Rational Equations Research Articles

Influence of fiber shapes on the compressive behaviors of ultra-high performance concrete containing coarse aggregate

The addition of coarse aggregate in ultra-high performance concrete (UHPC-CA) has the potential to reduce autogenous shrinkage and improve mechanical properties. However, it also affects the fiber distribution and crack propagation, thereby compressive relationship of UHPC. This study investigated the effects of different fiber shapes on the compressive behaviors of UHPC-CA. Three fiber shapes (straight fiber, hooked fiber, and hybrid fiber) and six coarse aggregate content (0–1200 kg/m3) are the main variables. The test results have indicated that the addition of coarse aggregate in UHPC rendered wider main cracks. Hybrid fiber was preferred over the hooked-end and straight fibers to optimize ductility in UHPC-CA. An increase in compressive strength of up to 17.7 % and elastic modulus of UHPC of up to 23.7 % after moderate addition of coarse aggregate contents, The fiber shape affected peak strain and ultimate strain values for UHPC-CA. Their peak and ultimate strains were improved by 12.8 %–15.3 % and 38.2%–127.4 % with the inclusion of 2 % hooked and hybrid fibers. Based on the experimental data and relevant equations, a proper rational equation from the existing literature is used to generate the complete compressive stress-strain curves of UHPC-CA with different fiber shapes. In summary, this study establishes that hybrid fiber shapes significantly enhance the compressive properties of UHPC-CA. The high elastic modulus and ductility of the UHPC-CA with hybrid fiber are particularly promising for blast and impact applications.

Study of different theories of thermoelasticity under the propagation of Rayleigh waves in thermoelastic medium

The present research is on the propagation of Rayleigh waves in a homogenous thermoelastic solid half-space by considering the compact form of six different theories of thermoelasticity. The medium is subjected to an insulated boundary surface that is free from normal stress, tangential stress, and a temperature gradient normal to the surface. After developing a mathematical model, a dispersion equation is obtained with irrational terms. To apply the algebraic method, this equation must be converted into a rational polynomial equation. From this, only those roots are filtered out, which has satisfied both of the above equations for the propagation of waves decaying with depth. With the help of these roots, different characteristics are computed numerically, like phase velocity, attenuation coefficient, and path of particles. Various particular cases are compared graphically by using phase velocity and attenuation coefficient. The elliptic path of surface particles in Rayleigh wave propagation is also presented for the different theories using physical constants of copper material for different depths and thermal conductivity.

МАТЕМАТИКА КУРСУН ОКУТУУДА КЫЙМЫЛ МАСЕЛЕЛЕРИН ЧЫГАРУУНУН МЕТОДИКАЛЫК КӨРСӨТМӨЛӨРҮ

At present, in accordance with the program of basic general education, a special place in mathematics teaching is occupied by the development of students' independence in solving motion problems. There is no doubt that the mathematical problem of motion enables the student to formulate mathematical concepts correctly, to clarify various aspects of his environment, and to apply theoretical concepts. Solving full-scale problems defined by the program contributes to knowledge building for students. In setting motion problems, students are exposed to facts of cognitive, educational significance. Motion problems-one of the most common types of algebra problems. Simple motion problems are taught in elementary school. In grades 6-7, the solution of motion problems is reduced to a linear equation or a system of linear equations. Here we will consider motion problems related to the rational fraction equation.

Application of X-ray principles to quantify sulfur oxidation states in concrete

This study introduces a novel mathematical approach rooted in X-ray spectroscopy principles to evaluate the oxidation state of sulfur, which is then applied in concrete and cementitious materials. Unlike previous studies that relied on an empirical quadratic relationship based on the ratios of characteristic X-ray lines (SKα, SKβ, and SKβ') for sulfur oxidation state determination, unique linear relationships between the intensities of the three X-ray lines for each oxidation state and total sulfur were utilized. Rational equations are then derived by combining these linear relationships for each species. Two independent rational calibration curves are constructed using the SKβ/SKα and SKβ'/SKβ ratios for samples containing sulfide-sulfate mixtures, irrespective of total sulfur concentration within the range of 0.3% to 2.0% by weight of concrete. These equations demonstrated interchangeability across three simulated matrices (Portland cement, concrete, and silica) and two sulfide species (pyrrhotite and pyrite). Absolute quantification of sulfide and sulfate species in concrete is achieved by independently measuring total sulfur. Compared to previous quadratic and linear equations, this approach is shown to minimize the difference between the experimental and calculated ratios of the calibration samples. Agreement was also observed between the sulfate ratio calculated through the rational approach and gravimetric analysis for concrete samples.

Today's Shifting Cultivation and its Impact on Forest and Environmental Damage in Routa, Konawe, Southeast Sulawesi

This article is intended to explore and analyze some of the impacts of shifting cultivation to the forest environmental degradation in Routa today. This study uses a combination of qualitative and quantitative research approaches, especially with the rational equation methods where the results of statistical calculations are then culturally interpreted. The results of this study indicate that current shifting cultivation has contributed greatly to the increasing damage to the forest environment and decreased hydro-urologic functions of forests in Routa as well as other ecological functions, including conversion of primary and secondary forest land to shifting cultivation areas, restoration of areas ex-shifting cultivation that are ongoing slower than usual, increased surface water flow, increased erosion, and sedimentation, increased local air temperature and decreased humidity and the potential for large carbon losses.

Difficulties and Errors of Grade 9 Learners in Solving Quadratic Equation: Input to the Development of Strategic Intervention Material

This study determined the difficulties and errors of Grade 9 learners in solving quadratic equations, it gauged the developmental of strategic intervention material. The researcher used the descriptive-developmental research design and used the 45 Grade 9 learners as respondents. Researcher-made test was used to gather the needed data. Frequency count, percentage, and mean were the statistical tools used. The results of the study showed that Grade 9 learners performed fairly satisfactory in characterizes the roots of quadratic equation using the discriminant, describes the relationship between the coefficients and the roots of a quadratic equation, solves equations transformable to quadratic equations (including rational algebraic equations), solves problems involving quadratic equations and rational algebraic equations, illustrates quadratic inequalities, solves quadratic inequalities, and solves problems involving quadratic inequalities. Additionally, the Grade 9 learners commit more errors in the reading errors and comprehension error. Further, the developed Strategic Intervention Materials on Solving Quadratic Equation for Mathematics 9 is very highly valid. Based from the findings, the following conclusions were drawn: The Grade 9 learners often encounter reading errors when solving quadratic equations, suggesting they have trouble comprehending the problem statements. This issue highlights a fundamental barrier to their mathematical proficiency. Additionally, they face specific challenges in areas such as characterizing the roots of equations, solving equations that can be transformed, and understanding quadratic inequalities, which are critical skills for mastering quadratic equations. To address these difficulties, strategic intervention material has been shown to be highly effective. These materials provide targeted support and practice, significantly improving the learners’ abilities to solve quadratic equations and enhancing their overall mathematical competence. It is therefore recommended that teachers use effective instructional strategies, including clear explanations, diverse problem-solving approaches, and real-world examples, to improve learners’ understanding and proficiency in quadratic equations. Learners should actively participate in problem-solving activities, seek help when needed, practice regularly, and develop a strong foundation in basic skills. The use of strategic intervention materials for teaching quadratic equations in Grade 9 is advised to enhance learners’ problem-solving competencies, and they study’s finding should be discussed in school Learning Action Cell (LAC) sessions. Future research should investigate the specific causes of learners’ difficulties, explore additional interventions and instructional approaches, and examine the long-term effects on learners’ mastery of quadratic equations, as well as the potential benefits of technology integration and gamification in teaching quadratic equations.

Symbol Sense Ability of Students to Solve Linear, Quadratic, and Rational Equations

In learning algebra, one of the skills needed to be acquired by students is solving equations, such as linear equations in one variable and quadratic equations. This study aims to analyze the symbol sense ability of students by investigating strategies used to solve algebraic equations that consist of linear, quadratic, and rational equations. A qualitative descriptive method was conducted. The instrument included a test with one linear equation, one quadratic equation, and three rational equations (involving linear or quadratic factors). The test was administered to 12 students from 12th-grade high school in Bandung. Each student was asked to solve it in two different ways of answers if possible. The results showed that students were able to use symbol sense strategies for solving problems. However, there were still mistakes in solving algebraic squared value concepts, affecting their final results of the answers. We conclude that even if students can use their symbol sense abilities properly, they need to pay attention to their mathematics concepts. Keywords: Symbol Sense Ability, Solve Linear, Quadratic, Rational Equations

Design of drainage channel for effective use of land on fully mechanized sugarcane plantations: A case study at Bone Sugarcane Plantation

Abstract Drainage in sugarcane cultivation in high-rainfall areas is of paramount importance because it affects sugarcane plants from planting, maintenance, and production. Poor drainage can damage plants and reduce productivity. This study was conducted at Bone Sugarcane Plantation, which typically has high annual rainfall; thus, drainage is necessary. The existing drainage channel cannot drain all surface runoff quickly and causes problems to sugarcane plants. Therefore, a study was conducted with the aim of designing the shape and size of drainage channels that can drain surface runoff quickly, have a strong structure, allows for an effective use of cultivated land by reducing headland for tractor turning, and maintains appropriate soil moisture. The research began by determining the saturated hydraulic conductivity of the soil using the Falling Head method. Surface runoff discharge was calculated using rational equations to determine the dimensions of the drainage channel. Rainfall intensity was determined from Intensity Duration Frequency curve which was constructed using Manonobe method. The results showed that the saturated hydraulic conductivity of the soil was 3.54 × 10−3 cm/s which is suitable with surface drainage. Rainfall intensity is estimated to be 201.33 mm/hour. The shape and dimensions of the drainage channel are parabolic with the largest width and depth of 1.70 m and 0.90, respectively. This study provides a practical method to solve drainage problems in sugarcane fields that apply full mechanization. In addition, the practical analysis used in this study can be adapted to analyze the design of drainage channel for other plantations or regions with similar constrains.

Strength Optimization of Nanocomposite Cementitious Materials Using Nanoscale Modifications

Represent Volume Element (RVE) is broadly used by investigators to control the properties of nano-cementitious materials. This paper focuses on analyzing a set of RVE data and proposes parametric equations for determining the compressive and flexural strength characteristics (σc and σf). Primarily, a parametric study is performed with RVE analysis. The essential design parameters are used to fit rational equations. The RVE data are also applied to train artificial neural networks of σc and σf. RVE, neural networks, and rational equations are correlated. Regression equations are validated with the experimental study. SEM, TEM, XRD, and FTIR results are carried out in the microscopic and mechanical analysis for carbon nanofiber cement composites, which can lift their strength, constancy, integrity, and density and reinforce the composite microstructure. Lastly, the Pareto-optimal design results are presented with a multi-objective optimization problem.

An efficient way to determine Liouvillian first integrals of rational second order ordinary differential equations

The Lie and Darboux methods to obtain Liouvillian first integrals of rational second order ordinary differential equations (rational 2ODEs) are very powerful. Nevertheless, there are cases where these procedures may encounter difficulties. Namely, when the Darboux polynomials present in the integrating factor of the 2ODE have a very high degree and/or when the 2ODE does not admit point symmetries. In [3,4] we developed a method (S-function method) that is successful in treating certain classes of rational second order ordinary differential equations that are particularly ‘resistant’ to canonical Lie methods and to Darbouxian approaches. However, although determining the S-function is in general much more efficient than determining Darboux polynomials or nonlocal symmetries, for very complicated rational 2ODEs, even finding the S-function itself can be quite hard. In this work we present a simple way of (in almost all cases) computing the S-function with a very efficient procedure. New version program summaryProgram Title: InSyDE – Invariants and Symmetries of (rational second order ordinary) Differential Equations.CPC Library link to program files:https://doi.org/10.17632/4ytft6zgk7.2Licensing provisions: CC by NC 3.0Programming language: Maple 17Journal reference of previous version: Comput. Phys. Commun. 234 (2019) 302–314Does the new version supersede the previous version?: Yes.Nature of problem: Determining Liouvillian first integrals of rational second order ordinary differential equations.Solution method: The method is explained in the body of the paper.Reasons for new version: The InSyDE package depends, in order to work, on determining the S-function. The problem is that, for very complicated 2ODEs, this can be very costly (computationally) or even unrealizable (within time and memory limits). We have developed a computationally cheaper way of determining the S-function that, in addition to being very efficient, is also surprisingly broad.Summary of revisions: We have, based on important new theoretical results presented on Appendix A: Supplementary Material, modified and improved the command Sfunction.

Adaptive design of tipping bucket flow meters for continuous runoff measurement

Introduction: Runoff measurement and monitoring is a laborious, time-consuming, and costly task. Additionally, common runoff monitoring usually primarily provide water level, requiring information on the stage-discharge relation. Automatic equipment such as flow meter tipping bucket (TB) is a potential option to simplify and provide continuous runoff monitoring in small catchments. However, a proper description of how to size and adapt the design under different flow conditions is still lacking.Methodology: In this paper we present a novel standardized framework for the design of TB that can be used for low-cost and real-time runoff monitoring under many different conditions. The framework consists of an estimation of the runoff peak rate using the rational equation and a volumetric capacity estimate of the cavity based on runoff rate, operation speed, and inclination angle of TB when at resting position. The proposed framework was implemented in a case study where four TBs were designed for continuous runoff monitoring from experimental plots (100 m2) with different land use (sugarcane, soybean, and bare soil).Results: During field tests (five months), the designed TBs had a recovery rate of actual runoff ranging from 61% to 81% and were able to capture features poorly studied (starting/ending time and peak flow) that have potential importance in hydrological models.Discussion: The proposed framework is flexible and can be used for different environmental conditions to provide continuous runoff data records.

Finding nonlocal Lie symmetries algorithmically

Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and nonlocal symmetries) admitted by a rational 2ODE if it presents a Liouvillian first integral. The procedure is based on an idea arising from the formal equivalence between the total derivative operator and the vector field associated with the 2ODE over its solutions (Cartan vector field). Basically, from the formal representation of a Lie symmetry it is possible to extract information that allows to use this symmetry practically (in the 2ODE integration process) even in cases where the formal operation cannot be performed, i.e., in cases where the symmetry is nonlocal. Furthermore, when the 2ODE in question depends on parameters, the procedure allows an analysis that determines the regions of the parameter space in which the integrable cases are located. We finish by applying the method to the 2ODE that represents a forced modified Duffing–Van der Pol oscillator (which presents chaos for arbitrary values of the parameters) and determining a region of the parameter space that leads to integrable cases, some not previously known.

Rational function neural networks for learning rock physics models from field data

Abstract Seismic wave velocity estimation is critical for understanding Earth's internal structure. Traditional rock physics models require careful physical assumptions and mathematical derivations, often facing challenges when applied to complex field data. Empirical formulas, while simple, lack a solid physical foundation. To address these limitations, we propose a data-driven approach using rational function neural networks (RafNN) for rock physics modelling. By analysing logging data, RafNN establishes a rational equation capturing the interdependencies among rock modulus, matrix stiffness, porosity, and fluid. The results show that RafNN accurately extracts the Gassmann's equation when the training data adheres to its constraints. Moreover, RafNN can derive general models from logging data that deviate from the Gassmann's equation. These data-driven models exhibit lower prediction errors while maintaining consistency with Gassmann's model. RafNN's adaptability to field data variability is a key advantage, facilitating better comprehension of the underlying mathematical and physical principles. Additionally, we explore the relationship between modulus, porosity, and compressibility, shedding light on the physical interpretation of RafNN models. Notably, RafNN derives analytical models directly from field data, reducing reliance on mathematical derivations and physical assumptions. Although further research is needed to understand the convergence theory of RafNN, this study presents a promising approach for data-driven rock physics modelling. It contributes to the exploration of Earth's heterogeneous structure and advances the field of seismic wave velocity estimation.

N-Oktanol karışımlarıyla çalışan bir dizel motorun ısı yayılımı oranının değişimi için yeni matematiksel korelasyonlar: Sayısal, deneysel ve karşılaştırmalı bir analiz

The current literature reports a few papers regarding the prediction of heat release rate of a diesel engine fuelled with n-octanol blends using numerical methods. To address this gap, the main objective of presented study is to derive new correlations for estimating the variation in heat release rate of a diesel engine fuelled with diesel fuel and diesel fuel-biodiesel-n-octanol blends. For this purpose, three different ternary blends were prepared by varying the n-octanol concentration to 6% (OCT6), 8% (OCT8), and 10% (OCT10) by volume. The estimation of heat release rate was accomplished using the least-squares regression (sine, piecewise, and rational equations) and the function approximation (Padé approximation) methods. The ignition delay and peak heat release rate increased by 3.8462%, 5.9501% for OCT6; 7.6923%, 3.7125% for OCT8; and 8.9744%, 3.0755% for OCT10, respectively, compared to diesel fuel. The peak cylinder pressure of OCT6 and OCT8 was observed to be higher by 2.4378% and 1.3982%, respectively, whereas that of OCT10 is found to be lower by 1.9458%, compared to diesel fuel. Compared to the others, the suggested rational equation qualitatively and quantitatively achieved the best correlation with all experimental heat release rate data measured by both the author and some other authors.

Nonlinear analysis of GFRP reinforced concrete beams using moment-rotation approach and conjugate beam method

The moment-rotation approach is a promising methodology aimed to predict the rotational capacity of reinforced concrete members. Recently, several authors have applied this method to evaluate the bending behavior of beams, replacing the moment–curvature approach. Considering the partial interaction phenomena, the moment-rotation approach can be used to develop parametric analyses and to assist in the formulation of rational equations for different material combinations. However, the techniques to integrate the moment-rotation approach into a nonlinear structural analysis have not been deeply explored, and the wider use of this methodology requires a rigorous validation. This study proposes and validates a modeling framework that integrates the moment-rotation approach into a reinforced concrete beam analysis based on the conjugate beam method. This model simulates the global stiffness decrease of beams as the cracking progresses. The numerical results were compared with experimental tests from literature performed on fiber-reinforced and plain concrete beams with glass‐fiber reinforced polymer (GFRP) bars. The experimental behavior of deflection, the crack-opening moment, and the moment–curvature were accurately simulated, especially for fiber-reinforced concrete beams. This accuracy indicates that the strategy adopted to integrate the moment-rotation approach into a structural analysis is definitely promising when modeling beams made of new materials.

Minute-Scale Models for the Diffuse Fraction of Global Solar Radiation Balanced between Accuracy and Accessibility

The separation models are tools used in solar engineering to estimate direct normal (DNI) and diffuse horizontal (DHI) solar irradiances from measurements of global solar irradiance (GHI). This paper proposes two empirical separation models that stand out owing to their simple mathematical formulation: a rational polynomial equation. Validation of the new models was carried out against data from 36 locations, covering the four major climatic zones. Five current top minute-scale separation models were considered references. The tests were performed on the final products of the estimation: DNI and DHI. The first model (M1) operates with eight predictors (evaluated from GHI post-processed measurements and clear-sky counterpart estimates) and constantly outperforms the already established models. The second model (M2) operates with three predictors based only on GHI measurements, which gives it a high degree of accessibility. Based on a statistical linear ranking method according to the models’ performance at every station, M1 leads the hierarchy, ranking first in both DNI and DHI estimation. The high accessibility of the M2 does not compromise accuracy; it is proving to be a real competitor in the race with the best-performing current models.

Sparse Triangular Decomposition for Computing Equilibria of Biological Dynamic Systems Based on Chordal Graphs.

Many biological systems are modeled mathematically as dynamic systems in the form of polynomial or rational differential equations. In this paper we apply sparse triangular decomposition to compute the equilibria of biological dynamic systems by exploiting the inherent sparsity of parameter-free systems via the chordal graph and by constructing suitable elimination orderings for parametric systems using the newly introduced block chordal graph. Our experiments with parameter-free systems provide practical information on suitable algorithms for chordal completion and verify the performance gains of sparse triangular decomposition against the ordinary one in the settings of computation of the equilibria. Then we establish full characterizations of block chordal graphs and propose algorithms for testing block chordality and constructing minimal block chordal completions. Based on these results, which are of their own merits in graph theory, we present a new algorithm of sparse triangular decomposition for parametric systems and apply it to detect the equilibria of parametric biological dynamic systems, with remarkable speedups against ordinary triangular decomposition verified by the experiments.

An assessment method of rail corrugation based on wheel–rail vertical force and its application for rail grinding

In practice, the assessment and treatment of rail corrugation are quantitatively based on the corrugation depth. Wheel–rail vertical forces (WRVF), as a direct reflection of wheel–rail interaction, can give expression to the corrugation depth and thus serve as a key parameter for assessing the corrugation. In this paper, we propose an evaluation method for rail corrugation based on the WRVF. First, a 3D wheel–rail dynamic finite element (FE) model was developed with typical parameters of CRTS II slab track and CRH3 vehicle for high-speed railways in China. The accuracy of the model was then validated with the measured WRVF data in the field. Second, using the validated model, the time–frequency domain distribution of WRVF (vehicle speed: 300 km/h) was obtained with consideration of the corrugation wavelength in the range of 40–180 mm. The non-linear least squares method and rational equation were used to fit the function between the large value of WRVF and the corrugation depth value under the conditions of different corrugation wavelengths. Next, effects of the Pinned–Pinned resonance frequency and vibration mode on the fitted parameters were analysed, by which an indicator for corrugation treatment (grinding) was proposed. Finally, the indicator was applied in the monitoring of rail corrugation for high-speed railway lines in the field. The results show that the misjudgement rate of rail grinding decisions (using the proposed indicator) is low with the accuracy at 92.5%. The proposed method can provide a basis for the rail corrugation evaluation and grinding decisions-making.

Dynamical behavior of one rational fifth-order difference equation

In this paper, we study the qualitative behavior of the rational recursive equation \begin{equation*} x_{n+1}=\frac{x_{n-4}}{\pm1\pm x_{n}x_{n-1}x_{n-2}x_{n-3}x_{n-4}}, \quad n \in \mathbb{N}_{0}:=\{0\}\cup\mathbb N, \end{equation*} where the initial conditions are arbitrary nonzero real numbers. The main goal of this paper, is to obtain the forms of the solutions of the nonlinear fifth-order difference equations, where the initial conditions are arbitrary positive real numbers. Moreover, we investigate stability, boundedness, oscillation and the periodic character of these solutions. The results presented in this paper improve and extend some corresponding results in the literature.

Diverse optical solitons to nonlinear perturbed Schrödinger equation with quadratic-cubic nonlinearity via two efficient approaches

In this study, the nonlinear perturbed Schrödinger equation(NPSE) with nonlinear terms as quadratic-cubic law nonlinearity media with the beta derivative is investigated and this investigated model is considered an icon in the field of optical fibers where it describes the wave function or state function of the optical system. Numerous solutions are extracted by engaging two novel schemes known as generalized -expansion method (GEEM) and rational extended sinh-Gordon equation expansion method (REShGEEM) in distinct forms such as bright, dark, singular and combinations of these solutions. In addition, plane wave, periodic and exponential solutions are also recovered. Using the computer application Wolfram Mathematica 11, the dynamic structure and physical characterization of some observed solutions are vividly depicted by sketching different plots. Comparing our new results with well-known literature is also given which justifies the novelty of this work. Obtained results will hold a significant place in the field of nonlinear optical fibers and suggest that the proposed methods are very influential and effective tools for solving more complex nonlinear partial differential equations in mathematical physics, engineering and nonlinear optics.