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Using a modification of the adapted Riccati transformation, we prove an oscillation criterion for generalizations of linear and half-linear Euler difference equations. Our main result complements a large number of previously known oscillation criteria about several similar generalizations of Euler difference equations. The major part of this paper is formed by the proof of the main theorem. To illustrate the fact that the presented criterion is new even for linear equations with periodic coefficients, we finish this paper with the corresponding corollary together with concrete examples of simple equations whose oscillatory properties do not follow from previously known criteria. [ABSTRACT FROM AUTHOR]

This study empirically investigates the causal relationship that exist between innovators and internal ecosystem factors driving technological innovation among small and medium-sized enterprises (SMEs) across the European region using the linear equation model. Our analysis is based on firm-level panel data from the European Innovation Survey conducted by the European Union from 2011–2018 with 296 SMEs across Europe. The results show that, human resources, research systems, financial support, firm investments, linkages, and employment impact was positively related to technological innovation performance. However, the outcome of an innovation-free environment and sales impacts as contextual factors had an insignificant influence on SMEs' innovation performance. The results imply that effective human resource approach coupled with good governmental policies are important factors that can increase SMEs' innovation performance. Based on this study results, SMEs' managers can provide effective and sustainable policies that would provide both direct and support roles on the main drivers of regional and economic development. The outcome of this research indicates that firms need to pay greater attention to external factors such as governmental policies governing SMEs, considering customers' needs, product specification and ensuring that they deliver quality product to satisfy consumers. [ABSTRACT FROM AUTHOR]

Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula for projecting onto the intersection of two hyperplanes. The answer turns out to be yes, as demonstrated recently by Behling, Bello-Cruz, and Santos, by López, by Needell and Ward, and by Ouyang. Most of these authors also provided formulas for projecting onto the intersection of an affine subspace and a hyperplane. In this note, we present an alternative approach which has the advantage of being more explicit and more elementary. Our results also provide useful information in the case when the two sets don't intersect. Luckily, the material is fully accessible to readers with a basic background in linear algebra and analysis. Finally, we demonstrate the computational efficiency of our formula when applied to an image reconstruction problem arising in Computed Tomography, and we also present a new formula for the projection onto the set of generalized bistochastic matrices with a moment constraint. [ABSTRACT FROM AUTHOR]

Heaviside proposed the telegrapher's equations as an alternative to the Maxwell equations. The telegrapher's equations demonstrated that the electromagnetic waves propagate in coaxial cables at the velocity of light c. The electromagnetic waves are used in Fourier analysis, Laplace transformation, and z transformation. Electromagnetic waves exhibit the physical phenomenon of simultaneous "partial reflection and refraction". Feynman found it tremendously important from the viewpoint of the physical phenomenon and advanced to the studies of quantum electrodynamics and quantum mechanics. In this session, we demonstrate that the physical phenomenon of partial reflection and refraction is an expression of the "law of conservation of energy" of electromagnetic waves, which falls into the category of classical physics. [ABSTRACT FROM AUTHOR]

Let p be a prime, n ∈ Z + and w ∈ (0 , 1) . Given a colouring χ : Z / p Z → { 1 , 2 , … , n } and a linear equation L : a 1 x 1 + a 2 x 2 + ⋯ + a n x n = b with a 1 , a 2 , … , a n ∈ (Z / p Z) ∗ and b ∈ Z / p Z , we denote by R (χ , L) the family of vectors (b 1 , b 2 , … , b n) ∈ (Z / p Z) n suchthat a 1 b 1 + a 2 b 2 + ⋯ + a n b n = b ad χ - 1 (i) ∩ { b 1 , b 2 , … , b n } ≠ ∅ for each i ∈ { 1 , 2 , … , n } . In this paper it is shown that there exists a constant c = c (w , n) > 0 with the following property: if min 1 ≤ i ≤ n | χ - 1 (i) | ≥ w p + 1 ≫ p 3 4 and if there exist coefficients a i and a j such that a i ∉ { ± a j } , then | R (χ , L) | ≥ c p n - 1. Moreover, this statement is sharp in different directions. A result about the solutions of L in a grid is used in its proof and it is interesting in its own right. [ABSTRACT FROM AUTHOR]

We study linear differential equations whose coefficients consist of products of powers of natural logarithm and general continuous functions. We derive conditions that guarantee the non-oscillation of all non-trivial solutions of the treated type of equations. The conditions are formulated as a non-oscillation criterion, which is the counterpart of a previously obtained oscillation theorem. Therefore, from the presented main result, it follows that the analysed equations are conditionally oscillatory. The used method is based on averaging techniques for the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This article is finished by new corollaries and examples.

Линейным уравнениям, т.е. уравнениям первой степени, а также системам из таких уравнений уделяется большое внимание как в алгебре, так в теории чисел. Наибольший интерес представляет случай таких уравнений с целыми коэффициентами и при этом их нужно решать в целых числах. Такие уравнения с указанными условиями называют линейными диофантовыми уравнениями. Еще Эйлер рассматривал способы решения линейных диофантовых уравнений с двумя неизвестными, причем один из этих способов был основан на применении алгоритма Евклида. Другой способ решения таких уравнений, основанный на цепных дробях, применялся также Лагранжем. Более удобным и перспективным оказался способ Эйлера, чем способ цепных дробей. В настоящей работе рассматривается один новый способ решения линейных уравнений над евклидовым кольцом, основанный на сравнениях по подходящим модулям. Известный ранее матричный метод решения таких уравнений с увеличением числа неизвестных является довольно громоздким в виду того, что он связан с нахождением обратных к унимодулярным целочисленным матрицам. Существенным в нашем способе решения линейных уравнений над евклидовым кольцом является использование алгоритма Евклида и линейного представления НОД элементов в евклидовом кольце. Доказанная в работе теорема применяется к нахождению решения линейного уравнения с тремя неизвестными над кольцом целых гауссовых чисел, являющимся, как известно, евклидовым кольцом. В заключении приводятся замечания о возможных путях дальнейшего развития изложенного исследования.

Various iterative algorithms for solving the linear equation using a quantum computer operating on the principle of quantum annealing are studied. Assuming that the result produced by the computer is described by the Boltzmann distribution, conditions under which these algorithms converge are obtained and an estimate of their convergence rate is provided. Application of this approach for algorithms that use an infinite number of qubits and a small number of qubits is considered. [ABSTRACT FROM AUTHOR]

Purpose. Improving the optimization method for predicting the efficiency of a robotic platform (using the gradient boosting method as an example). Methodology. The process of refining the optimization method for predicting efficiency has been investigated using robotic platforms as complex systems comprising hardware components, data exchange technology, security systems, and navigation, along with user interaction methods. The optimization method relies on a linear equation, whose mathematical model, through the triple interaction of factors, consolidates assessments of subsystem elements into an efficiency index for the robotic platform. The outcomes of the proposed optimization algorithm result in regression models from machine learning. These acquired models are employed for predicting the efficiency of a specific configuration of a robotic platform designed to perform particular practical tasks. Findings. The optimization method for predicting the efficiency of a robotic platform has been enhanced by utilizing evaluations of the robotic platform efficiency index as input data. In comparison to existing methods, the proposed index demonstrates minimal values of root mean square deviation at 0.1794, 0.14 and 0.1245, respectively. This particular characteristic has enabled the development of a more accurate optimization method for predicting the efficiency of robotic platforms. This assertion is supported both theoretically and empirically through criteria such as Root Mean Square Error, Mean Absolute Error, and Maximum Absolute Error on experimental datasets. Originality. The optimization method for predicting the efficiency of a robotic platform differs from existing approaches through its model-building process, which consists of two iterations and incorporates different sets of input evaluations. The first iteration involves primary and index-based evaluations of the robotic platform’s efficiency, while the second iteration incorporates primary, index-based evaluations, and predicted index-based evaluations. Practical value. Selection of the optimal configuration of a robotic platform for addressing tasks in the energy sector. Cost reduction through a finely tuned combination of robotic platforms. The proposed solutions will contribute to the Development Concept of Artificial Intelligence in Ukraine. [ABSTRACT FROM AUTHOR]

This paper investigates the impact of multiple representations-based instruction on the teaching and learning processes of linear equations among students in Standard VIII. It focuses on how different representations, such as graphs, tables, and equations, affect comprehension, retention, and overall mastery of linear equations in this educational context. An experimental design was employed, involving 159 students selected from Techiman Municipality in the Brong Ahafo Region of Ghana using a simple random sampling technique. Also, 86 mathematics teachers were randomly chosen to gather diverse viewpoints and valuable insights aimed at improving the teaching methods for this concept. Data collection instruments included a linear equations achievement test (pre-test & post-test) and a questionnaire. The collected data were analysed by using descriptive and inferential statistics. The study revealed that most teachers primarily relied on algebraic representation but only a few incorporated multiple representations due to various challenges such as time constraints, difficulty for students, lack of materials, and absence from the syllabus. It was further found that the implementation of multiple representations-based instructions resulted in a significant improvement in learners' scores in the linear equations achievement test, highlighting the effectiveness of this instructional approach. It was recommended that teachers teach linear equations in one variable using other representations such as manipulatives and graphics to enhance understanding. Also, students are encouraged to cultivate proficiency in integrating multiple representations when tackling problems related to linear equations. Further research should be conducted on equipping teachers with ample resources and designing robust training programs to enable them to adeptly incorporate multiple representations-based instruction for teaching linear equations. [ABSTRACT FROM AUTHOR]

Optimization that has not been implemented for learning media is the background ofthis Research, the medium that is needed will be used to support the learning process in the school. For this reason, media is needed which can later be used to support the learning process in schools. This research aims to: (1) describe the process of developing Powtoon-based animated videos on the material of the two-variable linear equation system (SPLDV). After teaching intern at SMP Negeri 31 Bandung, Bandung, West Java, (2) analyze the feasibility of Powtoon-based animated video media on the material of the two-variable linear equation system (SPLDV) at SMP Negeri 31 Bandung, and (3) see student responses in the use of Powtoon-based animated videos on two-variable linear equation system (SPLDV) materials at SMP Negeri 31 Bandung. The research method used is RND with ADDIE development model (analysis; design, development, implementation, evaluation). The research instruments used are interview sheets, material expert validation questionnaires, media experts and subject experts, and student responses. [ABSTRACT FROM AUTHOR]

Linear equations and linear functions are important materials for students in developing their in-depth knowledge of arithmetic and algebra. However, there is a gap between potential development and actual development, which causes obstacles for students in understanding the concepts of these two materials. This gap results in other obstacles, including a fixed mindset in mathematics. The results of interviews with secondary school teachers in Indonesia also show that students have problems in number operations and motivation to learn mathematics. The design used in this study uses a descriptive qualitative study based on the results of literature studies, interviews with subject teachers, and the author's experience as an educator. This study also uses the theory of zone proximal development and mathematical mindset as a frame for preparing alternative material for equations and linear functions based on visual representations in mathematics. The results of this study contribute to providing alternative materials to reduce these barriers using a visual representation approach in mathematics. The author also provides alternative teaching arts to anticipate obstacles that arise when implementing mathematics material based on visual representation.

This article introduces the idea of bicomplex valued S-metric space and deduces some of its features. Additionally, for bicomplex valued S-metric spaces, some fixed point results of contraction maps are shown to meet various categories of rational inequalities. Moreover, these results generalize certain significant, well-known results. An example is provided to highlight our major result. Furthermore, a theorem guaranteeing the existence of the one and only solution to the linear system of equations was developed using our main result.

Based on the methods of the theory of bifurcations, the problem of perturbation of linear equations by small analytic terms is considered. In contrast to the work by Trenogin [9], the case of an incomplete generalized Jordan set of a linear Fredholm operator acting from one Banach space to another Banach space is studied. A technique is proposed that uses the regularization of the Fredholm operator by a specially constructed finite-dimensional operator. [ABSTRACT FROM AUTHOR]

Maxmin- ω dynamical systems were previously introduced as an "all-in-one package" that can yield a solely min-plus, a solely max-plus, or a max-min-plus dynamical system by varying a parameter ω ∈ (0 , 1 ]. With such systems in mind, it is natural to introduce and consider maxmin- ω linear systems of equations of the type A ⊗ ω x = b. However, to our knowledge, such maxmin- ω linear systems have not been studied before and in this paper we present an approach to solve them. We show that the problem can be simplified by performing normalization and then generating a "canonical" matrix which we call the principal order matrix. Instead of directly trying to find the solutions, we search the possible solution indices which can be identified using the principal order matrix and the parameter ω. The fully active solutions are then immediately obtained from these solution indices. With the fully active solutions at hand, we then present the method to find other solutions by applying a relaxation, i.e., increasing or decreasing some components of fully active solutions. This approach can be seen as a generalization of an approach that could be applied to solve max-plus or min-plus linear systems. Our results also shed more light on an unusual feature of maxmin- ω linear systems, which, unlike in the usual linear algebra, can have a finite number of solutions in the case where their solution is non-unique. [ABSTRACT FROM AUTHOR]

This article introduces a new equation of the form 945(x2 + y2) − 1889xy + 10(x + y) + 100 = pzq and explores its solution by gradually changing the parameter q from 1 to 10 with generalized value q = m. A transformation x = u + v and y = u − v simplifies the equation which results (u + 10)2 + 3779v2 = pzq. We have employed multiple techniques to discover integral solutions for the equation and conducted a thorough analysis to examine the properties and characteristics of these solutions. [ABSTRACT FROM AUTHOR]

Studying the effect of boiler water released in rivers on some water properties such as density, viscosity was done in this research. Equations of momentum, conversation, turbulence and energy were used with the relation between the temperature and water properties to describe those properties along the study reach. Simplifying of partial equation was done to obtain system of linear equation. Verification of numerical model by conducting comparison of observed data from Euphrates River at Mussaib Thermal Power Station with those computed by the model. The comparison results of density and viscosity show an agreement with using statistical tests such as Chi square and RMSE. The results of the study showed that increasing the bed roughness from 0.04 m to 0.12 m causes a longitudinal retardation of the values of density and viscosity values about (26 and 22%) respectively and vertical advance of these values about (14 and 12% ) respectively. Increasing the water surface slope from (6 cm/km to 12 cm/km) causes a longitudinal advance of density and viscosity values about (19 and 12%) respectively and vertical retardation of these values about (16 and 13%) respectively. HIGHLIGHTS Effects of boiler water on water river properties such as water density and water viscosity were studied.; Significance and degree of freedom between observed data from the river and predicted data from the model were studied.; The effects of hydraulic river properties such as bed height and water surface slope on distribution of the values of these water river properties were explored.;

The increasing popularity of Multimedia courseware in education aims to address the challenges posed by abstract concepts and the consequent underachievement of students. It is also relevant to consider the limitations of working memory resources during multimedia courseware based instruction of abstract concepts in order to evaluate irrelevant factors that create unwanted cognitive drain and make retention difficult. The study investigated the effects of multimedia courseware based instruction on senior school students’ achievement and retention in linear equation word problems in Mathematics. The study adopted a quasi-experimental research design. In Gombe State Nigeria, 120 students (69 males and 51 females) were selected from three co-educational senior schools allocated either into experimental group or a control group. The results of the study indicated that students exposed to multimedia courseware based instruction achieved and retained linear equation word problems than their counterparts who were exposed to conventional teaching method. There was significant difference between the mean retention scores of male and female students exposed to multimedia courseware based instructions which the result is in favor of female students. High, medium and low achievers’ students exposed to multimedia courseware based instructions retained the concept of linear equations involving word problems and high scorer retained better. Based on the above findings it was recommended that multimedia courseware based instruction should be used in teaching and learning of Mathematics as it was found capable of improving students’ retention in linear equation word problems in Mathematics.

The important role of visual representations in supporting students' learning of geometry concepts and their reasoning is well established. GeoGebra is dynamic and interactive software that can assist students in exploring mathematical concepts effectively. This research aims to examine the effectiveness of GeoGebra in enhancing eighth-grade students' mathematical reasoning on the topic of linear equations. The study used a quantitative approach with a non-equivalent control group design by applying GeoGebra. Data were obtained through interviews and a mathematical reasoning ability test. The interviews were analyzed descriptively, and the reasoning test was analyzed using dependent t-tests. The findings revealed that GeoGebra significantly helps students' to have a better understanding of linear equation properties, such as the gradient of parallel, coincident, perpendicular, and intersecting lines. The mathematical reasoning test revealed that learning with GeoGebra can improve students' mathematical reasoning skills, specifically in linear equations. Teachers are encouraged to incorporate GeoGebra, as its visualization function enables students to explore linear equation properties. [ABSTRACT FROM AUTHOR]

Nowadays there are many approximate methods for thermal conductivity calculation, that lead to satisfactory outcomes in engineering practice. With the new approach, the solution of the nonlinear thermal conductivity equation was investigated. In addition to this, we reviewed the approximate solution of the nonlinear equation of thermal conductivity with cubic nonlinearity. To solve the problems of mathematical analysis, differential and integral equations, and boundary value problems of mathematical physics, difference, and interpolation are applied. Thus, for thinking of the effectiveness and reasonableness of these approaches, it is crucial for their theoretical investigation. The solution to these questions was found for each class of equation and each of its methods in their way and was often represented as significant difficulty, and in many cases was an obstacle for the current time. A natural approach to solving this issue is the use of the ideas of functional analysis. The variational principle initially was considered as a variational approach for solving linear functional equations and finding eigenvalues of linear operators. As in any variational approach, the problem of solving an equation will be brought to finding the extremum of the certain function of a special type, given over the entire space. It was found that the approach is useful in a way of minimizing functions of the more general type. [ABSTRACT FROM AUTHOR]

We continue our previous investigation of the Zelisko group of a matrix over Bézout domains. The explicit form of elements of this group over homomorphic image of Bézout domain of stable rank 1.5 is described. [ABSTRACT FROM AUTHOR]

Linear equations are commonly used in various fields of science such as engineering, physics, computer science, technology, business, and economics. A recent and promising extension of fuzzy set theory is represented by hesitant fuzzy sets (HFSs). The promising characteristic of HFSs is that they permit to model a type of imprecise human knowledge that cannot be captured by classical fuzzy sets. As no research has been performed in the field of hesitant fuzzy equations, in this paper, hesitant parametric fuzzy set (HPFS) is introduced and operations on this set are investigated. Also, fully hesitant parametric fuzzy equation (FHPFE) in form of a x = b is solved for the first time in which a and b are known HPFS and x is an unknown HPFS. Then, the existence of different approximate solutions of FHPFE is investigated and proved. Finally, some numerical examples are presented that verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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Ideological and political education as the basic curriculum of talent cultivation in China, to improve the effectiveness of ideological and political education in colleges and universities, education departments and professional colleges should explore the theoretical knowledge of ideological and political education in colleges and universities on the basis of empirical analysis combined with practical education results. Therefore, on the basis of understanding the analytical process of linear multivariate equation, this paper systematically studies the influence of students, teachers and teaching on the teaching effectiveness of ideological and political education in colleges and universities by using structural equation model analysis and questionnaire survey. The final results show that teaching plays a positive role in the effectiveness of ideological and political education. Therefore, in the educational innovation of the new era, based on the requirements of ideological and political education in colleges and universities, it is necessary to build an effective mechanism of subject interaction, continuously optimize practical teaching content, constantly innovate ideological and political education methods, and pay attention to the formation of symbiotic mechanism for teaching organization and learning atmosphere

The use of Multimedia courseware during teaching and learning is becoming progressively popular to overcome the abstract concepts within the subject and underachievement of the students. The study investigated the effects of multimedia courseware-based instruction on senior school students’ achievement and retention in linear equation word problems in Mathematics. In Gombe State Nigeria, 120 students (69 males and 51 females) were selected from three co-educational senior schools allocated either into experimental group or a control group. The study adopted a quasi-experimental research design. It used a pre-test, post-test, non-randomized, non-equivalent control group involving a factorial matrix. Mathematics Achievement Test on Linear Equations Word Problems in Mathematics (MAT) was used for collecting data and Multimedia Courseware on Linear Equations Word Problems in Mathematics (MCLEWPM) was used as a treatment instrument. The validated MAT was tested for reliability using Pearson Product Moment Correlation (PPMC), which yielded 0.78 at 0.05 alpha level of significance, indicating that the MAT is reliable with good construct validity. MAT was administered on students in experimental and control groups and data obtained were analyzed using T-test and Analysis of Covariance (ANCOVA) to test the hypotheses. The results of the study indicated that students exposed to multimedia courseware-based instruction achieved and retained linear equation word problems than their counterparts who were exposed to conventional teaching method. There was significant difference between the mean retention scores of male and female students exposed to multimedia courseware-based instructions which the result is in favor of female students. High, medium and low achievers’ students exposed to multimedia courseware-based instructions retained the concept of linear equations involving word problems and high scorer retained better. Based on the above findings it was recommended that multimedia courseware-based instruction should be used in teaching and learning of Mathematics as it was found capable of improving students’ retention in linear equation word problems in Mathematics.

This study aims to find out student errors in solving linear equations of one variable based on Kastolan's theory. This study is descriptive research. The data collection technics that are used are 1) a test to solve the story problems of linear equations of one variable based on the Kastolan theory, 2) an interview with all subjects, and 3) documentation. The subjects of this study were seventh-grade students of SMP Negeri 2 Boleng, West Manggarai Regency, NTT with 20 students categorized based on their errors in Kastolan theory. The study results showed that 1) 36.66% of students made conceptual errors in understanding the concept of the meaning of the problem, 2) 58.33% of students made procedural errors by not following the steps in solving the problem and not writing the conclusion at the end of the answer, and 3) 35% of students made operations technical errors in calculating the value of solving problems and errors in writing variables and constants. The factors that are causing these errors because students do not understand the meaning of the questions, are not serious in answering questions and do not review the results of the assignment

We establish a unified criterion for the reducibility of a linear homogeneous differential system with constant coefficients to a linear homogeneous differential equation with constant coefficients by means of both Lyapunov and periodic transformations. The resulting necessary and sufficient condition on a system is formulated in terms of properties of the Jordan normal form of its matrix. [ABSTRACT FROM AUTHOR]

Among many prevalent acoustic location techniques, the location problems can be modelled as solving a linear equation. Although many mature algorithms have been developed to solve the linear equation in acoustic location applications, few of them consider the inevitable noises in a real computing system that might degrade the convergence and accuracy of the algorithms or even lead to failure. Thus, to achieve promising performance when solving a linear equation in a noisy environment, a robust Newton iterative (RNI) algorithm is proposed in this paper based on control theory. Theoretical analyses indicated that the RNI algorithm can not only suppress the constant noise to zero but also maintain convergence against an increasing linear noise and random noise. In addition, extensive simulation results compared with the classic algorithms and their last variants are provided. Among these algorithms, the RNI algorithm achieves the best robustness and accuracy in the presence of noises, while it requires a longer convergence time. [ABSTRACT FROM AUTHOR]

This article introduces the idea of bicomplex valued S-metric space and deduces some of its features. Additionally, for bicomplex valued S-metric spaces, some fixed point results of contraction maps are shown to meet various categories of rational inequalities. Moreover, these results generalize certain significant, well-known results. An example is provided to highlight our major result. Furthermore, a theorem guaranteeing the existence of the one and only solution to the linear system of equations was developed using our main result. [ABSTRACT FROM AUTHOR]

In this paper, we study linear differential equations whose coefficients consist of products of powers of natural logarithm and very general continuous functions. Recently, using the Riccati transformation, we have identified a new type of conditionally oscillatory linear differential equations together with the critical oscillation constant. The studied equations are a generalization of these equations. Applying the modified Prüfer angle, we prove that they remain conditionally oscillatory with the same critical oscillation constant.

Objective To confirm whether a novel sagittal patellar angle linear equation used for evaluating patellar height by calculating expected sagittal patellar angle (SPA) at any degree of knee flexion angle is suitable for patients older than 17 years and its reliability compared with other commonly used methods. Methods From September 2016 to September 2019, a total number of 202 consecutive outpatients' knee lateral X‐ray radiographs were retrospectively measured and evaluated using a recently proposed linear equation Y = 1.94 + 0.74 × knee flexion(KF) angle. Patients were divided by ages into ayounger group, whose ages were between 17–49 years, and an older group, whose ages were older than 49 years, which has not been validated in the original study. Parameters such as KF, SPA, patella and patella tendon length and so on were measured on computer with picture archiving and communication system by two independent observers at an interval of 1 month. Insall‐Salvati (IS) index, Caton‐Deschamps (CD) index and Y value, correlation coefficients were calculated and compared using SPSS 22.0 software. Results In the younger group, 143 patients (165 knees) were included, ages were 17–49 (31.62 ± 11.38) years, males/females were 70 (48.95%)/73 (51.05%), left knees/right knees were 83 (50.30%)/82 (49.70%), mean value of Y was 31.50° ± 10.07°, and SPA was 34.38° ± 12.38°, mean value of IS was 1.06 ± 0.17, mean value of CD was 1.04 ± 0.18. While in older group, 59 patients (78 knees) were included, ages were 50–60 (mean 54.61 ± 2.99) years, there were 32 males (54.24%) and 27 females (45.76%), 42 knees were left (53.85%) and 36 knees were right (46.15%), mean values of Y and SPA were 25.90° ± 11.55° and 29.36° ± 14.22°, mean IS index in older group was 1.06 ± 0.18, mean CD index was 1.00 ± 0.16. Intra‐ and inter‐observer reliabilities of Y in younger and older groups were 0.999, 0.999, 1.000 and 0.999, meaning high reliability and reproducibility, but low Pearson's correlation coefficients with IS and CD index were showed as −0.213 and − 0.216 in younger group and − 0.113 and − 0.316 in older group. Conclusions In patients older than 17 years, the linear equation Y = 1.94 + 0.74 × KF is a reliable and practical method to evaluate SPA regardless of age and knee flexion angle, but has weak correlation coefficients with the IS and CD index.

In this paper, we analyse the oscillation of second-order half-linear differential equations whose coefficients are given by the products of (typically bounded) functions and power functions. Applying the generalized Riccati technique, we find a very general oscillation criterion for the studied equations. This criterion covers several half-linear equations with unbounded non-power parts of coefficients, but it is new also for linear equations with bounded non-power parts. This fact is appropriately documented by new corollaries. [ABSTRACT FROM AUTHOR]

The main idea of this article is to show the relation between students of a bilingual school who solved exercises in English and Spanish, indeed these exercises are focused on the first grade equation, and so it can determine if the mother’s tongue is involved by the resolution of a mathematical problem.

One of the challenging conditions in wellbore management is high-pressure, high-temperature wells that apply large expenditures and maintenance costs to petroleum industries. In this regard, drilling fluid rheology and its crucial factors would help engineers to have a fundamental understanding of the most appropriate wellbore management. Moreover, it can help engineers to control the fluid loss phenomenon. This paper aimed to consider different artificial intelligence and machine learning models to predict the drilling fluid density and select the optimum model, which can be extended to field applications. These models are ANFIS (adaptive neuro-fuzzy inference system), PSO-ANFIS (particle swarm optimization-adaptive​ neuro-fuzzy inference system), LSSVM-GA (least square support vector machine-genetics algorithm), and RBF (radial basis function) algorithm were modeled in the programming era. In LSSVM-GA model, it is evident that the proper linear equation for the prediction of drilling fluid density is “y = 1.0041x + 0.0019” with the correlation factor of (R2=0.9966). Due to the advantages of RBF algorithm to other genetics algorithm, this algorithm was used in this part to predict the drilling fluid density. Therefore, it is evident that the proper linear equation for the prediction of drilling fluid density is “y = 1.0009x + 0.0034” with the correlation factor of (R2=0.9999). Consequently, RBF model was selected as the optimum method in drilling fluid prediction. Furthermore, there is a proper match with training and experimental data and the maximum deviation is about -0.006. In ANFIS algorithm, exponential derivation would be preferred to predict the drilling fluid density instead of linear equation. Consequently, the RBF model provide has a good agreement with the experimental drilling fluid density. It is indicated that the model have appropriate accuracy and validity with experimental data.

A unified method for solving general polynomial equation of degree less than five that incorporate a computational formula that relate the coefficients of the depressed equation and the coefficient of the standard polynomial equation is proposed in this study. This is to ensure that this method is valid for all degree less than five. It shall apply the undetermined parameter method of auxiliary function to obtain solutions to these polynomial equations of degree less than five in one variable. In particular, the result of our work is a unification and improvement on the work of several authors in the sense that only applicable for the case of polynomial equation of degree one. Finally, our results improve and generalize the result by applying standard formula methods for solving higher degree polynomials. It is recommended that the effort should be made toward providing other variant methods that are simpler and friendly.

We generalize the well-known Dyson-Phillips theorem on perturbations of strongly continuous semigroups to the fundamental solutions of differential-difference equations in a Banach space. An application of our results to the stability theory of the considered equations is also discussed. [ABSTRACT FROM AUTHOR]

Let X, Y be linear spaces over fields K and F , respectively, and let f : X n → Y . For some fixed a ji ∈ K \ { 0 } , C i 1 … i n ∈ F , for j ∈ { 1 , … , n } , i , i j ∈ { 1 , 2 } , we consider the equation f (a 11 x 11 + a 12 x 12 , … , a n 1 x n 1 + a n 2 x n 2 ) = ∑ 1 ⩽ i 1 , … , i n ⩽ 2 C i 1 … i n f (x 1 i 1 , … , x n i n ) , (∗) for all x j i j ∈ X , j ∈ { 1 , … , n } , i j ∈ { 1 , 2 } . We determine the general solution of (∗) and as special cases we obtain some results for multi-Cauchy, multi-Jensen and multi-Cauchy-Jensen equations. [ABSTRACT FROM AUTHOR]

In this paper we continue the study of the linear equation with Stieltjes derivatives in [M. Frigon, R. López Pouso, Adv. Nonlinear Anal. 6(2017), 13–36]. Specifically, we revisit some of the results there presented, removing some of the required conditions as well as amending some mistakes. Furthermore, following the classical setting, we use the connection between the linear equation and the Gronwall inequality to obtain a new version of this type of inequalities in the context of Lebesgue–Stieltjes integrals. From there, we obtain a uniqueness criterion for initial value problems.

The critical volumeVc, the relation between the critical temperature and the pressure Tc / Pc and the product PcVc2 depend linearly on the number of carbons in a homologous series as a consequence of the theory of corresponding states. This suggests that additive and constitutive properties are suitable for the construction of equations of state. In this article, an equation of state of Soave–Redlich–Kwong type is proposed: an additive and constitutive equation of state because it contains among its constitutive parameters the parachor (℘ch) and the molar refraction (Rm). The homologous series of the n-alkanes is used to illustrate the applicability of the proposed equation.

Ultimate strength is an important design consideration for the safety of intact or damaged ship structures. The simplified progressive collapse method is a commonly used iterative method to obtain the accurate ultimate strength of ships. Since the accuracy of the neutral axis position directly affects the accuracy of the ultimate strength, the force equilibrium criterion and the force vector equilibrium criterion are adopted to search for the height and angle of the neutral axis, especially for damaged ships. However, the search for the neutral axis position based on the two criteria requires iterative computation which decreases the calculation efficiency. In this paper, the relationship between the results of the iterative process and the neutral axis position is studied, and it is found that the relationship is approximately linear. Then a new iterative method based on the linear equation is proposed to obtain the neutral axis position and is adopted to improve the simplified progressive collapse method. Finally, the new method is used to calculate the neutral axis position of a damaged VLCC. The comparison of the ultimate strength results shows that the improved simplified progressive collapse method based on the linear equation has improved efficiency and good accuracy. • It is found that the results of the equilibrium criterion vary approximately linearly for small movements of neutral axis. • An improved method based on the derived linear equation is proposed for the search of the location of neutral axis. • Comparison of the proposed method with existing methods shows that it is accurate and more computationally efficient. [ABSTRACT FROM AUTHOR]

We study the oscillation of Euler type half-linear difference equations with asymptotically periodic and bounded coefficients. Applying the adapted Riccati technique, we prove a non-oscillation criterion for the treated equations. Since the obtained criterion is new in several cases, we formulate many consequences explicitly. In particular, the criterion is new for intensively studied linear equations (e.g., with periodic coefficients). [ABSTRACT FROM AUTHOR]