEIP in a sample of teachers in BIH. The sample consisted of 239 elementary school teachers (204 females, 35 males; mean age‐ 43.5 years, SD‐ 8.3 years). The results of the CFA found that the original TEIP structure was not a good fit for our data. However, through some modifications on the item level, we created a satisfactory model that retained the same structure as the original TEIC. The composite reliability and construct maximal reliability were well above the recommended cut point of 0.70 for all factors. Our findings suggest that the modified version of the TEIP has the potential to be an effective and useful instrument for assessing teachers' efficacy in inclusive practices in BIH. This study represents an important step towards the cultural adaptation and validation of the TEIP in BIH, highlighting areas for future research and application in diverse educational contexts.