Binary Quadratic Equations Research Articles

Formulation of Special Pythagorean Triangles through Integer Solutions of the Hyperbola 𝑦𝟐 = (𝑘𝟐 + 𝟐𝑘)𝑥𝟐 + 𝟏

Objectives: The objective of this research paper is to formulate Pythagorean triangles, each of which satisfies the relation: Hypotenuse = (𝑘 + 1) times the leg with even values added with unity, through employing the integer solutions to the Hyperbola 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1. Methods: The sides of the Pythagorean triangle satisfying the requirement are obtained by suitably choosing its generators consisting of the integer solutions to the considered hyperbola. Findings : There are plenty of Pythagorean triangles satisfying the given characterization for each value of k > 0 in the binary quadratic equation given by 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1.The characterizations of special pattern of the Pythagorean triangle for 𝑘 = 2is illustrated. Novelty: From the integer solutions of a binary quadratic equation, namely, a hyperbola (Two-dimensional geometrical representation), it is possible to obtain integer solutions to a ternary quadratic equation, namely, Pythagorean triangle (Threedimensional geometrical representation). Keywords: Binary quadratic equation; Ternary quadratic equation; Hyperbola; Pythagorean triangle; Integer solutions

Effect of freeze-thaw cycles on the mechanical performance of wheat straw biomass ash modified soil

In cold regions, freeze-thaw cycles (FTCs) can alter the properties of soil used as a foundation filler, leading to failures in foundation engineering. The increase in biomass power plants has resulted in a significant amount of waste biomass ash, causing negative environmental impacts. To address these issues, waste wheat straw biomass ash (WSBA) is harnessed to enhance the properties of silty clays. This study examines how WSBA affects the mechanical properties and microstructure of silty clay after FTCs through FTCs tests, Triaxial tests, and Scanning Electron Microscopy (SEM) tests. The findings reveal that incorporating WSBA significantly enhances the mechanical properties and microstructure of the soil by filling internal pores and strengthening its structure. The mechanical properties of all soil samples exhibit significant deterioration after 1 FTC, with gradual stabilization ensuing after 6 FTCs. Notably, WSBA-modified samples show better resistance to freeze-thaw weathering compared to unmodified samples, particularly at a WSBA content of 10%. Furthermore, the study establishes empirical formulas linking mechanical parameters, freeze-thaw cycles, and WSBA content using binary quadratic equations. The investigation results could serve as a valuable reference for projects involving roadway subgrade backfill materials in regions with seasonal frozen soil.

Effect of freeze-thaw cycles on mechanical properties of compacted volcanic ash

Volcanic ash is a potential fill material for subgrade to replace traditional materials which contributes to resource conservation and sustainable development. In seasonally frozen regions volcanic ash inevitably is exposed to the effects of freeze and thaw, such as in China’s Changbai Mountains. In order to investigate the mechanical behavior of compacted volcanic ash during temperature variations, 72 groups of closed-system freeze–thaw cycle tests and triaxial shear tests with different water contents were conducted in this study. The stress–strain relationship for volcanic ash is strain-softening, and the softening degree decreases with the number of freeze–thaw cycles and water content. The results show that the mechanical parameters of volcanic ash decrease firstly with the number of freeze–thaw cycles and remain relatively stable when 10 freeze–thaw cycles elapsed. Mechanical degradation is primarily caused by the loss of inter-particle bonding forces in volcanic ash by freezing and thawing. The maximum drops in elastic modulus, peak friction angle, cohesion and internal friction angle were 54%, 7%, 20% and 4% respectively. Empirical formulations are fitted by binary quadratic equations that describe the relationship between mechanical parameters of volcanic ash and the number of freeze–thaw cycles as well as the water content. The findings can provide a reference for the construction of projects related to volcanic ash as a potential building material in seasonally frozen regions.

Integrated Growth Model of Typical Crops in China with Regional Parameters

The analysis of common properties of growth for crops is the basis for further understanding crop growth in different regions. We used four typical crops of China, winter wheat, summer maize, rice, and cotton, to build an integrated model suitable for simulating the growth of different crops. The rates and characteristics of crop growth were systematically analysed based on semirelative and fully relative logistic models of crop growth, and a comprehensive, fully relative logistic model for the four crops was established. The spatial distributions of the maximum leaf area index (LAImax) and maximum dry-matter accumulation (DMAmax) for the four crops were analysed. The semirelative and fully relative growth models exhibited different characteristics of crop growth. The essential characteristics of growth and the characteristics of the crops at each stage of growth were better represented by the fully relative logistic growth model than by the semirelative model. The comprehensive, fully relative logistic model fitted the growth of all four crops well. LAImax and DMAmax varied greatly amongst the four crops and were strongly regionally distributed. These indicators for the same crop were differentially spatially variable, and the two indicators were not significantly correlated, except for rice. LAImax and DMAmax in different regions could be obtained using a binary quadratic equation of water consumption and growing degree days for the crops. This study provides a novel method for quantitatively judging the status of crop growth, predicting crop yields, and planning for regional agricultural planting.

Experimental study of the dynamic characteristics of Songhua River silt with fine grains under freeze-thaw cycles using asymmetric hysteresis

The contact and bonding of soil particles are affected by the fine particle content of silty sand and the freeze-thaw cycle effects in the seasonal frozen zone, which are reflected as the dynamic characteristics of silty sand from the macroscopic point of view. The dynamic stress, dynamic shear modulus, dynamic shear modulus ratio, and damping ratio of Songhua River silt with different contents of fine soil particle (less than 0.075 mm) after repeated freeze-thaw cycles were studied respectively using a low-temperature dynamic triaxial test system and were compared with those of quartz sand. In order to approach the real test results, a piecewise equation was used to solve for the dynamic parameters. The results show that repeated freeze-thaw cycles have a great influence on reducing the dynamic stress, the dynamic shear modulus, and the dynamic shear modulus ratio, while they increase the damping ratio. For a given fine particle content and number of freeze-thaw cycles, the dynamic stress, dynamic shear modulus, and damping ratio of the Songhua River silt with fine particles are larger than those of quartz sand when the dynamic strain reaches 0.2%, and the difference gradually decreases as the number of freeze-thaw cycles increases. The dynamic shear modulus of the Songhua River silt with fine grains decreases more slowly with increasing dynamic strain than that of quartz sand. Therefore, a normalized empirical model was established to describe the relationship between the dynamic shear modulus ratio, the damping ratio, and the dynamic strain. This model is suitable for Songhua River silt with different fine particle contents and numbers of freeze-thaw cycles, and it provides good prediction results. Furthermore, the evolution of the model parameters with the experimental conditions was analyzed, and the limit value of the fine particle content was determined to be 10%. After repeated freezing and thawing, the influence of the fine particle content on parameters a and c weakened. The regression relationship between the model parameters and the test conditions can be described by a binary quadratic equation. It is suggested that the soil parameters should be used for the dynamic response analysis of Songhua River silt with fine grains after five freeze-thaw cycles.

Optimization of irrigation and nitrogen fertilization increases ash salt accumulation and ions absorption of drip-fertigated sugar beet in saline-alkali soils

• Irrigation and nitrogen effects on dry matter, ash salt accumulation, irons uptake of sugar beet were explored. • Dry matter and ash salt accumulation, Na + , K + , Ca 2+ and Mg 2+ uptake of W 3 were smaller than those of W 1 and W 2 . • Ash salt accumulation, Na + , K + , Ca 2+ and Mg 2+ uptake were proportional to dry matter accumulation. • Binary quadratic equations were fitted between water-nitrogen inputs and dry matter, ash salt accumulation and irons uptake. Sugar beet ( Beta vulgaris L.), as a salt-tolerant plant, has been widely cultivated in saline-alkali soils, e.g. southern Xinjiang of China, due to its effective capability of reducing soil salinity. A two-year field experiment was conducted on drip-fertigated sugar beet under plastic mulch to explore to effects of three irrigation amounts (W 1 : 0.60 ET c , W 2 : 0.80 ET c and W 3 : 1.00 ET c , where ET c is the crop evapotranspiration) and six nitrogen application rates (N 1 : 25 kg N ha −1 , N 2 : 60 kg N ha −1 , N 3 : 120 kg N ha −1 , N 4 : 240 kg N ha −1 , N 5 : 360 kg N ha −1 and N 6 : 480 kg N ha −1 ) on sugar beet growth, ash salt accumulation, plant Na + , K + , Ca 2+ and Mg 2+ uptake. The results showed that the year and the two-way interaction of irrigation and nitrogen fertilization had significant effects on dry matter, ash salt accumulation, plant Na + , K + , Ca 2+ and Mg 2+ uptake, but there was no significant effect of the three-way interaction of year, irrigation and nitrogen fertilization on ash salt accumulation, plant Ca 2+ and Mg 2+ uptake. The dry matter, ash salt accumulation, Na + , K + , Ca 2+ and Mg 2+ uptake of N 1 were significantly lower than those of other treatments. Nitrogen fertilization increased the dry matter and ash salt accumulation, Na + , K + , Ca 2+ and Mg 2+ uptake of sugar beet. The ash salt, K + and Ca 2+ were mainly distributed in the leaf and petiole, while Na + and Mg 2+ were mainly distributed in the taproot. Among the four ions, the average contents of K + and Na + were larger in all treatments, which were 1.79 % and 1.62 % in the two years, respectively. The average content of Mg 2+ was 0.36 % in all treatments in the two years, and the average content of Ca 2+ was lowest in all treatments, which was 0.14 % in the two years. Compared to W 1 and W 2 , W 3 decreased the dry matter and ash salt accumulation, plant Na + , K + , Ca 2+ and Mg 2+ uptake. The ash salt accumulation, plant Na + , K + , Ca 2+ and Mg 2+ uptake increased with the increasing dry matter accumulation. Based on the response surface methodology, the relationships between water-nitrogen inputs and dry matter, ash salt accumulation, plant Na + , K + , Ca 2+ and Mg 2+ uptake were established. The results can provide a theoretical basis for the phytoremediation of sugar beet in southern Xinjiang of China and maybe elsewhere with similar arid and semi-arid climates.

Building a skeleton-based 3D body model with angle sensor data

Human body activity recognition is beneficial for health monitoring, and wearable angle sensors are one of the most significant techniques to provide accurate body motion information, like joint angles and bone lengths. However, angle sensor data are partial and separated, which is inconvenient to describe a whole body motion. In the paper, we built a skeleton-based 3D body model to recover whole body motions with angle sensor data. We took advantage of two sub models: a multi-output regression model to expand body information and a skeleton recovery model to produce joint positions. The multi-output regression model included Gradient Boosting and Random Forest learning regressions. The skeleton recovery model made use of the law of cosines and binary quadratic equations. In addition, we developed an Android application prototype to show the practical scenarios of the 3D body model. In the evaluation, we measured the performance of the multi-output regression model by calculating statistic metrics, including mean absolute error (MAE), mean square error (MSE) and R2 score. Moreover, we computed the absolute errors and error percentages of the 3D body model for different joints and different motions. In addition, we evaluated the impact of bone lengths to make the evaluation complete. In total, the accuracy of this 3D body model achieves about eighty percent, while most joints’ bias are limited into 10 cm in real world datasets.

On the Positive Pell Equation

The binary quadratic equation () representing the hyperbola is studied for its non-zero distinct integer solutions. A few interesting properties among the solutions are presented. Employing the integer solutions of the equation under consideration, integer solutions for special straight lines, hyperbolas and parabolas are exhibited.

Observation on the Binary Quadratic Equation

Observation on the Binary Quadratic Equation

Dynamical phase transition of two-component Bose–Einstein condensate with nonlinear tunneling in an optomechanical cavity-mediated double-well system

We investigate the dynamical phase transition of two-component Bose–Einstein condensate with nonlinear tunneling, which is trapped inside a double-well and dispersively coupled to a single mode of a high-finesse optical cavity with one moving end mirror driven by a single mode standing field. In the mean-field approximation, the Hamiltonian of the system with nonlinear tunneling is characterized by the population difference of the particles and relative phase in double wells. The characteristics of the system’s stability points are determined by the corresponding Hessian matrices. However, the number of the system’s stability points depends on the roots of the first derivative of the energy. In the absence of the nonlinearity, the Hamiltonian of the system is a binary quadratic equation. When there is nonlinear tunneling, the Hamiltonian of the system is a binary quartic equation. Therefore, from a mathematical point of view, it can be found that the roots of the equation will increase, which means that the number of stability points in the system will increase. We also build the phase portrait from classical analysis. Considering the strong pair tunneling case, we confirm that nonlinear tunneling interaction leads to an increment of stability points and enriches the phase diagram of the system. We further analyze the phase portrait with the moving end mirror. It is shown that the semiclassical dynamics of the system undergoes a bifurcation of stability points with an increase in the coupling strength or detuning. In other words, the system experiences a phase transition.

ON THE POSITIVE PELL EQUATION y^2 = 72x^2 + 36

The binary quadratic equation represented by the positive pellian Y2 = 72X2 + 36 is analysed for its distinct integer solutions. A few interesting relations among the solutions are given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas, parabolas and special Pythagorean triangle.

ON THE NEGATIVE PELL EQUATION y^2=90*x^2-9

The negative pell equation represented by the binary quadratic equation is analysed for its non-zero distinct integer solutions. A few interesting relations among the solutions are presented. Employing the solutions of the equation under consideration, the integer solutions for a few choices of hyperbola and parabola are obtained.

On the Negative Pell Equation y<sup>2</sup>=45x<sup>2</sup>-11

The binary quadratic equation represented by the negative pellian y2=45x2-11 is analyzed for its distinct integer solutions. A few interesting relations among the solutions are also given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas, parabolas and special Pythagorean triangle.

Comparison of electrogenic capabilities of microbial fuel cell with different light power on algae grown cathode

Electricity generation capabilities of microbial fuel cell with different light power on algae grown cathode were compared. Results showed that microbial fuel cell with 6 and 12W power of light always produced higher voltage and power density than with 18 and 26W. Similarly, microbial fuel cell with 6 and 12W of light power always displayed higher Coulombic efficiency and specific power than the one with 18 and 26W. The results also showed that microbial fuel cell with covered anodic chamber always displayed higher voltage, power density, Coulombic efficiency and specific power than the one without covered anodic chamber. Binary quadratic equations can be used to express the relationships between the light power and the voltage, power density, Coulombic efficiency and specific power. Although lower power of light on algae grown cathode and covering anodic chamber will increase system’s electricity production, they will not significantly reduce its internal resistance.

Based on Quadratic Equations of Tyre Lifting Mechanism before Connecting Rod Combined-Type Modeling Method

The method of using two equations, a push connecting rod type dump before lifting mechanism of mathematical model, the use of mathematical and mechanical knowledge, establishing geometric equations, establish the statics equation, calculation and analysis, solution binary quadratic equations, get oil cylinder's thrust, and thus for study, so as to further optimized.

Power Generation Capabilities of Microbial Fuel Cells with Different Oxygen Supplies in the Cathodic Chamber

Two microbial fuel cells (MFCs) inoculated with activated sludge of a wastewater treatment plant were constructed. Oxygen was provided by mechanical aeration in the cathodic chamber of one MFC, whereas it was obtained by the photosynthesis of algae in the other. Electrogenic capabilities of both MFCs were compared under the same operational conditions. Results showed that the MFC with mechanical aeration in the cathodic chamber displayed higher power output than the one with photosynthesis of algae. Good linear relationship between power density and chemical oxygen demand (COD) loading rate was obtained only on the MFC with mechanical aeration. Furthermore, the relationships between power density and effluent COD and between Coulombic efficiency and COD loading rate can only be expressed as binary quadratic equations for the MFC with mechanical aeration and not for the one with photosynthesis of algae.

Effects of flow rate and chemical oxygen demand removal characteristics on power generation performance of microbial fuel cells

Two microbial fuel cells with different oxygen supplies in the cathodic chamber were constructed. Electrogenic capabilities of both cells were compared under the same operational conditions. Results showed that binary quadratic equations can express the relationships between chemical oxygen demand degradation rate and chemical oxygen demand loading and between chemical oxygen demand removal rate and chemical oxygen demand loading in both cells. Good linear relationships between power output (voltage or power density) and flow rate and between power output and chemical oxygen demand degradation rate were only found on the cell with mechanical aeration in the cathodic chamber, but not on the cell with algal photosynthesis in the cathodic chamber. The relationships between power output and chemical oxygen demand removal rate and between power output and effluent chemical oxygen demand concentration on both cells can be expressed as binary quadratic equations. The optimum flow rates to obtain higher power density and higher Coulombic efficiency in the cell with mechanical aeration in the cathodic chamber (=0.85 mW/m2 and 0.063%) and in the cell with algal photosynthesis in the cathodic chamber (=0.65 mW/m2 and 0.05%) are about 1000 and 1460 μL/min, respectively. The optimum chemical oxygen demand removal rates to obtain higher power density and higher Coulombic efficiency in the cell with mechanical aeration in the cathodic chamber (=1.2 mW/m2 and 0.064%) and in the cell with algal photosynthesis in the cathodic chamber (=0.81 mW/m2 and 0.051%) are about 40.5 and 36.5%, respectively.