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Sathiyapriya, R., та M. A. Gopalan. "Homogeneous Quadratic Equation with Four Unknowns 𝑥2 + 𝑥𝑦 + 𝑦2 = 𝑧2 + 𝑧𝑤 + 𝑤2". Indian Journal Of Science And Technology 17, № 27 (2024): 2841–47. http://dx.doi.org/10.17485/ijst/v17i27.1710.
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Objectives: Diophantine research focuses on various ways to tackle multi variable and multi-degree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to Polynomial Diophantine equation . Methods: Diophantine equations may have finite, infinite, or no solutions in integers. There is no universal method for finding solutions to Diophantine equations. Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Findings: Many distinct patterns of integer solutions are obtained. Novelty: The main thrust is to illustrate different ways of obtaining various choices of solutions in integers to second-degree equations with four variables . Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Utilization of substitution strategy reduces the given equation to a ternary quadratic equation for which solutions can be found easily. Mathematics Subject Classification:11D09 Keywords: Homogeneous second degree with four variables, Solutions in integers, Factorization method, Linear transformation, Polynomial diophantine equation
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S.Vidhyalakshmi1, M.A.Gopalan2 S. Aarthy Thangam*3. "REAL AND GAUSSIAN INTEGER SOLUTIONS TO." GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 5, no. 9 (2018): 46–53. https://doi.org/10.5281/zenodo.1409631.
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The quadratic equation with four unknowns given by is analysed for its non-zero distinct integer solutions and Gaussian integer solutions. Different choices of solutions in real and Gaussian integers are obtained. A general formula for obtaining sequence of solutions (real and complex) based on its given solution is illustrated.
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R, Sathiyapriya, та A. Gopalan M. "Homogeneous Quadratic Equation with Four Unknowns 𝑥2 + 𝑥𝑦 + 𝑦2 = 𝑧2 + 𝑧𝑤 + 𝑤2". Indian Journal of Science and Technology 17, № 27 (2024): 2841–47. https://doi.org/10.17485/IJST/v17i27.1710.
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Abstract <strong>Objectives:</strong> Diophantine research focuses on various ways to tackle multi variable and multi-degree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to Polynomial Diophantine equation .<strong> Methods:</strong> Diophantine equations may have finite, infinite, or no solutions in integers. There is no universal method for finding solutions to Diophantine equations. Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method.<strong> Findings:</strong> Many distinct patterns of integer solutions are obtained. <strong>Novelty:</strong> The main thrust is to illustrate different ways of obtaining various choices of solutions in integers to second-degree equations with four variables . Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Utilization of substitution strategy reduces the given equation to a ternary quadratic equation for which solutions can be found easily. Mathematics Subject Classification:11D09 <strong>Keywords:</strong> Homogeneous second degree with four variables, Solutions in integers, Factorization method, Linear transformation, Polynomial diophantine equation
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Sangeetha, P., and T. Divyapriya. "On The Ternary Quadratic Diophantine Equation." International Journal for Research in Applied Science and Engineering Technology 11, no. 3 (2023): 726–29. http://dx.doi.org/10.22214/ijraset.2023.49479.
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Abstract: The non-zero unique integer solutions to the quadratic Diophantine equation with three unknowns 2 2 2 x 14 xy y z are examined. We derive integral solutions in four different patterns. A few intriguing relationships between the answers and a few unique polygonal integers are shown
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Kooij, Robert E. "Cubic systems with four real line invariants." Mathematical Proceedings of the Cambridge Philosophical Society 118, no. 1 (1995): 7–19. http://dx.doi.org/10.1017/s0305004100073412.
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A polynomial system is a real autonomous system of ordinary differential equations on the plane with polynomial nonlinearities:with aij, bij ∈ ℝ and where x = x(t) and y = y(t) are real-valued functions.The problem of analysing limit cycles (isolated periodic solutions) in polynomial systems was first discussed by Poincaré[16]. Then, in the famous list of 23 mathematical problems stated in 1900, Hilbert[9] asked in the second part of the 16th problem for an upper bound for the number of limit cycles for nth degree polynomial systems, in terms of n. Recently, it has been proved that, given a particular choice of coefficients for a system of form (1·1), the number of limit cycles is finite. This result is known as Dulac's theorem, see Ecalle[8] or Il'yashenko[10]. However, it is unknown whether or not there exists an upper bound for the number of limit cycles in system (1·1) in terms of n. Even for quadratic systems (i.e. polynomial systems with quadratic nonlinearities) this remains an open question.
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Chen, Jian, Jian-Hua Yin, and C. F. Lee. "Upper bound limit analysis of slope stability using rigid finite elements and nonlinear programming." Canadian Geotechnical Journal 40, no. 4 (2003): 742–52. http://dx.doi.org/10.1139/t03-032.
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In this paper, the development and application of a new upper bound limit method for two- and three-dimensional (2D and 3D) slope stability problems is presented. Rigid finite elements are used to construct a kinematically admissible velocity field. Kinematically admissible velocity discontinuities are permitted to occur at all inter-element boundaries. The proposed method formulates the slope stability problem as an optimization problem based on the upper bound theorem. The objective function for determination of the minimum value of the factor of safety has a number of unknowns that are subject to a set of linear and nonlinear equality constraints as well as linear inequality constraints. The objective function and constrain equations are derived from an energywork balance equation, the MohrCoulomb failure (yield) criterion, an associated flow rule, and a number of boundary conditions. The objective function with constraints leads to a standard nonlinear programming problem, which can be solved by a sequential quadratic algorithm. A computer program has been developed for finding the factor of safety of a slope, which makes the present method simple to implement. Four typical 2D and 3D slope stability problems are selected from the literature and are analysed using the present method. The results of the present limit analysis are compared with those produced by other approaches reported in the literature.Key words: limit analysis, upper bound, rigid finite element, nonlinear programming, sequential quadratic algorithm, slope stability.
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Diachuk, Maksym, and Said M. Easa. "Using Inverse Dynamics Technique in Planning Autonomous Vehicle Speed Mode Considering Physical Constraints." Highlights of Vehicles 1, no. 1 (2023): 29–53. http://dx.doi.org/10.54175/hveh1010003.
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The study aims at improving the technique of planning the autonomous vehicles’ (AV) speed mode based on a kinematic model with physical restrictions. A mathematical model relates the derivatives of kinematic parameters with ones of the trajectory’s curvature. The inverse approach uses an expanded vehicle model considering the distribution of vertical reactions, wheels’ longitudinal reactions according to a drive type, and lateral forces ensuring motion stability. For analysis of the drive type, four options are proposed: front-wheel drive (FWD), rear-wheel drive (RWD), permanent engaged all-wheel drive (AWD), and 4-wheel drive with torque vectoring (4WD-TV). The optimization model is also built by the inverse scheme. The longitudinal speed’s higher derivatives are modeled by the finite element (FE) functions with nodal unknowns. The sequential integrations ensure the optimality and smoothness of the third derivative. The kinematic restrictions are supplemented by the tire-road critical slip states. Sequential quadratic programming (SQP) and the Gaussian N-point scheme for quadrature integration are used to minimize the objective function. The simulation results show a significant difference in the mode forecasts between four types of AV drives at the same initial conditions. This technique allows redistributing the traction forces strictly according to the wheels’ adhesion potentials and increases the optimization performance by about 40% compared to using the kinematic model based on the same technique without physical constrains.
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Shao, Guanghao, Yucai Shi, Di Chang, Wensheng Zhang, and Zeqiang Zhang. "The Scaling Ellipsoid Method to Calculate Wellbore Separation Factors." Journal of Physics: Conference Series 2520, no. 1 (2023): 012017. http://dx.doi.org/10.1088/1742-6596/2520/1/012017.
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Abstract The wellbore separation factor is the most common index to evaluate wellbore collision risk. However, all existing calculation methods of separation factor can only evaluate the separation degree of two error ellipses on some plane, and cannot directly evaluate the separation degree of two error ellipsoids in 3D space. These methods will lead to over-optimistic or over-conservative assessments inevitably, and cannot satisfy the wellbore collision avoidance requirements. Therefore, a new calculation method for the wellbore separation factor is presented, which is named as Scaling Ellipsoid Method (SEM). The new method references the Expansion Ellipse Method (EEM) and expands or contracts two error ellipsoids by the same scaling factor up to the exterior contact condition. Then the scaling factor is selected as the wellbore separation factor. SEM can directly and accurately evaluate the separation degree of two error ellipsoids in 3D space. A mathematical model to calculate the SEM separation factor (SESF) characterized by a quadratic equation set with four unknowns is established, and its numerical solution method and optimization tactics are presented to enhance its solution efficiency. Instance analyses have shown that SEM can preferably evaluate wellbore collision risk, contribute to pinpointing dangerous intervals, and decrease the anti-collision workloads.
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Vaidyanathan, S. "A 3 - D Novel Highly Chaotic System with Four Quadratic Nonlinearities, its Adaptive Control and Anti - Synchronization with Unknown Parameters." Journal of Engineering Science and Technology Review 8, no. 2 (2015): 96–115. http://dx.doi.org/10.25103/jestr.082.15.
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Devi, P. Jamuna, та K. S. Araththi. "On the Ternary Cubic Diophantine Equation 𝑥3 + 𝑦3 = 2(𝑧 + 𝑤)2(𝑧 − 𝑤)". Indian Journal Of Science And Technology 17, № 33 (2024): 3473–80. http://dx.doi.org/10.17485/ijst/v17i33.2186.
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The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. Objectives: The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. Method: Solving Diophantine equation is obtained by the method of Decomposition. The structure of decomposition: like , where and Z. By the decomposing method in primary terms of a, we achieve a countable number of decompositions in k full factors . Each decomposition of this kind leads to a system of equations similar to: , . We get multitude of solutions for a given equation, by determining the system of equations. Findings: By the method of linear transformations, the ternary cubic equation with four unknowns is solved for its integral solutions. The equation is researched for its attributes and correlation among the solutions for its non – zero unique integer points. In each of the transformations taken, the cubic equation yields different solutions. The properties of the solutions and their relationship with the special numbers are also exhibited. Novelty: Mathematician’s interest towards solving Pell’s equation has been so much not because they approximate with a value for . The main importance of the Pell’s equation is due to that most of the common questions have answers in this equation which can be sorted by 2 variables in the Quadratic equations. This document is about the research on higher degree Cubic Diophantine equation which gives the integral solutions of this equation, taken into consideration. Keywords: Integral solutions, Ternary Cubic, Oblong number, Polygonal number
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Ahmed, Mariwan Rashid, Shazad Shawki Ahmed, and Pishtiwan Othman Sabir. "Utilizing quadratic B-spline functions to solve nonlinear systems of Volterra integro-fractional differential equations with variable coefficients." Physica Scripta 100, no. 5 (2025): 055223. https://doi.org/10.1088/1402-4896/adca67.
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Abstract The primary objective of this article is to develop new methods based on quadratic B-spline functions to numerically solve the general nonlinear system of Volterra integro-fractional differential equations of Hammerstein type (NSVIFDE-H) described in the Caputo sense. Using the Gauss quadrature method for numerical integration and quadratic B-spline functions, we propose four novel strategies for solving the considered equations. A key advantage of the proposed schemes is their ability to transform the NSVIFDE-H into a nonlinear system of ordinary algebraic equations, which can be effectively solved using appropriate techniques. By deriving the general form of the Caputo fractional derivative of the quadratic B-spline curve, we convert the system into a set of nonlinear ordinary algebraic equations involving unknown control points. Newton’s method is then applied to solve the resulting system and determine the control points. Numerical examples, along with corresponding graphs and tables, are presented to demonstrate the effectiveness and efficiency of the new algorithms. These results are compared with exact solutions, and most of the general programs were developed using MATLAB software 9.2 to implement the solutions for the test examples. Furthermore, these mathematical methods can be applied to solve complex physical problems, improve system performance, and address physics-related challenges such as optimizing optical systems, enhancing signal processing for antennas, and refining image analysis.
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Moldovyan, Dmitriy, Alexander Moldovyan, and Nikolay Moldovyan. "A NEW CONCEPT FOR DESIGNING POST-QUANTUM DIGITAL SIGNATURE ALGORITHMS ON NON-COMMUTATIVE ALGEBRAS." Voprosy kiberbezopasnosti, no. 1(47) (2022): 18–25. http://dx.doi.org/10.21681/2311-3456-2022-1-18-25.
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Purpose of work is the development of a new approach to designing post-quantum digital signature algorithms that are free from the shortcomings of known analogs – large sizes of the signature and public key. Research method is the use of power vector equations with multiple occurrences of the signature S as a signature verification equation. The computational difficulty of solving equations of the said type relatively the unknown value of S ensures the resistance of the signature scheme to attacks using S as a fitting parameter. The possibility of calculating the value of S by the secret key is provided by using the public key in the form of a set of secret elements of the hidden group, masked by performing left and right multiplications by matched invertible vectors. Results of the study include a new proposed concept for the development of post-quantum digital signature algorithms on non-commutative algebras, which use a hidden commutative group. One of its main differences is the use of a secret key in the form of a set of vectors, the knowledge of which makes it possible to calculate the correct signature value for the random powers present in the verification equation. The form of the latter defines a system of quadratic vector equations connecting the public key with the secret, which is reduced to a system of many quadratic equations with many unknowns, given over a finite field. The computational difficulty of finding a solution to the latter system determines the security of the algorithms developed within the framework of the proposed concept. A quantum computer is ineffective for solving this problem, therefore, the said algorithms are post-quantum. As analogs in construction, digital signature algorithms based on the computational difficulty of the hidden discrete logarithm problem are considered, however, the use of a hidden group and exponentiation operations represent only a general technique for ensuring the correctness of the signature schemes developed within the framework of the concept, and not for specifying a basic computationally difficult problem. To improve the performance of the signature generation and verifications procedures, the four-dimensional algebras defined by sparse basis vector multiplication tables are used as an algebraic support. The proposed concept is confirmed by the development of a specific post-quantum algorithm that provides a significant reduction in the size of the public key and signature in comparison with the finalists of the NIST global competition in the nomination of post-quantum digital signature algorithms. Practical relevance: The developed new concept for constructing post-quantum digital signature algorithms expands the areas of their application in conditions of limited computing resources
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Thuan, Mai Viet, Dinh Cong Huong, Nguyen Huu Sau, and Quan Thai Ha. "Unknown input fractional-order functional observer design for one-side Lipschitz time-delay fractional-order systems." Transactions of the Institute of Measurement and Control 41, no. 15 (2019): 4311–21. http://dx.doi.org/10.1177/0142331219856190.
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This paper addresses the problem of unknown input fractional-order functional state observer design for a class of fractional-order time-delay nonlinear systems. The nonlinearities consist of two parts where one part is assumed to satisfy both the one-sided Lipschitz condition and the quadratically inner-bounded condition and the other is not necessary to be Lipschitz and can be regarded as an unknown input, making the wider class of considered nonlinear systems. By taking the advantages of recent results on Caputo fractional derivative of a quadratic function, we derive new sufficient conditions with the form of linear matrix inequalities (LMIs) to guarantee the asymptotic stability of the systems. Four examples are also provided to show the effectiveness and applicability of the proposed method.
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Fu, Qidi, Jianwei Wu, Chuanyun Yu, Tao Feng, Ning Zhang, and Jianrun Zhang. "Linear Quadratic Optimal Control with the Finite State for Suspension System." Machines 11, no. 2 (2023): 127. http://dx.doi.org/10.3390/machines11020127.
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The control algorithm could greatly help the suspension system improve the comprehensive performance of the vehicle. Existing control methods need to obtain the intermediate states, which are difficult to obtain directly or accurately when estimated by filters or observers. Thus, this paper proposed a new practical finite state LQR control method to deal with this problem. By combining with the output state of the finite sensor of the vehicle suspension system and weakening the unknown state as the goal, an optimization model is established with the design variables as the LQR weight coefficients. Then, the direct relationship between the current control input and the finite sensor output is obtained, and the finite state LQR control is realized. Taking the quarter-car suspension model as an example, the corresponding noise is added considering sensor accuracy, and the control performance of the four control methods is studied considering the uncertainties of suspension system parameters. In addition, the acceleration of sprung mass and the dynamic travel coefficient of suspension have been separately calculated by methods of finite state LQR control, LQR control, and PID control. The results show that there is not much difference between them under shock excitation or random excitation. However, the finite state LQR control method has the best comprehensive control performance in that its dynamic tire load coefficient is better than other methods; it could take into account the suspension work stroke coefficient, dynamic tire load coefficient, and sprung mass’ acceleration of the vehicle suspension system at the same time. In order to realize the optimal control effect with limited sensor arrangement, the finite state LQR control method only needs to obtain the current sensor output and the current control input, without estimating the unknown intermediate state. By this means, the proposed control method greatly simplifies the design of the control system and has great advantages on practical value.
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Panda, Mahesh Kumar, and Lipsa Rani Bhoi. "E-Bayesian Estimation of Rayleigh Distribution and Its Evaluation Standards: E-posterior Risks and E-MSEs under Progressive Type-II Censoring." Austrian Journal of Statistics 54, no. 4 (2025): 82–116. https://doi.org/10.17713/ajs.v54i4.2047.
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The present study considers the problem of estimating the scale parameter, reliability function, and hazard function of Rayleigh distribution using the E-Bayesian estimation approach when progressively Type-II censored data are available. The evaluation standards of these estimates are accessed through the definition of E-posterior risk (expected posterior risk) and E-MSE (expected mean square error). These estimations are carried out using conjugate prior distributions of the unknown parameters under four different loss functions i.e. quadratic, weighted squared error, Degroot, and entropy loss functions. Further, we perform Monte Carlo simulations to compare the performances of these proposed methods and use a real dataset for illustration purposes.
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Steihaug, Trond. "A Story of Computational Science: Colonel Titus’ Problem from the 17th Century." Axioms 11, no. 6 (2022): 287. http://dx.doi.org/10.3390/axioms11060287.
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Experimentation and the evaluation of algorithms have a long history in algebra. In this paper we follow a single test example over more than 250 years. In 1685, John Wallis published A treatise of algebra, both historical and practical, containing a solution of Colonel Titus’ problem that was proposed to him around 1650. The Colonel Titus problem consists of three algebraic quadratic equations in three unknowns, which Wallis transformed into the problem of finding the roots of a fourth-order (quartic) polynomial. When Joseph Raphson published his method in 1690, he demonstrated the method on 32 algebraic equations and one of the examples was this quartic equation. Edmund Halley later used the same polynomial as an example for his new methods in 1694. Although Wallis used the method of Vietè, which is a digit–by–digit method, the more efficient methods of Halley and Raphson are clearly demonstrated in the works by Raphson and Halley. For more than 250 years the quartic equation has been used as an example in a wide range of solution methods for nonlinear equations. This paper provides an overview of the Colonel Titus problem and the equation first derived by Wallis. The quartic equation has four positive roots and the equation has been found to be very useful for analyzing the number of roots and finding intervals for the individual roots, in the Cardan–Ferrari direct approach for solving quartic equations, and in Sturm’s method of determining the number of real roots of an algebraic equation. The quartic equation, together with two other algebraic equations, have likely been the first set of test examples used to compare different iteration methods of solving algebraic equations.
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P, Jamuna Devi, та S. Araththi K. "On the Ternary Cubic Diophantine Equation 𝑥3 + 𝑦3 = 2(𝑧 + 𝑤)2(𝑧 − 𝑤)". Indian Journal of Science and Technology 17, № 33 (2024): 3473–80. https://doi.org/10.17485/IJST/v17i33.2186.
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Abstract The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. <strong>Objectives:</strong> The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. <strong>Method:</strong> Solving Diophantine equation is obtained by the method of Decomposition. The structure of decomposition: like , where and Z. By the decomposing method in primary terms of a, we achieve a countable number of decompositions in k full factors . Each decomposition of this kind leads to a system of equations similar to: , . We get multitude of solutions for a given equation, by determining the system of equations. <strong>Findings:</strong> By the method of linear transformations, the ternary cubic equation with four unknowns is solved for its integral solutions. The equation is researched for its attributes and correlation among the solutions for its non – zero unique integer points. In each of the transformations taken, the cubic equation yields different solutions. The properties of the solutions and their relationship with the special numbers are also exhibited. <strong>Novelty</strong>: Mathematician’s interest towards solving Pell’s equation has been so much not because they approximate with a value for . The main importance of the Pell’s equation is due to that most of the common questions have answers in this equation which can be sorted by 2 variables in the Quadratic equations. This document is about the research on higher degree Cubic Diophantine equation which gives the integral solutions of this equation, taken into consideration. <strong>Keywords:</strong> Integral solutions, Ternary Cubic, Oblong number, Polygonal number
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Vaidyanathan, Sundarapandian, and Christos Volos. "Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system." Archives of Control Sciences 25, no. 3 (2015): 333–53. http://dx.doi.org/10.1515/acsc-2015-0022.
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AbstractFirst, this paper announces a seven-term novel 3-D conservative chaotic system with four quadratic nonlinearities. The conservative chaotic systems are characterized by the important property that they are volume conserving. The phase portraits of the novel conservative chaotic system are displayed and the mathematical properties are discussed. An important property of the proposed novel chaotic system is that it has no equilibrium point. Hence, it displays hidden chaotic attractors. The Lyapunov exponents of the novel conservative chaotic system are obtained as L1= 0.0395,L2= 0 and L3= −0.0395. The Kaplan-Yorke dimension of the novel conservative chaotic system is DKY=3. Next, an adaptive controller is designed to globally stabilize the novel conservative chaotic system with unknown parameters. Moreover, an adaptive controller is also designed to achieve global chaos synchronization of the identical conservative chaotic systems with unknown parameters. MATLAB simulations have been depicted to illustrate the phase portraits of the novel conservative chaotic system and also the adaptive control results.
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Sathiyapriya, R., та M. A. Gopalan. "A Glimpse on Integral Solutions of Ternary Uniform Second Degree Equation 𝟓 (𝐱𝟐 + 𝐲𝟐) − 𝟑𝑥𝑦 = 𝟐𝟎𝟎𝟎𝑧𝟐". Indian Journal Of Science And Technology 17, № 38 (2024): 4005–9. http://dx.doi.org/10.17485/ijst/v17i38.2575.
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Objectives: Many integer solutions to ternary homogeneous quadratic given by are determined. Various choices of integer solutions are secure from beginning to end employing linear modifications and used to simplify expressions. Methods: Different transformations are utilized to determine many solutions in integers for the considered polynomial equation. Findings: Five distinct transformations are applied to obtain choices of integral solutions for the considered second-degree equation having three unknowns. Novelty: The equation in the title has been reduced to a solvable ternary quadratic equation by employing suitable transformation. Mathematics Subject Classification: 11D09 Keywords: Quadratic having three unknowns, Homogeneous quadratic, Integer Solutions, Ternary Quadratic Equation, Pair of equations
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Choi, Stephen Kwok-Kwong, and Angel V. Kumchev. "Quadratic equations with five prime unknowns." Journal of Number Theory 107, no. 2 (2004): 357–67. http://dx.doi.org/10.1016/j.jnt.2004.03.007.
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R, Sathiyapriya, та A. Gopalan M. "A Glimpse on Integral Solutions of Ternary Uniform Second Degree Equation 𝟓 (𝐱𝟐 + 𝐲𝟐) − 𝟑𝑥𝑦 = 𝟐𝟎𝟎𝟎𝑧𝟐". Indian Journal of Science and Technology 17, № 38 (2024): 4005–9. https://doi.org/10.17485/IJST/v17i38.2575.
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Abstract <strong>Objectives:</strong> Many integer solutions to ternary homogeneous quadratic given by are determined. Various choices of integer solutions are secure from beginning to end employing linear modifications and used to simplify expressions. <strong>Methods:</strong> Different transformations are utilized to determine many solutions in integers for the considered polynomial equation. <strong>Findings:</strong> Five distinct transformations are applied to obtain choices of integral solutions for the considered second-degree equation having three unknowns. <strong>Novelty:</strong> The equation in the title has been reduced to a solvable ternary quadratic equation by employing suitable transformation. Mathematics Subject Classification: 11D09 <strong>Keywords:</strong> Quadratic having three unknowns, Homogeneous quadratic, Integer Solutions, Ternary Quadratic Equation, Pair of equations
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Thiruniraiselvi, N., та M. A. Gopalan. "A Portrayal of Integer Solutions to Transcendental Equation with Six Unknowns 𝟑√ 𝐗 + 𝐘 + √ 𝐱𝐲 = 𝑧𝟐 + 𝑤𝟐". Indian Journal Of Science And Technology 17, № 29 (2024): 2992–3001. http://dx.doi.org/10.17485/ijst/v17i29.1705.
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Objectives: This research article focuses on finding non-zero integer solutions to the Transcendental equation with six unknowns represented by . Methods: There is no universal method for finding integer solutions to Diophantine equations involving surds. Various substitutions strategies are employed to obtain non-zero distinct integer solutions to the surd equation under consideration. Findings: Five different choices of substitutions are utilized to determine many non-zero distinct integer solutions to the transcendental equation with six unknowns in title. Novelty: In this analysis, the transcendental equation with six unknowns given by has been reduced to either Space Pythagorean equation or non-homogeneous ternary quadratic equation through suitable transformations and for which integer solutions can be found elegantly. Keywords: Transcendental Equation With Six Unknowns, Surd Equation, Integer Solutions, Space Pythagorean Equation, Ternary Quadratic Equation
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Kulkarni, Raghavendra G. "Intersect a quartic to extract its roots." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 16, no. 1 (2017): 73–76. http://dx.doi.org/10.1515/aupcsm-2017-0006.
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AbstractIn this note we present a new method for determining the roots of a quartic polynomial, wherein the curve of the given quartic polynomial is intersected by the curve of a quadratic polynomial (which has two unknown coefficients) at its root point; so the root satisfies both the quartic and the quadratic equations. Elimination of the root term from the two equations leads to an expression in the two unknowns of quadratic polynomial. In addition, we introduce another expression in one unknown, which leads to determination of the two unknowns and subsequently the roots of quartic polynomial.
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Vidhya, S. "Intrinsic Solution on Quadratic Diophantine Equation." International Journal for Research in Applied Science and Engineering Technology 12, no. 3 (2024): 631–35. http://dx.doi.org/10.22214/ijraset.2024.58885.
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Di Polito, Nicola, Anthea A. Stylianakis, Rick Richardson, and Kathryn D. Baker. "Real-World Intake of Dietary Sugars Is Associated with Reduced Cortisol Reactivity Following an Acute Physiological Stressor." Nutrients 15, no. 1 (2023): 209. http://dx.doi.org/10.3390/nu15010209.
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There is increasing academic and clinical interest in understanding the nature of the relation between diet and response to stress exposure as a risk factor for mental illness. Cross-species evidence shows that conditions of chronic and acute stress increase the intake of, and preference for, caloric-dense palatable foods, a phenomenon thought to be explained by the mitigating effects of comfort foods on the activity of the stress-response network. It is largely unknown whether and how real-world dietary intake of saturated fat and sugars impacts stress responsivity in humans. Therefore, here we examined whether real-world dietary intake of saturated fat and sugars predicted salivary cortisol reactivity following an acute physiological stressor. Multilevel modelling of four salivary cortisol measures collected up to 65 min after the stressor on 54 participants (18–49 years old) were analyzed using a quadratic growth curve model. Sugar intake significantly predicted a weaker cortisol response following the Cold Pressor Test (CPT) controlling for BMI and gender, revealing an inhibitory effect of caloric-dense diets on cortisol reactivity to stress. As the consumption of sugar rose individuals had lower post-stressor cortisol levels, a smaller rate of increase in cortisol 20 and 35 min after the CPT, a lower cortisol peak, and an overall weaker quadratic effect. These observations add to a growing body of evidence reporting suppressive effects of high-energy foods on stress-associated glucocorticoids reactivity and are consistent with the comfort food hypothesis, where people are seen as motivated to eat palatable foods to alleviate the detrimental repercussions of stressor exposure.
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N, Thiruniraiselvi, та A. Gopalan M. "A Portrayal of Integer Solutions to Transcendental Equation with Six Unknowns 𝟑√ 𝐗 + 𝐘 + √ 𝐱𝐲 = 𝑧𝟐 + 𝑤𝟐". Indian Journal of Science and Technology 17, № 29 (2024): 2992–3001. https://doi.org/10.17485/IJST/v17i29.1705.
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Abstract <strong>Objectives:</strong> This research article focuses on finding non-zero integer solutions to the Transcendental equation with six unknowns represented by . <strong>Methods:</strong> There is no universal method for finding integer solutions to Diophantine equations involving surds. Various substitutions strategies are employed to obtain non-zero distinct integer solutions to the surd equation under consideration.<strong> Findings:</strong> Five different choices of substitutions are utilized to determine many non-zero distinct integer solutions to the transcendental equation with six unknowns in title. <strong>Novelty:</strong> In this analysis, the transcendental equation with six unknowns given by has been reduced to either Space Pythagorean equation or non-homogeneous ternary quadratic equation through suitable transformations and for which integer solutions can be found elegantly. <strong>Keywords:</strong> Transcendental Equation With Six Unknowns, Surd Equation, Integer Solutions, Space Pythagorean Equation, Ternary Quadratic Equation
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Liu, Hao, Jia Chen Ma, Ming Li Yang, and Yu De Sun. "Adaptive Observer of One-Sided Lipschitz Nonlinear Systems." Applied Mechanics and Materials 336-338 (July 2013): 417–22. http://dx.doi.org/10.4028/www.scientific.net/amm.336-338.417.
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Aiming to design an adaptive observer for one-sided Lipschitz nonlinear systems, the paper focuses on extensions to the restrictive nature of the Lipschitz continuity condition and the conservativeness of approaches proposed in the past. Main contributions include the following four aspects. Firstly, new sufficient conditions for the existence of observer and asymptotical stability are developed by using an LMI approach, which can be easily solved via standard numerical software. Then, the design scheme presented can cope with the situation where the one-sided Lipschitz constant is unknown, making the complexity of this algorithm reduced significantly compared to almost all existing results. Thirdly, we remove the constraint of quadratic inner-boundedness which is a limitation imposed extensively by the existing works. Finally, by integrating an adaptation law the conservativeness is dropped sharply, which makes the applicable systems larger. In the end, the effectiveness and less conservativeness of results are tested in a series of numerical examples.
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Liang, Wei, Jin Huang, and Jie Zhang. "A New Adaptive System for Suppression of Transient Wind Disturbance in Large Antennas." International Journal of Antennas and Propagation 2019 (September 4, 2019): 1–13. http://dx.doi.org/10.1155/2019/2015341.
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Under transient wind disturbance, vibration deformation of the large antenna surface profile causes deterioration of pointing performance. This paper presents a new adaptive system to suppress unknown transient wind disturbance. First, to monitor the vibration, based on the acceleration measurement and a low-order flexible model considering equivalent identification of forces, the real-time estimation of the vibration state is obtained in an unbiased minimum-variance way. Next, a novel four-cable-actuator mechanism with a circular slide track is proposed for suppressing the vibration, in which the locations of the cable drivers on the slide track are determined according to the attitude of the antenna, and the expected tension distribution of the cables is found by the vibration state and the optimal gain of the linear quadratic regulator (LQR). In the end, the simulation implementation of a 7.3 m antenna under the transient wind disturbance is used to verify the effectiveness of the proposed method.
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J. Shanthi, S. Vidhyalakshmi, and M. A. Gopalan. "OBSERVATION ON THE BIQUADRATIC EQUATION WITH FIVE UNKNOWNS 2(x - y)(x3 + y3) + x4 - y4 = 2(z2 - w2)p2." jnanabha 53, no. 02 (2023): 65–68. http://dx.doi.org/10.58250/jnanabha.2023.53207.
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This paper focuses on obtaining non-zero integer quintuples (x,y, z,w, p) satisfying the bi-quadratic equation with ve unknowns given by 2(x - y)(x3 + y3) + x4 - y4 = 2(z2 - w2)p2. Various patterns of solutions are obtained by reducing the given bi-quadratic equation to solvable ternary quadratic equation through employing linear transformations.
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Janaki, G. "Integer Solutions on Ternary Quadratic Diophantine Equation." International Journal for Research in Applied Science and Engineering Technology 12, no. 3 (2024): 761–65. http://dx.doi.org/10.22214/ijraset.2024.58905.
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Abstract: The non-zero distinct integer solution to the Quadratic Diophantine equation with three unknowns are examined. We obtain the various patterns of integer points that satisfy the ternary Quadratic equation. A few interesting relationship between the solution and the unique pattern.
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Radha, Dr R., and Ms K. Saranya. "Integral Solutions of the Bi-Quadratic Equation." International Journal for Research in Applied Science and Engineering Technology 12, no. 3 (2024): 165–70. http://dx.doi.org/10.22214/ijraset.2024.58706.
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Abstract: The biquadratic with six unknowns of the form has been studied for its non-trivial distinct integral solutions. A few interesting relations among the solutions and polygonal numbers are presented.
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Hema, K. "A Review on Second Degree Homogeneous Diophantine Equation with Three Unknowns x2+y2=122z2." International Journal for Research in Applied Science and Engineering Technology 11, no. 5 (2023): 4724–27. http://dx.doi.org/10.22214/ijraset.2023.52661.
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A, Gowri Shankari, and Sangavi S. "Integral Solutions of the Ternion Quadratic Equation." International Journal for Research in Applied Science and Engineering Technology 11, no. 3 (2023): 730–35. http://dx.doi.org/10.22214/ijraset.2023.49490.
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Abstract: In order to find its non zero unique integral solutions for the quadratic diophantine equation with three unknowns given by is analysed. The equation under consideration exhibits multiple patterns of solutions. The solutions are presented with a few fascinating aspects.
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Chen, Qiang, Chuang Xia, Yinyan Shi, Xiaochan Wang, Xiaolei Zhang, and Ye He. "A Novel Approach for Asparagus Comprehensive Classification Based on TOPSIS Evaluation and SVM Prediction." Agronomy 14, no. 6 (2024): 1175. http://dx.doi.org/10.3390/agronomy14061175.
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As a common vegetable variety, asparagus is rich in B vitamins, vitamin A, and trace elements such as folate, selenium, iron, manganese, and zinc. With the increasing market demand, China has become the world’s largest cultivated area for asparagus production and product exportation. However, traditional asparagus grading mostly relies on manual visual judgment and needs a lot of manpower input to carry out the classification operation, which cannot meet the needs of large-scale production. To address the high labor cost and labor-intensive production process resulting from the large amount of manpower input and low accuracy of existing asparagus grading devices, this study proposed an improved asparagus grading system and method based on TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) objective evaluation and SVM (support vector machine) prediction. The key structure of classification device was analyzed first, the key components were designed, and the structural parameters were determined by theoretical calculation. Through analysis of the factors affecting asparagus quality, three key attributes were determined: length, diameter, and bruises, which were used as reference attributes to conduct experimental analysis. Then, the graded control groups were set up, combining the TOPSIS principle with weighting, and a score for each asparagus sample was determined. These scores were compared with those of a graded control group to derive the grade of each asparagus, and these subsets of the dataset were used as the training set and the test set, excluding the error caused by the subjectivity of the manual judgment. Based on a comparison of the accuracies of different machine learning models, the support vector machine (SVM) was determined to be the most accurate, and four SVM methods were used to evaluate the test set: linear SVM, quadratic SVM, cubic SVM, and medium Gaussian SVM. The test results showed that the grading device was feasible for asparagus. The bruises had a large influence on asparagus quality. The training accuracy of the medium Gaussian SVM method was high (96%), whereas its test accuracy was low (86.67%). The training accuracies and test accuracy of the quadratic and cubic SVM methods were 93.34%. The quadratic SVM and cubic SVM were demonstrated to have better generalization ability than the medium Gaussian SVM method for predicting unknown grades of asparagus and meeting the operational requirements of the asparagus grading.
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Meldi, Nadya Febriani, Ahmad Yani T., and Dede Suratman. "Penyelesaian Persamaan Bentuk Kuadrat Berbantuan Aplikasi Photomath Berdasarkan Sistem Bilangan Real." Variabel 5, no. 2 (2022): 83. http://dx.doi.org/10.26737/var.v5i2.3224.
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<p><em>Penelitian ini bertujuan mendeskripsikan ketidaksesuaian istilah dalam prosedur pengerjaan matematika materi persamaan kuadrat berdasarkan aplikasi photomath yang disesuaikan dalam objek kajian dalam analisis real. Ketidaksesuaian yang dimaksud adalah prosedur pengerjaan yang kurang lengkap serta beberapa penamaan yang tidak dikenal dalam matematika. Penelitian ini merupakan penelitian kualitatif dengan metode deskriptif. Deskriptif dilakukan untuk mengungkap temuan ketidaksesuaian dari prosedur pengerjaan persamaan bentuk kuadrat yang dianalisis dari langkah pengerjaan yang tersaji pada </em><em>aplikasi </em><em>photomath. Langkah penyelesaian pada aplikasi photomath merupakan objek kajian serta aksioma lapangan yang menjadi dasar utama dalam proses kajian. Teknik analisis yang digunakan adalah tematik analisis. Hasil penelitian menunjukkan bahwa telah diperoleh empat langkah penyelesaian yang tidak bersesuaian pada disiplin ilmu matematika dalam menyelesaikan persamaan kuadrat pada aplikasi photomath, yaitu </em><em>“keluarkan faktor”, “pindahkan satu suku ke ruas lainnya dan ubahlah tandanya (pindah ruas)”, ”hasil kali faktor-faktor adalah nol, setidaknya salah satu faktor adalah nol”, </em><em>“bagi kedua ruas dengan suatu bilangan”.</em><em></em></p><p><strong><em>Kata Kunci</em></strong><em>: </em><em></em></p><p>Penyelesaian Persamaan Bentuk Kuadrat; Aplikasi <em>Photomath</em>; Sistem Bilangan Real</p><p> </p><p><strong>Solving Quadratic Shape Equations Assisted Photomath Application Based on Real Number Systems</strong></p><p><strong><em>ABSTRACT</em></strong><strong><em></em></strong></p><p><em>This study aims to describe the discrepancy of terms in the procedure for mathematical work of quadratic equation material based on the application of photomath adjusted in the object of study in real analysis. The discrepancies in question are incomplete workmanship procedures as well as some unknown naming in mathematics. This research was a qualitative research with a descriptive method. Descriptive is carried out to reveal findings of discrepancies from the procedure for working on quadratic shape equations analyzed from the work steps presented in the photomath application. The completion step in the photomath application was the object of study and the field axiom that was the main basis in the study process. The analysis technique used was thematic analysis. The results showed that four non-conforming solution steps have been obtained in the mathematical discipline in solving quadratic equations in the photomath application, namely "issue a factor", "move one tribe to another and change the sign (moving segments)", "the product of factors is zero, at least one of the factors is zero", "divide both fields by a number".</em></p>
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Bramley, RGV, and RE White. "The effect of pH, liming, moisture and temperature on the activity of nitrifiers in a soil under pasture." Soil Research 27, no. 4 (1989): 711. http://dx.doi.org/10.1071/sr9890711.
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A short-term nitrification assay (SNA) was used to measure the activity of soil nitrifiers in the field in relation to soil pH and seasonal changes in soil temperature and moisture content. At roughly two-monthly intervals over two successive years, samples of the Tokomaru Silt Loam which had been limed in 1982 and/or 1987 were analysed in addition to an unlimed control. The SNA analysis was carried out for a range of pH values between 4.5 and 7.5, obtained by amending the incubation medium with small amounts of HCl or KOH. A quadratic curve was fitted to a plot of SNA value v, incubation pH. The fitted equations were used to calculate the pH optimum for nitrification (pHopt), the SNA value at pHopt (SNAopt) and the SNA value at the soil pH at sampling (SNApH). Values of pHopt and the mean soil pH over the year were higher in soil limed in 1982 than control soil; neither variable showed marked change in either soil over the first 320 days of observation. The addition of lime in 1987 raised the mean soil pH and pHopt in unlimed soil, but had negligible effect on either variable in soil limed five years previously. A covariance-type analysis demonstrated that the same quadratic equation could be fitted to each plot of SNA v. incubation pH for each soil treatment by changing the intercept parameter (C), i.e. the relationship between nitrifier activity and pH in the four soils remained constant over the year. No significant relationships could be found for the four soils between soil pH, pHopt, SNA, soil moisture content, soil temperature and C. However, the slight fluctuation in pHopt tended to follow the variation in soil pH. Values of SNAopt and SNApH showed a more obvious seasonal trend and showed a 1 : 1 relationship over a range of values from 0.015 to 0.110 �mol g-1 h-l; i.e. the nitrifier activity in the soil, irrespective of variations that were random (unknown influences) or associated with seasonal variables (temperature and moisture), was near the optimum with respect to pH. Nitrification activity is dynamic and changes quickly in response to changes in the soil environment. The implications of this with respect to nitrate leaching models include the likelihood that it may not be possible to produce a definitive model which works for all soil types.
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Wood, Carly J., Jo L. Barton, and Claire L. Wicks. "The Impact of Therapeutic Community Gardening on the Wellbeing, Loneliness, and Life Satisfaction of Individuals with Mental Illness." International Journal of Environmental Research and Public Health 19, no. 20 (2022): 13166. http://dx.doi.org/10.3390/ijerph192013166.
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Background: literature on the mental health benefits of therapeutic community gardening is not specific to individuals with mental illness and reports short-term outcomes. The impact of the coronavirus pandemic on intervention effectiveness is also unknown. This study examined the impact of therapeutic community gardening prior to and across the pandemic on the wellbeing of individuals referred for support with their mental illness. Methods: garden members (n = 53; male = 36, female = 17) aged 47.38 ± 13.09 years reported their wellbeing at baseline and four follow-up points (FU1–FU4) across the pandemic. Results: there was significant quadratic growth in wellbeing (−1.248; p < 0.001) that varied between genders (p = 0.021). At baseline, male wellbeing scores were significantly lower (p = 0.020) than the UK population norm, but there were no significant differences at any other follow-up point. Female wellbeing was significantly lower than the UK population norm at baseline (p < 0.001), FU1 (p = 0.012) and FU2 (p < 0.001), but not FU3 and FU4. Conclusion: therapeutic community gardening can improve and maintain the wellbeing of individuals with mental illness, even when wellbeing is deteriorating nationally. Future research should further demonstrate the long-term and cost-effectiveness of interventions.
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Sands, Timothy. "Virtual Sensoring of Motion Using Pontryagin’s Treatment of Hamiltonian Systems." Sensors 21, no. 13 (2021): 4603. http://dx.doi.org/10.3390/s21134603.
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To aid the development of future unmanned naval vessels, this manuscript investigates algorithm options for combining physical (noisy) sensors and computational models to provide additional information about system states, inputs, and parameters emphasizing deterministic options rather than stochastic ones. The computational model is formulated using Pontryagin’s treatment of Hamiltonian systems resulting in optimal and near-optimal results dependent upon the algorithm option chosen. Feedback is proposed to re-initialize the initial values of a reformulated two-point boundary value problem rather than using state feedback to form errors that are corrected by tuned estimators. Four algorithm options are proposed with two optional branches, and all of these are compared to three manifestations of classical estimation methods including linear-quadratic optimal. Over ten-thousand simulations were run to evaluate each proposed method’s vulnerability to variations in plant parameters amidst typically noisy state and rate sensors. The proposed methods achieved 69–72% improved state estimation, 29–33% improved rate improvement, while simultaneously achieving mathematically minimal costs of utilization in guidance, navigation, and control decision criteria. The next stage of research is indicated throughout the manuscript: investigation of the proposed methods’ efficacy amidst unknown wave disturbances.
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Ghassemieh, M., A. A. Gholampour, and S. R. Massah. "Application of Weight Functions in Nonlinear Analysis of Structural Dynamics Problems." International Journal of Computational Methods 13, no. 01 (2016): 1650005. http://dx.doi.org/10.1142/s0219876216500055.
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This paper presents a weighted residual method with several weight functions for solving differential equation of motion in nonlinear structural dynamics problems. Order of variation of acceleration is assumed to be quadratic in each time step in which polynomial of displacement would contain five unknown coefficients. Five equations are required for determination of these coefficients in each time step. These equations are obtained from initial conditions, satisfying equation of motions at both ends, and weighted residual integration. In this study, four procedures are considered for weight function to be used in the weighted residual integration as; unit weight function, Petrov–Galerkin’s weight function, least square weight function, and collocation weight function. Due to higher order of acceleration in the proposed method, the results indicate better and more accurate responses. Among the tested functions, the unit weighted function method demonstrated to be non-dissipative and its numerical dispersion showed to be clearly less than the common Newmark’s linear acceleration method. Also critical time step duration in stability investigation for weighted function procedure showed to be larger than the critical time step duration obtained by other methods used in the nonlinear structural dynamics problems.
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Shreemathi, Adiga1* N. Anusheela2 &. M.A. Gopalan3. "INTEGER SOLUTIONS OF HOMOGENEOUS BI-QUADRATIC EQUATION WITH FIVE UNKNOWNS." GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 5, no. 8 (2018): 145–49. https://doi.org/10.5281/zenodo.1344365.
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The homogeneous biquadratic equation represented by is considered and analysed for its non-zero distinct integer solutions. It is observed that by introducing the linear transformations in the given equation, one obtains infinitely many non-zero distinct integer solutions.
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CIARLET, PHILIPPE G., and LILIANA GRATIE. "A NEW APPROACH TO LINEAR SHELL THEORY." Mathematical Models and Methods in Applied Sciences 15, no. 08 (2005): 1181–202. http://dx.doi.org/10.1142/s0218202505000704.
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We propose a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary. Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korn's inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincaré. The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.
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Kim, Sang Dong. "Preconditioning collocation method using quadratic splines with applications to 2nd-order separable elliptic equations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 37, no. 4 (1996): 549–70. http://dx.doi.org/10.1017/s0334270000010869.
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AbstractIn this paper we propose a P1 finite element preconditioning using the so-called ‘hat-function’, to a collocation scheme constructed by quadratic splines for a 2nd-order separable elliptic operator and we show that the resulting preconditioning system of equations is well conditioned with the condition number independent of the number of unknowns.
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Savkina, Marta. "SUFFICIENT CONDITION FOR COINCIDENCE OF THE LS AND AITKEN ESTIMATIONS OF PARAMETER OF QUADRATIC REGRESSION IN CASE HETEROSCEDASTIC DEVIATIONS." Journal of Numerical and Applied Mathematics, no. 2 (134) (2020): 45–58. http://dx.doi.org/10.17721/2706-9699.2020.2.03.
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In the paper in case heteroscedastic independent deviations a regression model whose function has the form $f(x) = ax^2+bx+c$, where $a$, $b$ and $c$ are unknown parameters, is studied. Approximate values (observations) of functions $f(x)$ are registered at equidistant points of a line segment. The theorem which is proved at the paper gives a sufficient condition on the variance of the deviations at which the Aitken estimation of parameter $a$ coincides with its estimation of the LS in the case of odd number of observation points and bisymmetric covariance matrix. Under this condition, the Aitken and LS estimations of $b$ and $c$ will not coincide. The proof of the theorem consists of the following steps. First, the original system of polynomials is simplified: we get the system polynomials of the second degree. The variables of both systems are unknown variances of deviations, each of the solutions of the original system gives a set variances of deviations at which the estimations of Aitken and LS parameter a coincide. In the next step the solving of the original system polynomials is reduced to solving an equation with three unknowns, and all other unknowns are expressed in some way through these three. At last it is proved that there are positive unequal values of these three unknowns, which will be the solution of the obtained equation. And all other unknowns when substituting in their expression these values will be positive.
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Savkina, Marta. "SUFFICIENT CONDITION FOR COINCIDENCE OF THE LS AND AITKEN ESTIMATIONS OF PARAMETER OF QUADRATIC REGRESSION IN CASE HETEROSCEDASTIC DEVIATIONS." Journal of Numerical and Applied Mathematics, no. 2 (134) (2020): 45–58. http://dx.doi.org/10.17721/2706-9699.2020.2.03.
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In the paper in case heteroscedastic independent deviations a regression model whose function has the form $f(x) = ax^2+bx+c$, where $a$, $b$ and $c$ are unknown parameters, is studied. Approximate values (observations) of functions $f(x)$ are registered at equidistant points of a line segment. The theorem which is proved at the paper gives a sufficient condition on the variance of the deviations at which the Aitken estimation of parameter $a$ coincides with its estimation of the LS in the case of odd number of observation points and bisymmetric covariance matrix. Under this condition, the Aitken and LS estimations of $b$ and $c$ will not coincide. The proof of the theorem consists of the following steps. First, the original system of polynomials is simplified: we get the system polynomials of the second degree. The variables of both systems are unknown variances of deviations, each of the solutions of the original system gives a set variances of deviations at which the estimations of Aitken and LS parameter a coincide. In the next step the solving of the original system polynomials is reduced to solving an equation with three unknowns, and all other unknowns are expressed in some way through these three. At last it is proved that there are positive unequal values of these three unknowns, which will be the solution of the obtained equation. And all other unknowns when substituting in their expression these values will be positive.
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Zambrano, Elena, Luis A. Reyes-Castro, Guadalupe L. Rodríguez-González, et al. "Developmental Programming-Aging Interactions Have Sex-Specific and Developmental Stage of Exposure Outcomes on Life Course Circulating Corticosterone and Dehydroepiandrosterone (DHEA) Concentrations in Rats Exposed to Maternal Protein-Restricted Diets." Nutrients 15, no. 5 (2023): 1239. http://dx.doi.org/10.3390/nu15051239.
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The steroids corticosterone and dehydroepiandrosterone (DHEA) perform multiple life course functions. Rodent life-course circulating corticosterone and DHEA trajectories are unknown. We studied life course basal corticosterone and DHEA in offspring of rats fed protein-restricted (10% protein, R) or control (20% protein, C), pregnancy diet first letter, and/or lactation second letter, producing four offspring groups—CC, RR, CR, and RC. We hypothesize that 1. maternal diet programs are sexually dimorphic, offspring life course steroid concentrations, and 2. an aging-related steroid will fall. Both changes differ with the plastic developmental period offspring experienced R, fetal life or postnatally, pre-weaning. Corticosterone was measured by radioimmunoassay and DHEA by ELISA. Steroid trajectories were evaluated by quadratic analysis. Female corticosterone was higher than male in all groups. Male and female corticosterone were highest in RR, peaked at 450 days, and fell thereafter. DHEA declined with aging in all-male groups. DHEA: corticosterone fell in three male groups but increased in all-female groups with age. In conclusion, life course and sexually dimorphic steroid developmental programming-aging interactions may explain differences in steroid studies at different life stages and between colonies experiencing different early-life programming. These data support our hypotheses of sex and programming influences and aging-related fall in rat life course serum steroids. Life course studies should address developmental programming-aging interactions.
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Hidano, Kunio, and Kazuyoshi Yokoyama. "Global Existence for a System of Quasi-Linear Wave Equations in 3D Satisfying the Weak Null Condition." International Mathematics Research Notices 2020, no. 1 (2018): 39–70. http://dx.doi.org/10.1093/imrn/rny024.
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Abstract We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for the scalar unknowns is also refined.
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Alatriste-Jiménez, Tania S., Fidel Blanco-Macías, Catarina Loredo-Osti, Sergio Beltrán-López, José Luis Lara-Mireles, and Ricardo David Valdez Cepeda. "Boundary-Line Approach micro-nutrient optimum concentrations and sufficiency ranges for Opuntia ficus-indica (L.) Miller fruiting." Journal of the Professional Association for Cactus Development 23 (November 19, 2021): 150–64. http://dx.doi.org/10.56890/jpacd.v23i.472.
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Optimum concentrations and/or sufficiency ranges of nutrients are useful for correct diagnosis and improvements of nutrient status of cultivated plants. However, Opuntia ficus-indica nutrient storage at plant and cladode levels and best fit plant requirements for fruiting remain practically unknown. Then, this research work aimed to identify the Boundary-Line Approach (B-LA) micronutrient optimum concentrations linked to maximum fruit yield per one-year old fructification cladode and sufficiency ranges at 90% maximum fruit yield for O. ficus-indica variety ‘Rojo Pelón’. Four years (2012 – 2015) data of fruit yield per cladode and micronutrient concentrations (B, Cu, Fe, Mn, and Zn) were used for the elaboration of scatter diagrams (n = 228) and selection of 7 to 10 points to estimate the B-LA quadratic functions. Then, the vertices allowed estimation of the optimum micronutrient concentrations: B = 33.62 mg kg–1, Cu =11.74 mg kg–1, Fe = 108.51 mg kg–1, Mn = 149.33 mg kg–1 and Zn = 414.91 mg kg–1 as linked to estimated maximum yield per cladode; and the sufficiency ranges at 90% maximum fruit yield: B = 24.44 to 46.25 mg kg–1, Cu = 8.68 – 15.87 mg kg–1, Fe = 73.73 – 143.3 mg kg–1, Mn = 114.89 – 183.77 mg kg–1 and Zn = 334.31 – 514.95 mg kg–1. The linked estimated maximum fruit yield per cladode varies from 1999.48 g to 2139.59 g. The proposed B-LA standards can be used to perform reliable micronutrient diagnosis and proper fertilization recommendations.
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Wilson, T. A., M. J. Hill, and R. D. Hubmayr. "Regional lung volume trajectories during expiratory flow in dogs." Journal of Applied Physiology 80, no. 4 (1996): 1144–48. http://dx.doi.org/10.1152/jappl.1996.80.4.1144.
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Parenchymal markers were placed in the left caudal lobes of anesthetized dogs, and the three-dimensional positions of the markers were tracked by biplane videofluoroscopy during expiration from total lung capacity (TLC) to approximately 30% TLC at steady expiratory flows of 0.3-7% TLC/s. Regional volumes of samples of parenchyma with volumes at TLC of 1-5 cm3 were obtained by computing the volumes of tetrahedral defined by taking the markers, four at a time, as apices of the tetrahedra, Regional volume (Vr), as a fraction of volume at TLC, was plotted against average volume (VL), as a fraction of volume at TLC, and Vr was fit by a quadratic function of VL. The initial slopes of the plots, dVr/dVL at VL = 1, varied by +/- 26% from the mean slope of 1. The curvatures were highly correlated with the slopes, and the Vr vs. VL plots formed a nest of “onion skins”. The initial slopes were weakly correlated with the vertical position of the centroid of the tetrahedron, with the slope increasing by approximately 20%/cm, on average, in the dorsal direction in the supine dog. The vertical gradient in Vr accounted for approximately 30% of the total variability; small-scale heterogeneity contributed the remaining 70%. These results confirm earlier quasistatic measurements of nonuniform Vr and provide the first data on curvature of Vr trajectories. The mechanisms that cause the nongravitational component of Vr variability and the curvatures of the trajectories are unknown.
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Nahil, Mohamad A., Omar Aboelazayem, Scott Wiseman, et al. "Production and Optimisation of Oxygenated Biofuel Blend Components via the Ethanolysis of Lignocellulosic Biomass: A Response Surface Methodology." Energies 18, no. 11 (2025): 2985. https://doi.org/10.3390/en18112985.
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In this study, a response surface methodology (RSM) using a central composite design (CCD) was implemented to investigate the influence of process variables on ethyl levulinate (EL) production from the ethanolysis of waste corn cob samples, using sulphuric acid as a catalyst. The effects of four independent variables, namely, the temperature (A), the corn cob content (B), corn cob/H2SO4 mass ratio (C) and the reaction time (D) on the yields of EL (Y1), diethyl ether (DEE) (Y2) and solid residue (Y3) were explored. Using multiple regression analysis, the experimental results were fitted to quadratic polynomial models. The predicted yields based on the fitted models were well within the experimental uncertainties. Optimum conditions for maximising the EL yield were found to be 176 °C, 14.6 wt. %, 21:1 and 6.75 h for A to D, respectively. A moderate-to-high EL yield (29.2%) from corn cob was achieved in optimised conditions, a result comparable to those obtained from model C6 carbohydrate compounds. Side products were also produced, including diethyl ether, furfural, levulinic acid, 5-hydroxymethyl furfural, ethyl acetate, ethyl formate and water. Total unknown losses of only 5.69% were reported after material balancing. The results suggest that lignocellulosic waste such as corn cob can be used as a potential feedstock for the production of ethyl levulinate by direct acid-catalysed ethanolysis, but that the treatment of side products will need to be considered.
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Vijayasankar, A., Sharadha Kumar, and M. A. Gopalan. "A Peer Search on Integer Solutions to Quadratic Diophantine Equation with Three Unknowns (Equation)." Academic Journal of Applied Mathematical Sciences, no. 68 (September 10, 2020): 166–71. http://dx.doi.org/10.32861/ajams.68.166.171.
Full textAbstract:
The non- homogeneous ternary quadratic diophantine (Equation) is analyzed for its patterns of non-zero distinct integral solutions. Various interesting relations between the solutions and special numbers namely polygonal, Pronic and Gnomonic numbers are exhibited.