Search Results (62)

Sampling with replacement occurs when drawing without removing individuals from finite populations. It is a common distribution technique used in physics, biology, and medicine. It is used in state analysis of qubits, the physics of particle interactions, studies of genetic variation and variability, and analyzing the treatment effects from clinical trial analyses. When applied, sample statistics should be accompanied by confidence intervals. The major difficulty in expressing the confidence intervals in sampling with replacement consists of discreetness regarding the probability distribution. As a result, no mathematical formula can handle an optimum solution. Using a simple algorithm is proposed in order to obtain confidence intervals for sampling with replacement variables (x from m trials with replacement) and their proportion ($x/m$). A question-based discussion is presented. Traditional confidence intervals often require large sample sizes. Confidence intervals, constructed in a deterministic way provided by the proposed algorithm for sampling with replacement, allow constructing intervals without constraints. Full article

While many continuous distributions are known, the list of discrete ones (usually derived from counting) often reported is relatively short. This list becomes even shorter when dealing with dichotomous observables: binomial, hypergeometric, negative binomial, and uniform. Binomial distribution is important for medical studies, since a finite sample from a population included in a medical study with yes/no outcome resembles a series of independent Bernoulli trials. The problem of calculating the confidence interval (CI, with conventional risk of 5% or otherwise) is revealed to be a problem of combinatorics. Several algorithms dispute the exact calculation, each according to a formal definition of its exactness. For two algorithms, four previously proposed case studies are provided, for sample sizes of 30, 50, 100, 150, and 300. In these cases, at 1% significance level, ordered sequences defining the confidence bounds were generated for two formal definitions. Images of the error’s alternation are provided and discussed. Both algorithms propose symmetric solutions in terms of both CIs and actual coverage probabilities. The CIs are not symmetric relative to the observed variable, but are mirrored symmetric relative to the middle of the observed variable domain. When comparing the solutions proposed by the algorithms, with the increase in the sample size, the ratio of identical confidence levels is increased and the difference between actual and imposed coverage is shrunk. Full article

Some optimal and non-optimal iterative approaches for computing multiple zeros of nonlinear functions have recently been published in the literature when the multiplicity $\theta$ of the root is known. Here, we present a new family of iterative algorithms for multiple zeros that are distinct from the existing approaches. Some special cases of the new family are presented and it is found that existing Liu-Zhou methods are the special cases of the new family. To check the consistency and stability of the new methods, we consider the continuous stirred tank reactor problem, isentropic supersonic flow problem, eigenvalue problem, complex root problem, and standard test problem in the numerical section and we find that the new methods are more competitive with other existing fourth-order methods. In the numerical section, the error of the new methods confirms their robust character. Full article

Healthcare resources have changed fundamentally compared to decades ago. Modern bio-food products and sustainable solutions for their production have increased the attention of researchers, taking into account the current level of pollution of the earth and atmosphere along with modern technologies applied to processed foods. Therefore, this review aims to highlight: (1) the impact and relationship between the physiological parameters of the atmosphere, solar radiation and soil, (in terms of their composition and stages of formation and organization) along with the evolution to modern life; (2) the environmental impacts on algae, living organisms, food, and human health and sustainability. In addition, we address the significant impact of algae as a sustainable resource in reducing environmental pollution contributing to a healthier life. Full article

In the current era, the economic model that measures the dependence of the training offered by companies to their employees on certain variables allows for identifying the steps taken on account of digital transformations, given the fact that companies want to be competitive, to develop sustainably and the positive effect to it spreads globally. However, how digital transformation contributes remains unclear in both the literature and practice. Five descriptors of information on the economy in relation to the digital economy were extracted from the Eurostat database, and data on eight Eastern European countries in the period 2012–2020 served as primary data in the analysis. A generalized linear model was used as a statistical tool to infer the data series. Following the statistical regression analysis, it was found that the variable measuring the share of companies that offered training for the development/improvement of information and communication technology (ICT) skills is influenced by the combined effect of several other variables: ‘country’, ‘country × year’, ‘country × share of ICT personnel in total employees’, year × “share of ICT sector in GDP”. Based on the results, we noticed that the studied countries are included in two groups with distinct features, which influence the obtained GLZ model, showing the increase in the dependency effect or, on the contrary, the decrease in this effect. Full article

In this article, we present a novel three-step with-memory iterative method for solving nonlinear equations. We have improved the convergence order of a well-known optimal eighth-order iterative method by converting it into a with-memory version. The Hermite interpolating polynomial is utilized to compute a self-accelerating parameter that improves the convergence order. The proposed uni-parametric with-memory iterative method improves its R-order of convergence from 8 to $8.8989$. Additionally, no more function evaluations are required to achieve this improvement in convergence order. Furthermore, the efficiency index has increased from $1.6818$ to $1.7272$. The proposed method is shown to be more effective than some well-known existing methods, as shown by extensive numerical testing on a variety of problems. Full article

Some polynomials find their way into chemical graph theory less often than others. They could provide new ways of understanding the origins of regularities in the chemistry of specific classes of compounds. This study’s objective is to depict the place of polynomials in chemical graph theory. Different approaches and notations are explained and levelled. The mathematical aspects of a series of such polynomials are put into the context of recent research. The directions in which this project was intended to proceed and where it stands right now are presented. Full article

Linear relationships, expressing the electrochemical properties of molecules as functions of structure, give insight into the associated electrochemical process and are a tool for prediction. Many biological activities rely on water-based dissociation, making electrochemical properties a bridge between structure and activity. Motivated by a previous study, a replica is made here on a different dataset in order to validate/invalidate the previously reported results. There are several methods for obtaining structure-based descriptors. Some of the methods have been devised to account for molecular topology, some to account for molecular geometry, and others to account for both. Two methods are involved here to derive structure-based descriptors and further obtain structure–property relationships (FMPI and ChPE). In order to express structure descriptors, both FMPI and ChPE express first the topology of the molecule, using the heavy atoms identity matrix and the heavy atoms adjacency matrix, both square symmetric matrices in the belief that symmetry is one major factor of molecular stability. A set of 2,6-dimethyl-1,4-dihydropyridine derivatives with oxidation peak potentials and coulometrically determined number of electrons experimental data is subjected to the search for structure–activity relationships. Even if the 2,6-dimethyl-1,4-dihydropyridine is a symmetric compound (of C_s point group), their derivatives are generally not symmetric (9 out of 24 are asymmetric). The dataset is subjected to descriptive and inferential statistics in order to filter out the most relevant structure–activity relationship. The geometry is built using three levels of theory (one from molecular mechanics and two others from density functionals, of which one accounts for the interaction with water as solvent). One challenge of picking one out of two reported measured values is dealt with by calculating the likelihood associated with the two choices. Relevant structure–activity models are extracted and discussed. The use of in vivo (in water, SM8 model) models in geometry optimization (from MMFF94 and B3LYP and to M06 + Water SM8) results in a precision gain, but this is, in most of the cases, not statistically significant, and this can be considered a negative result. Full article

We present a new family of optimal eighth-order numerical methods for finding the multiple zeros of nonlinear functions. The methodology used for constructing the iterative scheme is based on the approach called the ‘weight factor approach’. This approach ingeniously combines weight functions to enhance convergence properties and stability. An extensive convergence analysis is conducted to prove that the proposed scheme achieves optimal eighth-order convergence, providing a significant improvement in efficiency over lower-order methods. Furthermore, the applicability of these novel methods to some real-world problems is demonstrated, showcasing their superior performance in terms of speed and accuracy. This is illustrated through a series of three examples involving basins of attraction with reflection symmetry, confirming the dominance of the new methods over existing counterparts. The examples highlight not only the robustness and precision of the proposed methods but also their practical utility in solving the complex nonlinear equations encountered in various scientific and engineering domains. Consequently, these eighth-order methods hold great promise for advancing computational techniques in fields that require the resolution of multiple roots with high precision. Full article

Approaches to the sustainable development of enterprises are a continuous concern of EU countries, contributing to the achievement of national well-being. Companies today face the combined effects of a multitude of factors that affect their results. This study was carried out to analyse the factors that influence the enterprises in the non-financial sector (industry, trade, and services). The analysis used the related data from eight Eastern European countries in the period of 2018–2020, and includes companies by country, size, number of employees, number of companies by number of employees, turnover, added value and average productivity per employee in generating turnover and added value, respectively. To carry out the study, four research questions and four possible hypotheses were used. For data analysis, generalized linear models were used, and four models were obtained and statistically validated. The obtained results led to the identification of the factors associated with the dependent variables that were the basis for the creation of the models. Full article

In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to $10.5208$. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from $1.5157$ to $1.6011$. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms. Full article

In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method. Full article

Nonlinear equations are frequently encountered in many areas of applied science and engineering, and they require efficient numerical methods to solve. To ensure quick and precise root approximation, this study presents derivative-free iterative methods for finding multiple zeros with an ideal fourth-order convergence rate. Furthermore, the study explores applications of the methods in both real-life and academic contexts. In particular, we examine the convergence of the methods by applying them to the problems, namely Van der Waals equation of state, Planck’s law of radiation, the Manning equation for isentropic supersonic flow and some academic problems. Numerical results reveal that the proposed derivative-free methods are more efficient and consistent than existing methods. Full article

The transition to a circular economy determines benefits at the level of each country by ensuring the premises for a sustainable economy in the future. The purpose of this study is to analyze indicators that measure the results of the implementation of circular economy through five study directions: Production and Consumption, Waste Management, Secondary Raw Materials, Competitiveness and Innovation, and Global Sustainability and Resilience. The analysis used 11 indicators that allow for the measurement of the evolution of eight Eastern European countries in the years 2012–2020. The study used three working hypotheses, which were all verified. The obtained results allowed for the classification of countries using a composite index and the grouping of countries with the help of cluster analysis using Pearson’s correlation coefficient. In a nutshell, sustainable development requires both economic development and environmental sustainability, and the developed composite index groups the countries’ evolution towards this goal. Full article

Small-scale photovoltaic (PV) systems are essential for the local energy supply. The most commonly known PV cell is configured as a large-area p–n junction made from silicon, but PV systems today include PV cells of various manufactures and origins. The dependence relationship between current and voltage is nonlinear, known as the current–voltage characteristic. The values of the characteristic equation’s parameters define the working regime of the PV cell. In the present work, the parameter values are iteratively obtained by nonlinear regression for an explicit model. The acceleration of the convergence of these values is studied for an approximation simplifying the iterative calculation in the case of perpendicular offsets. The new estimations of parameters allow for a much faster estimate of the maximum power point of the PV system. Full article