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Six derivatives of 4-(azulen-1-yl)-2,6-divinylpyridine were the subject of experimental determination of oxidation and reduction potentials being reported elsewhere. In this paper, a computational study was employed in order to obtain a function of structure for these potentials. The geometry was optimized at three theory levels (MMFF94, B3LYP and M06), and the following analysis was conducted with the separately saved optimum geometry in each instance. Two families of molecular descriptors (FMPI and EChP) were used to derive structure-based descriptors. Simple linear regressions were extracted with the best of descriptors for each family and level of theory for both potentials. The study revealed that the MMFF94 optimum geometries best explained the selected electro­chemical properties. Furthermore, the EChP family of descriptors, much bigger than FMPI (about 64 times), was able to better explain the connection between the structure and the property. Once more, it has been shown that the eigenproblem has deep roots in structural chemistry.

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $ p $ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $ p+3 $ (for $ p\geq3 $) and have extended it to a generalized $ (m+2) $-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $ p+3m $, $ m\in\mathbb{N} $. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.

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.

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.

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 Cs 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.

Nowadays, the challenge is to transform dehydrated sewage sludge resulting from wastewater treatment plants from waste into resource. Following this objective, the sludge was further dried and submitted to X-ray diffraction (XRD) and FTIR analysis. The sludge was first dried in ventilated and unventilated spaces at 50∘C and 100∘C, for 60 and 100 minutes (min) in each case. The final mass and evaporation degree of the sludge depends on the initial mass, ventilation type, drying time, and temperature. The ventilated drying space is preferred for temperature control, homogeneity, and higher evaporation degree. The influence of the drying process on the structure and behavior of the sewage sludge was emphasized through X-ray diffraction (XRD) and FTIR analysis. The XRD shows good structural properties of the sludge samples given by the reduction of the particle size through evaporation. According to FTIR, evaporation influences the depolymerization of the silicate network. The hydroxyl units and metallic ion modifiers can improve the sludge structure, but its intensity decreases through evaporation. With high content of solid substance, and good relation between the composition of the sludge and its structure and behavior, the dried sewage sludge can be used in: (i) agriculture, (ii) construction, (iii) the energy sector.

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.

In this paper, a derivative-free one-point iterative technique is proposed, with memory for finding multiple roots of practical problems, such as van der Waals and continuous stirred tank reactor problems, whose multiplicity is unknown in the literature. The new technique has an order of convergence of 1.84 and requires two function evaluations. It can be used as a seed to produce higher-order methods with similar properties, and it increases the efficiency of a similar procedure without memory due to Schröder. After studying its order of convergence, its stability is checked by applying it to the considered problems and comparing with the technique of the same nature for finding multiple roots. The geometrical behavior of the numerical results of the techniques is also studied.

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.

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.

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.

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.

In this paper, we propose a new fifth-order family of derivative-free iterative methods for solving nonlinear equations. Numerous iterative schemes found in the existing literature either exhibit divergence or fail to work when the function derivative is zero. However, the proposed family of methods successfully works even in such scenarios. We extended this idea to memory-based iterative methods by utilizing self-accelerating parameters derived from the current and previous approximations. As a result, we increased the convergence order from five to ten without requiring additional function evaluations. Analytical proofs of the proposed family of derivative-free methods, both with and without memory, are provided. Furthermore, numerical experimentation on diverse problems reveals the effectiveness and good performance of the proposed methods when compared with well-known existing methods.

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.

Counting polynomials find their way into chemical graph theory through quantum chemistry in two ways: as approximate solutions to the Schrödinger equation or by storing information in a mathematical form and trying to find a pattern in the roots of these expressions. Coefficients count how many times a property occurs, and exponents express the extent of the property. They help understand the origin of regularities in the chemistry of specific classes of compounds. Our objective is to accelerate the research of newcomers into chemical graph theory. One problem in understanding these concepts is in the different approaches and notations of each research study; some researchers provide online tools for computing these mathematical concepts, but these need to be maintained for functionality. We take advantage of similar mathematical aspects of 14 such polynomials that merge theoretical chemistry and pure mathematics; give examples, differences, and similarities; and relate them to recent research.

In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of 4 and 8, respectively. To further enhance their convergence capabilities, the with-memory methods incorporate accelerating parameters, elevating their convergence orders to 7.5311 and 15.5156, respectively, without introducing extra function evaluations. As such, they exhibit exceptional efficiency indices of 1.9601 and 1.9847, respectively, nearing the maximum efficiency index of 2. The convergence domains are also analysed using the basins of attraction, which exhibit symmetrical patterns and shed light on the fascinating interplay between symmetry, dynamic behaviour, the number of diverging points, and efficient root-finding methods for nonlinear equations. Numerical experiments and comparison with existing methods are carried out on some nonlinear functions, including real-world chemical engineering problems, to demonstrate the effectiveness of the new proposed methods and confirm the theoretical results. Notably, our numerical experiments reveal that the proposed methods outperform their existing counterparts, offering superior precision in computation.

This article considers the iterative approach for finding the Moore–Penrose inverse of a matrix. A convergence analysis is presented under certain conditions, demonstrating that the scheme attains third-order convergence. Moreover, theoretical discussions suggest that selecting a particular parameter could further improve the convergence order. The proposed scheme defines the special cases of third-order methods for β=0,1/2, and 1/4. Various large sparse, ill-conditioned, and rectangular matrices obtained from real-life problems were included from the Matrix-Market Library to test the presented scheme. The scheme’s performance was measured on randomly generated complex and real matrices, to verify the theoretical results and demonstrate its superiority over the existing methods. Furthermore, a large number of distinct approaches derived using the proposed family were tested numerically, to determine the optimal parametric value, leading to a successful conclusion.

In the study of systems’ dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local symmetry, dictated the dynamics of gravity, of space-time itself. In certain instances, especially in the presence of symmetry, we end up having to deal with an equation with multiple roots. A variety of optimal methods have been proposed in the literature for multiple roots with known multiplicity, all of which need derivative evaluations in the formulations. However, in the literature, optimal methods without derivatives are few. Motivated by this feature, here we present a novel optimal family of fourth-order methods for multiple roots with known multiplicity, which do not use any derivative. The scheme of the new iterative family consists of two steps, namely Traub-Steffensen and Traub-Steffensen-like iterations with weight factor. According to the Kung-Traub hypothesis, the new algorithms satisfy the optimality criterion. Taylor’s series expansion is used to examine order of convergence. We also demonstrate the application of new algorithms to real-life problems, i.e., Van der Waals problem, Manning problem, Planck law radiation problem, and Kepler’s problem. Furthermore, the performance comparisons have shown that the given derivative-free algorithms are competitive with existing optimal fourth-order algorithms that require derivative information.

The methods that use memory using accelerating parameters for computing multiple roots are almost non-existent in the literature. Furthermore, the only paper available in this direction showed an increase in the order of convergence of 0.5 from the without memory to the with memory extension. In this paper, we introduce a new fifth-order without memory method, which we subsequently extend to two higher-order with memory methods using a self-accelerating parameter. The proposed with memory methods extension demonstrate a significant improvement in the order of convergence from 5 to 7, making this the first paper to achieve at least a 2-order improvement. In addition to this improvement, our paper is also the first to use Hermite interpolating polynomials to approximate the accelerating parameter in the proposed with memory methods for multiple roots. We also provide rigorous theoretical proofs of convergence theorems to establish the order of the proposed methods. Finally, we demonstrate the potential impact of the proposed methods through numerical experimentation on a diverse range of problems. Overall, we believe that our proposed methods have significant potential for various applications in science and engineering.

It is more effective to maintain good health than to regain it after losing it. This work focuses on the biochemical defense mechanisms against free radicals and their role in building and maintaining antioxidant shields, aiming to show how to balance, as much as possible, the situations in which we are exposed to free radicals. To achieve this aim, foods, fruits, and marine algae with a high antioxidant content should constitute the basis of nutritional elements, since natural products are known to have significantly greater assimilation efficiency. This review also gives the perspective in which the use of antioxidants can extend the life of food products, by protecting them from damage caused by oxidation as well as their use as food additives.

It is known that, for paired measurements subjected to experimental error, better suited linear regression is obtained by using perpendicular offsets. Even so, the great majority of statistical software still uses classical vertical offsets for reasons of convenience. The same convenience leads to the preference of the least squares method in the favor of maximum-likelihood estimation. The treatise for perpendicular offsets for simple linear regression is slightly trickier than the corresponding one for vertical offsets. However, there is no general treatise for perpendicular offsets for nonlinear cases to date. In this work, a typical case of nonlinear dependence—potential versus intensity of current produced by a photovoltaic cell—is subjected to study. A series of paired potential/current data was collected from a commercial photovoltaic device and served for introducing the perpendicular offsets approach in the case of a nonlinear regression. Photovoltaic cell parameters—internal resistance, short-circuit current intensity, potential of open-circuit, and the maximum power point—have been determined by using the perpendicular offsets approach. Several issues were addressed in this work, such as exploring the intrinsic symmetry in the treatment of current versus potential diagrams, the suitability of perpendicular offsets in obtaining of the regression coefficients, and the implementation of nonlinear regression models with perpendicular offsets. Even if both the treatises of perpendicular offsets and nonlinear regression are known for some time now, there is no report in the literature of using both. Furthermore, since both potential and current measurements are affected by errors, it is more natural to use the proposed approach of perpendicular offsets.

We have developed a two-point iterative scheme for multiple roots that achieves fifth order convergence by using two function evaluations and two derivative evaluations each iteration. Weight function approach is utilized to frame the scheme. The weight function named as R(υt) is used, which is a function of υt, and υt is a function of ωt, i.e., υt=ωt1+aωt, where a is a real number and ωt=g(yt)g(xt)1m˜ is a multi-valued function. The consistency of the newly generated methods is ensured numerically and through the basins of attraction. Four complex functions are considered to compare the new methods with existing schemes via basins of attraction, and all provided basins of attraction possess reflection symmetry. Further, five numerical examples are used to verify the theoretical results and to contrast the presented schemes with some recognized schemes of fifth order. The results obtained have proved that the new schemes are better than the existing schemes of the same nature.

Abstract A series of difficulties make inaccessible precise experimental determinations of solubilities in standard conditions for monosaccharides; in water, the monosaccharides may switch from the acyclic to cyclic form and also the cyclic forms can undergo mutarotation. There are many ways to express the structural information as structure descriptors, but the alternatives become fewer when looking for invariants with good identification abilities, and the characteristic polynomial is one of them. The disadvantage of the characteristic polynomial resides in the fact that is defined with disregard of the chemical information coming from the type of the element and the type of the bond. Here, an extension of the characteristic polynomial was used accounting for the chemical information. In water, monosaccharides exist in all forms, but only one is an invariant for all the acyclic form. If it is something in structure which associates the structure information with the solubility, then it is present in all its form including the acyclic one, and therefore, the acyclic forms can be used to derive structure–property relationships. A search for linear relationships expressing the solubility as a function of the structure of the acyclic forms of monosaccharides was conducted by using the extension of the characteristic polynomial. The search used the experimental data available to select the models that are able to estimate the solubility, with each different to the other in terms of the effects considered. Considering the obtained results, the extended characteristic polynomial provides a very good estimation capability for the solubilities of monosaccharides.

Probably the best example to start with with regard to structures with high symmetry (SHS) is C60 fullerene (buckminsterfullerene) [...]

In this paper, we describe iterative derivative-free algorithms for multiple roots of a nonlinear equation. Many researchers have evaluated the multiple roots of a nonlinear equation using the first- or second-order derivative of functions. However, calculating the function’s derivative at each iteration is laborious. So, taking this as motivation, we develop second-order algorithms without using the derivatives. The convergence analysis is first carried out for particular values of multiple roots before coming to a general conclusion. According to the Kung–Traub hypothesis, the new algorithms will have optimal convergence since only two functions need to be evaluated at every step. The order of convergence is investigated using Taylor’s series expansion. Moreover, the applicability and comparisons with existing methods are demonstrated on three real-life problems (e.g., Kepler’s, Van der Waals, and continuous-stirred tank reactor problems) and three standard academic problems that contain the root clustering and complex root problems. Finally, we see from the computational outcomes that our approaches use the least amount of processing time compared with the ones already in use. This effectively displays the theoretical conclusions of this study.

Structural symmetry, anti-symmetry, or asymmetry represent a foundational property that, for chemical compounds, often determines their chemical and biological activity [...]

When collecting experimental data, the observable may be dichotomous. Sampling (eventually with replacement) thus emulates a Bernoulli trial leading to a binomial proportion. Because the binomial distribution is discrete, the analytical evaluation of the exact confidence interval of the sampled outcome is a mathematical challenge. This paper proposes three alternative confidence interval calculation methods that are characterized and exemplified.

Various methods (Hartree–Fock methods, semi-empirical methods, Density Functional Theory, Molecular Mechanics) used to optimize a molecule structure feature the same basic approach but differ in the mathematical approximations used. The geometry optimization procedure calculates the energy at an initial geometry of a molecule and then proceeds to search a new geometry with a lower energy. Using the 3D structures collected from the PubChem database, 20 amino acid geometry optimization calculations were performed with several methods. The purpose of the study was to analyze these methods (39) to find the relationship between them and to determine which to use under different circumstances. Cluster analysis and principal component analysis were performed to evaluate the similarities between the different methods. The results after the analysis can classified into three main groups and can be selected accordingly to solve different types of problems.

Medical studies often involve a comparison between two outcomes, each collected from a sample. The probability associated with, and confidence in the result of the study is of most importance, since one may argue that having been wrong with a percent could be what killed a patient. Sampling is usually done from a finite and discrete population and it follows a Bernoulli trial, leading to a contingency of two binomially distributed samples (better known as 2×2 contingency table). Current guidelines recommend reporting relative measures of association (such as the relative risk and odds ratio) in conjunction with absolute measures of association (which include risk difference or excess risk). Because the distribution is discrete, the evaluation of the exact confidence interval for either of those measures of association is a mathematical challenge. Some alternate scenarios were analyzed (continuous vs. discrete; hypergeometric vs. binomial), and in the main case—bivariate binomial experiment—a strategy for providing exact p-values and confidence intervals is proposed. Algorithms implementing the strategy are given.

Protein alignment finds its application in refining results of sequence alignment and understanding protein function. A previous study aligned single molecules, making use of the minimization of sums of the squares of eigenvalues, obtained for the antisymmetric Cartesian coordinate distance matrices Dx and Dy. This is used in our program to search for similarities between amino acids by comparing the sums of the squares of eigenvalues associated with the Dx, Dy, and Dz distance matrices. These matrices are obtained by removing atoms that could lead to low similarity. Candidates are aligned, and trilateration is used to attach all previously striped atoms. A TM-score is the scoring function that chooses the best alignment from supplied candidates. Twenty essential amino acids that take many forms in nature are selected for comparison. The correct alignment is taken into account most of the time by the alignment algorithm. It was numerically detected by the TM-score 70% of the time, on average, and 15% more cases with close scores can be easily distinguished by human observation.

Polymerase chain reaction (PCR ) techniques development allows elaboration of many assays for identification of bacteria’s resistance mechanisms to antibiotics. Following this idea, the results of molecular level investigation of bacteria’s resistance mechanisms to antibiotics may give many opportunities to find more rapid methods for identifying the genes which are responsible for antibiotic resistance induction. The aim of this study was to investigate antibiotic resistance genes in Staphylococcus bacteria on molecular level. As classes of antibiotics it was used macrolides-lincosamides-streptogramin B (MLSB) and beta-lactams. In the proposed study the bacterial strains are represented by 50 isolates of Staphylococcus. The bacterial strains were analyzed using polymerase chain reaction to identify the nuc, tuf, tst, sea, pathogenic activity genes. After this, the bacteria were tested for ermA, ermB, ermC genes and for mecA, femA which are involved in resistance to macrolides, lincosamides, streptogramin B and to beta-lactams, respectively. The presence or the absence of these genes confirms that tested strains are resistant to specific antibiotic or not. Bacteria pathogenic activity was emphasized by genes as follows: sea (enterotoxin) which was found at all isolates, tst (toxic shock toxin) gene was not detected in any of isolates and tuf gene (elongation factor) was obtained with one pair of primers. Resistance to beta-lactams was evidenced by the presence of mecA in all isolates and femA in some strains. Each of ermC, ermA and ermB, macrolides-lincosamides-streptogramin B resistance genes, were detected.

A series of counting, sequence and layer matrices are considered precursors of classifiers capable of providing the partitions of the vertices of graphs. Classifiers are given to provide different degrees of distinctiveness for the vertices of the graphs. Any partition can be represented with colors. Following this fundamental idea, it was proposed to color the graphs according to the partitions of the graph vertices. Two alternative cases were identified: when the order of the sets in the partition is relevant (the sets are distinguished by their positions) and when the order of the sets in the partition is not relevant (the sets are not distinguished by their positions). The two isomers of C28 fullerenes were colored to test the ability of classifiers to generate different partitions and colorings, thereby providing a useful visual tool for scientists working on the functionalization of various highly symmetrical chemical structures.

Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. But contrarily, derivative free optimal order techniques for multiple root are almost nonexistent. By this as an inspirational factor, here we present a family of optimal fourth order derivative-free techniques for computing multiple roots of nonlinear equations. At the beginning the convergence analysis is executed for particular values of multiplicity afterwards it concludes in general form. Behl et. al derivative-free method is seen as special case of the family. Moreover, the applicability and comparison is demonstrated on different nonlinear problems that certifies the efficient convergent nature of the new methods. Finally, we conclude that our new methods consume the lowest CPU time as compared to the existing ones. This illuminates the theoretical outcomes to a great extent of this study.

A number of optimal order multiple root techniques that require derivative evaluations in the formulas have been proposed in literature. However, derivative-free optimal techniques for multiple roots are seldom obtained. By considering this factor as motivational, here we present a class of optimal fourth order methods for computing multiple roots without using derivatives in the iteration. The iterative formula consists of two steps in which the first step is a well-known Traub–Steffensen scheme whereas second step is a Traub–Steffensen-like scheme. The Methodology is based on two steps of which the first is Traub–Steffensen iteration and the second is Traub–Steffensen-like iteration. Effectiveness is validated on different problems that shows the robust convergent behavior of the proposed methods. It has been proven that the new derivative-free methods are good competitors to their existing counterparts that need derivative information.

In the subject of statistics for engineering, physics, computer science, chemistry, and earth sciences, one of the sampling challenges is the accuracy, or, in other words, how representative the sample is of the population from which it was drawn. A series of statistics were developed to measure the departure between the population (theoretical) and the sample (observed) distributions. Another connected issue is the presence of extreme values—possible observations that may have been wrongly collected—which do not belong to the population selected for study. By subjecting those two issues to study, we hereby propose a new statistic for assessing the quality of sampling intended to be used for any continuous distribution. Depending on the sample size, the proposed statistic is operational for known distributions (with a known probability density function) and provides the risk of being in error while assuming that a certain sample has been drawn from a population. A strategy for sample analysis, by analyzing the information about quality of the sampling provided by the order statistics in use, is proposed. A case study was conducted assessing the quality of sampling for ten cases, the latter being used to provide a pattern analysis of the statistics.

im: The aim of our research was to design and implement data collection instruments to be use in context of an observational prospective clinical study with follow-up conducted on new born with intrauterine growth restriction. Methods: The structure of the data collection forms (paper based and electronic based) was first identified and for each variable the best type to accomplish the research aim was established. The code for categorical variables has also been decided as well as the units of measurements for quantitative variables. In respect of good practice, a set of confounding factors (as gender, date of birth, etc.) have also been identified and integrated in data collection instruments. Data-entry validation rules were implemented for each variable to reduce data input errors when the electronic data collection instrument was created. Results: Two data collection instruments have been developed and successfully implemented: a paper-based form and an electronic data collection instrument. The developed forms included demographics, neonatal complications (as hypoglycemia, hypocalcemia, etc.), biochemical data at birth and follow-up, immunological data, as well as basal and follow-up echocardiographic data. Data-entry validation criteria have been implemented in electronic data collection instrument to assure validity and precision when paper-based data are translated in electronic form. Furthermore, to assure subject’s confidentiality a careful attention was given to HIPPA identifiers when electronic data collection instrument was developed. Conclusion: Data collection instruments were successfully developed and implemented as an a priori step in a clinical research for assisting data collection and management in a case of an observational prospective study with follow-up visits.

To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the first Fréchet-derivative only. Such estimates are not introduced in the earlier procedures employing Taylor’s expansion of higher derivatives that may not exist or may be expensive to compute. The convergence domain of the method is also shown by a visual approach, namely basins of attraction. Theoretical results are endorsed via numerical experiments that show the cases where earlier results cannot be applicable.

Molecular conformation as a subproblem of the geometrical shaping of the molecules is essential for the expression of biological activity. It is well known that from the series of all possible sugars, those that are most naturally occurring and usable by living organisms as a source of energy—because they can be phosphorylated by hexokinase, the first enzyme in the glycolysis pathway—are D-sugars (from the Latin dextro). Furthermore, the most naturally occurring amino acids in living cells are L-sugars (from the Latin laevo). However, a problem arises in dealing with the comparison of their conformers. One alternative way to compare sugars is via their molecular alignment. Here, a solution to the eigenproblem of molecular alignment is communicated. The Cartesian system is rotated, and eventually translated and reflected until the molecule arrives in a position characterized by the highest absolute values of the eigenvalues observed on the Cartesian coordinates. The rotation alone can provide eight alternate positions relative to the reflexes of each coordinate.

We propose a derivative-free iterative method with fifth order of convergence for solving systems of nonlinear equations. The scheme is composed of three steps, of which the first two steps are that of third order Traub-Steffensen-type method and the last is derivative-free modification of Chebyshev’s method. Computational efficiency is examined and comparison between the efficiencies of presented technique with existing techniques is performed. It is proved that, in general, the new method is more efficient. Numerical problems, including those resulting from practical problems viz. integral equations and boundary value problems, are considered to compare the performance of the proposed method with existing methods. Calculation of computational order of convergence shows that the order of convergence of the new method is preserved in all the numerical examples, which is not so in the case of some of the existing higher order methods. Moreover, the numerical results, including the CPU-time consumed in the execution of program, confirm the accurate and efficient behavior of the new technique.

In this paper, a model has been developed that can estimate the composition of the phenol compounds, based on censored data and the total equivalent antioxidant capacity (TEAC) measured by three different methods. A contingency of 32 plants was analyzed: total phenolic content (TPC), caffeic acid, p-coumaric acid, ferulic acid, neochlorogenic acid and TEAC. They were measured by three different methods: ABTS (2,20-azinobis-(3-ethylbenzthiazoline- 6-sulfonic acid)), DPPH (1,1-diphenyl-2-picrylhydrazyl radical) and FRAP (ferric reducing/antioxidant power). Five values of caffeic-, thirteen of p-coumaric-, seven of ferulic-, and nineteen neochlorogenic acids were missing. Due to the complexity of the compounds, data mining and computational methods are required to determine the missing data. The method developed for independent variables was used to estimate the missing data. The contingency was filled with the calculated values obtained with all alternatives. The performance of each approach is shown in the estimation and/or prediction of the phenolic composition compared to the approaches used. The results indicated a strong correlation and mutual influence between the data analyzed.

One of the pillars of experimental science is sampling. Based on the analysis of samples, estimations for populations are made. There is an entire science based on sampling. Distribution of the population, of the sample, and the connection among those two (including sampling distribution) provides rich information for any estimation to be made. Distributions are split into two main groups: continuous and discrete. The present study applies to continuous distributions. One of the challenges of sampling is its accuracy, or, in other words, how representative the sample is of the population from which it was drawn. To answer this question, a series of statistics have been developed to measure the agreement between the theoretical (the population) and observed (the sample) distributions. Another challenge, connected to this, is the presence of outliers - regarded here as observations wrongly collected, that is, not belonging to the population subjected to study. To detect outliers, a series of tests have been proposed, but mainly for normal (Gauss) distributions—the most frequently encountered distribution. The present study proposes a statistic (and a test) intended to be used for any continuous distribution to detect outliers by constructing the confidence interval for the extreme value in the sample, at a certain (preselected) risk of being in error, and depending on the sample size. The proposed statistic is operational for known distributions (with a known probability density function) and is also dependent on the statistical parameters of the population—here it is discussed in connection with estimating those parameters by the maximum likelihood estimation method operating on a uniform U(0,1) continuous symmetrical distribution.

The Chi-Square test (χ2 test) is a family of tests based on a series of assumptions and is frequently used in the statistical analysis of experimental data. The aim of our paper was to present solutions to common problems when applying the Chi-square tests for testing goodness-of-fit, homogeneity and independence. The main characteristics of these three tests are presented along with various problems related to their application. The main problems identified in the application of the goodness-of-fit test were as follows: defining the frequency classes, calculating the X2 statistic, and applying the χ2 test. Several solutions were identified, presented and analyzed. Three different equations were identified as being able to determine the contribution of each factor on three hypothesizes (minimization of variance, minimization of square coefficient of variation and minimization of X2 statistic) in the application of the Chi-square test of homogeneity. The best solution was directly related to the distribution of the experimental error. The Fisher exact test proved to be the “golden test” in analyzing the independence while the Yates and Mantel-Haenszel corrections could be applied as alternative tests.

The present paper comes up with an experimental design meant to point out the factors interferingin octylphenol’s degradation in surface waters under solar radiation, underlining each factor’sinfluence on the process observable (concentration of p-octylphenol). Multiple linear regressionanalysis and artificial neural network (Multi-Layer Perceptron type) were applied in order to obtaina mathematical model capable to explain the action of UV-light upon synthetic solutions of OP inultra-pure water (MilliQ type). Neural network proves to be the most efficient method in predictingthe evolution of OP concentration during photodegradation process. Thus, determination in neuralnetwork’s case has almost double value versus the regression analysis.

This paper aimed to present a methodology for the geometry optimization of molecules in order to analyze structure-activity/property relationships. The main objective was to reduce the probability of carrying out misleading analyses while preparing the molecules for modeling. An algorithm that used HyperChem 8.0 software was developed and implemented in a series of PHP programs. Asample of 319 drug-like compounds was used to test the implemented algorithm. The program executed the geometry optimization protocols efficiently and quickly on both small and large sample sizes. The results obtained in the investigated sample of 319 compounds showed that our programs performed well on the investigated compound collection thus increasing the reliability of compound preparation for quantitative structure-activity relationships analysis.

The benefits of moderate consumption of wines consist in the protective effects against cardiovascular diseases and anticarcinogenic is associated with their antioxidants content. There are plenty of analytical methods for assessing the antioxidant content of wines but unfortunately, there is not a standardized method. The antioxidant content of ‘Cabernet Sauvignon’, ‘Merlot’ and ‘Pinot Noir’ wines obtained from cultivated grapes varieties from Recas and Minis Romanian vineyard from different harvesting years were investigated. The electron paramagnetic resonance spectroscopy, using the 2,2,6,6-tetramethyl-4-hydroxypiperidine-N-oxyl (Tempol) free stable radical, was used to determine the antioxidant content of wines. Measurements were done in order to analyze the effects of the grape varieties, the harvesting year and the vineyard on the antioxidant content of wines. The obtained results revealed that the antioxidant content of studied red wines depends on the harvesting year of grape, the grape variety and on the vineyard. Using the observed values of the antioxidant content, a relationship was identified between the antioxidant content of wines and the harvesting year of grapes. Furthermore, the designed generalized nonlinear model revealed that the antioxidant content of wine depends on the combined effect of the grape variety and the vineyard, besides the year, the grape variety and the vineyard

Genetic algorithms are built as abstract populations of a number of candidate solutions, each of it being evaluated for accomplish a desired performance. Populations evolve from one generation to another through mutation, crossover and selection in order to obtain an acceptable solution. Genetic algorithms applications cover the subject of decision, classification, optimization and simulation of hard problems. The quality of a genetic algorithm is evaluated in terms of speed, accuracy and domain of applicability. The use of all genetic operators could assure the convergence towards the optimum solution for a specific hard problem. The approaches used to construct the search space and the objective function (survival of the fittest, natural selection) assure the diversity of genetic algorithms. Studies on the development and use of genetic algorithms in solving hard problems in the field of agricultural systems were identified, analyzed and are presented here.

The efficiency of a genetic algorithm is frequently assessed using a series of operators of evolution like crossover operators, mutation operators or other dynamic parameters. The present paper aimed to review the main results of evolution supervised by genetic algorithms used to identify solutions to agricultural and horticultural hard problems and to discuss the results of using a genetic algorithms on structure-activity relationships in terms of behavior of evolution supervised by genetic algorithms. A genetic algorithm had been developed and implemented in order to identify the optimal solution in term of estimation power of a multiple linear regression approach for structure-activity relationships. Three survival and three selection strategies (proportional, deterministic and tournament) were investigated in order to identify the best survival-selection strategy able to lead to the model with higher estimation power. The Molecular Descriptors Family for structure characterization of a sample of 206 polychlorinated biphenyls with measured octanol-water partition coefficients was used as case study. Evolution using different selection and survival strategies proved to create populations of genotypes living in the evolution space with different diversity and variability. Under a series of criteria of comparisons these populations proved to be grouped and the groups were showed to be statistically different one to each other. The conclusions about genetic algorithm evolution according to a number of criteria were also highlighted.

Interspecific hybridizations represent one of the apple breeding methods by which a wide variability can be achieved, useful for creating new cultivars. The study of 2190 interspecific hybrids, obtained from 25 combinations among crab apple species (Malus coronaria, M. floribunda, M. niedzwetzkyana, M. zumi, M. prunifolia) and different apple cultivars, points out a large variability of the F1 seedlings for several traits, with significant importance in apple breeding programs. The first year of fructification, as mean per hybrid combination, varied from 6 (M. zumi x ‘Jonathan’) to 9.3 years; the average hybrid’s age for fructification was 7.4 years. The size of fruits among families varied from 1.5 (‘Starkrimson’ x M. prunifolia) to 4.0 (‘Reinette Baumann’ x M. zumi), the mean being settled at 2.8 (therefore below mark 3, meaning ‘small fruits’). The lowest infection rate both for apple scab and powdery mildew attack was noticed at hybrids from M. coronaria x ‘Reinette Baumann’. On the whole, the hybrids with genitors of ‘species x cultivars’ type have had an early fruiting and a better resistance to scab and powdery mildew, compared to the ones from ‘cultivars x species’ combinations. The large variability of the studied traits gave the possibility to identify offsprings with desirable characteristics on nine hybrid combinations. Among these, 53 elite plants were selected, with a strength of selection of 2.42%, therefore a relative low value, correlated with the peculiarities of the interspecific population and the selection criteria for dessert apple.