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Academic literature on the topic 'Quadratic with four unknowns'

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Published: 3 June 2025

Last updated: 23 June 2025

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Journal articles on the topic "Quadratic with four unknowns"

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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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Dissertations / Theses on the topic "Quadratic with four unknowns"

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Öhman, Johan. "Four-dimensional Quadratic Division Algebras." Thesis, Uppsala University, Department of Mathematics, 2000. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-122597.

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QI, G., G. Chen, Wyk MA van, Wyk BJ van, and Y. Zhang. "A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system." Elsevier, 2007. http://encore.tut.ac.za/iii/cpro/DigitalItemViewPage.external?sp=1000759.

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This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic attractors and a pair of diagonal double-wing chaotic attractors. More importantly, the system can generate a fourwing chaotic attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic attractors, major and minor diagonal double-wing chaotic attractors, and a four-wing chaotic attractor. Poincare´-map analysis shows that the system has extremely rich dynamics. The physical existence of the fourwing chaotic attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.

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Cang, S., G. Qi, and Z. Chen. "A four-wing hyper-chaotic attractor and transient chaos generated from a new 4-D quadratic autonomous system." Springer Science, 2009. http://encore.tut.ac.za/iii/cpro/DigitalItemViewPage.external?sp=1001264.

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Abstract This paper presents a new four-dimensional (4-D) smooth quadratic autonomous chaotic system, which can present periodic orbit, chaos, and hyperchaos under the conditions on different parameters. Importantly, the system can generate a four-wing hyper-chaotic attractor and a pair of coexistent doublewing hyper-chaotic attractors with two symmetrical initial conditions. Furthermore, a four-wing transient chaos occurs in the system. The dynamic analysis approach- in the paper involves time series, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents, to investigate some basic dynamical behaviors of the proposed 4-D system.

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Henriksson, Johan. "Molecular Quadratic Response Properties with Inclusion of Relativity." Doctoral thesis, Linköping : Department of Physics, Chemistry and Biology, Linköping University, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11035.

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Gouzien, Élie. "Optique quantique multimode pour le traitement de l'information quantique." Thesis, Université Côte d'Azur (ComUE), 2019. http://www.theses.fr/2019AZUR4110.

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Cette thèse étudie l’optique quantique multimode, aussi bien du point de vue de la génération que celui de la détection. Elle s’articule autour de trois volets. Nous étudions la génération de lumière comprimée multimode dans une cavité. Pour cela nous considérons la forme la plus générale de hamiltonien quadratique, permettant entre autres de décrire l’utilisation de plusieurs pompes dans un matériau effectuant du mélange à quatre ondes. Une approche combinant fonctions de Green et décompositions de matrices symplectiques est décrite. Cette théorie générique est appliquée à des cas particuliers. Dans un premier temps, des exemples en basse dimension sont donnés. Ensuite, une configuration d’oscillateur paramétrique optique pompé de manière synchrone (SPOPO) est décrite et étudiée ; les résultats obtenus montrent que ce système a un comportement très différent de celui du SPOPO utilisant une non-linéarité d’ordre 2. Ces travaux ouvrent la voie à la réalisation de peignes de fréquences quantiques avec des micro-résonateurs en anneau gravés sur silicium. Un autre problème examiné est celui de prendre en compte l’information temporelle obtenue lors du clic d’un détecteur de photon unique. Pour cela nous utilisons un formalisme multimodal temporel afin d’expliciter les opérateurs décrivant la mesure. Les principaux défauts des détecteurs réels, dont la gigue temporelle, l’efficacité finie et les coups d’obscurité sont pris en compte. L’utilisation des opérateurs est illustrée par la description d’expériences usuelles de l’optique quantique. Enfin, on montre que la lecture du temps de clic du détecteur permet d’améliorer la qualité de l’état généré par une source de photons annoncés. En troisième partie nous présentons un schéma de génération d’intrication hybride entre variables continues et discrètes, pour laquelle la partie discrète est encodée temporellement. Ce schéma est analysé en détail vis-à-vis de sa résistance aux imperfections expérimentales<br>This thesis studies multimode quantum optics, from generation to detection of light. It focuses on three main parts. Multimode squeezed states generation within cavity is studied. More specifically, we take into account general quadratic Hamiltonian, which allows describing experiments involving arbitrary number of modes and pumps within a medium performing four-wave mixing. We describe a generic approach combining Green functions and symplectic matrix decomposition. This general theory is illustrated on specific cases. First, low-dimensional examples are given. Then, a synchronously pumped optical parametric oscillator (SPOPO) is described and studied; it shows a very distinct behavior from that of the SPOPO using second order non-linearity. This work opens way to the realization of quantum frequency combs with ring micro-resonators engraved on silicon. Single-photon detectors are described taking into account temporal degrees of freedom. We give positive-valued measurement operators describing such detectors including realistic imperfections such as timing-jitter, finite efficiency and dark counts. Use of those operators is illustrated on common quantum optics experiments. Finally, we show how time-resolved measurement allows improving the quality of state generated by single-photon heralded source. In the third part we propose a protocol for generating a hybrid state entangling continuous and discrete variables parts, for which the discrete part is time-bin encoded. This scheme is aanalysed in detail with respect to its resilience to experimental imperfections

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He, Guo-Zhen, and 何國禎. "Optimal Linear Quadratic Estimator and Tracker Designs for Linear Systems with Unknown Disturbances." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/43941320731659090214.

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碩士<br>國立中央大學<br>電機工程學系<br>104<br>Improved robust observer-based servo designs are proposed in this thesis for the linear systems subject to unknown disturbances. First, the poles of the error dynamic system of the state observer integrated with the unknown input estimator for the continuous-time minimum phase system subject to unknown input disturbance (UID) are optimally assigned to lie to the left of some vertical line in the s-plane with prescribed degree of relative stability. Similar merit has been also applied to the servo design. Consequently, restrictions on the estimation of UID with low frequencies and servo control for slow time-varying command inputs presented in literature have been released to the cases for the UID with high frequencies and drastic time-varying command inputs, so that a more wide range unknown input estimations and servo designs can be achieved. In contrast with the above-mentioned merits, the proposed approach for the continuous-time systems has been also extended to the discrete-time version for the minimum phase and/or non-minimum phase systems. Especially, the new current-output observer/UID esitmator-based servo design for the discrete-time system with an unknown disturbance is proposed. Furthermore, based on the equivalent input disturbance (EID) principle, the proposed approaches are applicable to the class of mismatched input disturbances.

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PAN, CHIEN YUAN, and 潘建源. "Chinese Four Tone Recognition Based on Karhunen-Loeve Transform and Quadratic Classifier." Thesis, 1996. http://ndltd.ncl.edu.tw/handle/45764390257821211947.

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Wu, Pei-Lun, and 吳培綸. "Design of CMOS Quadratic Polynomial Circuit Using Best Fit Method and Four-Segment Method." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/95497873543099295336.

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碩士<br>中華大學<br>電機工程學系碩士班<br>100<br>In this thesis, we design CMOS quadratic polynomial circuit using best fit method and four-segment method. The propose of this thesis is based on CMOS current-mode quadratic function circuits, its W/L ratio and construction could be adjusted by the relative error that the users needed. First we explain how to implement CMOS current-mode quadratic circuits and design the proposed circuit in the way of multiple corrections. We use the best fit CMOS quadratic polynomial circuit to realization exponential function. The imitative result based on the range of -48μA~22μA, and the relative error within±3%. The output dynamic range of 14.1dB, and linearity error less than ±0.3dB. Four-segment CMOS quadratic polynomial circuit utilize Taylor to fitting sigmoid function. The circuit is designed by four sets of quadratic polynomial circuit and three sets of segmentation control circuit. The imitative result based on the range of -277μA~480μA, and the relative error within±3%. The PSRR are 34.3. The imitative tool is HSPICE and MATLAB. Circuit layout software used the CADANCE which provided with CIC and TSMC 0.35μm process.

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Chih-HaoLee and 李至浩. "A New Linear Quadratic Observer and Tracker under Input Constraint for the Unknown System with a Direct Feedthrough Term: Digital Redesign Approach." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/n57z7d.

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碩士<br>國立成功大學<br>電機工程學系碩博士班<br>101<br>This thesis develops a new linear quadratic observer and tracker for the linear unknown system with a direct transmission term from input to output. The off-line observer/Kalman filter identification (OKID) method is used to determine the sampled-data linear model with a direct feedthrough term. Based on this model, a high-gain optimal linear quadratic analogue observer-based tracker are proposed, so that it can effectively induce a high quality performance on the state estimation and the trajectory tracking design. Besides, the prediction-based digital redesign method is utilized to obtain a relatively low-gain and implementable observer and digital tracker from the theoretically well-designed high-gain analogue observer and tracker for the linear system with a direct transmission term from input to output. To reduce the magnitude of control input, which is caused by the high-gain property to fit the requirement of the input constraint, the modified linear quadratic analogue tracker (LQAT) is proposed. Thus, the control input can be compressed effectively without losing the original high performance of tracking much.

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Liang-ChangChen and 陳良政. "New Quadratic Tracker under Input Constraint for the Unknown System with a Direct Transmission Term from Input to Output: An Alternative Digital Redesign Approach." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/46d55g.

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碩士<br>國立成功大學<br>電機工程學系碩博士班<br>101<br>This thesis proposes a new quadratic digital tracker under input constraint to efficiently restrict control input for the unknown system with a direct transmission term from input to output, without losing the good tracking performance as possible. First, the observer/Kalman filter identification (OKID) is utilized to identify the unknown linear/nonlinear system with a feed-through term into the equivalent mathematical model containing a feed-through term. The equivalent mathematical model is used as an analytic and designed tool for the controller and observer. The linear analogue quadratic performance index is modified to contain the term of input constraint. By discretizing the linear analogue quadratic performance index with input constraint into an equivalent discrete one directly, we propose a new quadratic suboptimal digital tracker under input constraint for the unknown system with a feed-through term. This design procedure is also called an alternative digital redesign approach. Further, the analog and digital observers are respectively proposed for the system when the system states are immeasurable. Illustrative examples demonstrate the effectiveness of the proposed design.

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Books on the topic "Quadratic with four unknowns"

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The jade mirror of the four unknowns by Zhū Shìjié: An early fourteenth century mathematics manual for teaching the derivation of systems of polynomial equations in up to four unknowns : a study. Mingming Bookroom, 2007.

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Cheng, Russell. Non-Standard Problems: Some Examples. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198505044.003.0002.

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This chapter provides motivation for the rest of the book by giving a selection of simple examples showing how non-standard behaviour can occur. The well-known maximum likelihood estimator is used throughout this book to estimate an unknown vector of parameters. Its behaviour is standard if the log-likelihood is a concave quadratic function with the maximum in the neighbourhood of the true parameter value, but is otherwise non-standard. Examples of non-standard situations given in this chapter include the true parameter value not being an internal point of the parameter space, but being on a fixed boundary that may not even be finite, or where the mathematical form of the log-likelihood is different with non-estimable indeterminate parameters, or where the true model is an embedded model. Other examples given include where the log-likelihood is unbounded at a finite parameter point, is discontinuous, or is no longer quadratic.

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Book chapters on the topic "Quadratic with four unknowns"

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Wright, Steve. "Four Interesting Applications of Quadratic Reciprocity." In Lecture Notes in Mathematics. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-45955-4_4.

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Goiński, Adam, Tomasz Zawadzki, and Sławomir Nikiel. "Automatic Shape Generation Based on Quadratic Four-Dimensional Fractals." In Computer Vision and Graphics. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-33564-8_9.

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Artés, Joan C., Jaume Llibre, Dana Schlomiuk, and Nicolae Vulpe. "Quadratic systems with finite singularities of total multiplicity four." In Geometric Configurations of Singularities of Planar Polynomial Differential Systems. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-50570-7_11.

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Bragg, Nate F. F., Jeffrey S. Foster, Cody Roux, and Armando Solar-Lezama. "Program Sketching by Automatically Generating Mocks from Tests." In Computer Aided Verification. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-81685-8_38.

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AbstractSketch is a popular program synthesis tool that solves for unknowns in a sketch or partial program. However, while Sketch is powerful, it does not directly support modular synthesis of dependencies, potentially limiting scalability. In this paper, we introduce Sketcham, a new technique that modularizes a regular sketch by automatically generating mocks—functions that approximate the behavior of complete implementations—from the sketch’s test suite. For example, if the function f originally calls g, Sketcham creates a mock $$g_m$$ g m from g’s tests and augments the sketch with a version of f that calls $$g_m$$ g m . This change allows the unknowns in f and g to be solved separately, enabling modular synthesis with no extra work from the Sketch user. We evaluated Sketcham on ten benchmarks, performing enough runs to show at a 95% confidence level that Sketcham improves median synthesis performance on six of our ten benchmarks by a factor of up to 5$$\times $$ × compared to plain Sketch, including one benchmark that times out on Sketch, while exhibiting similar performance on the remaining four. Our results show that Sketcham can achieve modular synthesis by automatically generating mocks from tests.

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Chen, Yan, Wenqiang Chen, Xingmin Wei, and Fuquan Zhao. "Linear Quadratic Gaussian Optimal Control Strategy for Four-Wheel Steering Vehicle." In Lecture Notes in Electrical Engineering. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-33835-9_59.

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Kozera, Ryszard, Lyle Noakes, and Piotr Szmielew. "Sharpness in Trajectory Estimation for Planar Four-points Piecewise-Quadratic Interpolation." In Computer Algebra in Scientific Computing. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-10515-4_20.

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Rao, Somala Arjuna, and H. R. Ramesh. "Linear Quadratic Regulator Controlled Four-Bus Closed Loop Micro Grid System." In Emerging Technologies & Applications in Electrical Engineering. CRC Press, 2024. http://dx.doi.org/10.1201/9781003505181-5.

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Misevicius, Alfonsas, and Evaldas Guogis. "Computational Study of Four Genetic Algorithm Variants for Solving the Quadratic Assignment Problem." In Communications in Computer and Information Science. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-33308-8_3.

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Vaidyanathan, Sundarapandian, and Ahmad Taher Azar. "A Novel 4-D Four-Wing Chaotic System with Four Quadratic Nonlinearities and Its Synchronization via Adaptive Control Method." In Advances in Chaos Theory and Intelligent Control. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30340-6_9.

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Pillonetto, Gianluigi, Tianshi Chen, Alessandro Chiuso, Giuseppe De Nicolao, and Lennart Ljung. "Regularization in Reproducing Kernel Hilbert Spaces." In Regularized System Identification. Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-95860-2_6.

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AbstractMethods for obtaining a function g in a relationship $$y=g(x)$$ y = g ( x ) from observed samples of y and x are the building blocks for black-box estimation. The classical parametric approach discussed in the previous chapters uses a function model that depends on a finite-dimensional vector, like, e.g., a polynomial model. We have seen that an important issue is the model order choice. This chapter describes some regularization approaches which permit to reconcile flexibility of the model class with well-posedness of the solution exploiting an alternative paradigm to traditional parametric estimation. Instead of constraining the unknown function to a specific parametric structure, the function will be searched over a possibly infinite-dimensional functional space. Overfitting and ill-posedness are circumvented by using reproducing kernel Hilbert spaces as hypothesis spaces and related norms as regularizers. Such kernel-based approaches thus permit to cast all the regularized estimators based on quadratic penalties encountered in the previous chapters as special cases of a more general theory.

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Conference papers on the topic "Quadratic with four unknowns"

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Tan, Lei, and Xinmin Song. "Quadratic Estimation for Discrete Time System with Packet Dropping and Unknown Input." In 2024 14th International Conference on Information Technology in Medicine and Education (ITME). IEEE, 2024. https://doi.org/10.1109/itme63426.2024.00153.

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Chai, Shijie, Yueheng Ding, Yuchen Dai, Wei Hua, and Wenfei Yu. "Quadratic Programming-based Coordinated Torque Vectoring Allocation Strategy for Four-Wheel Independently Driven Electric Vehicles." In 2025 4th International Conference on Green Energy and Power Systems (ICGEPS). IEEE, 2025. https://doi.org/10.1109/icgeps65133.2025.11034606.

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Sun, Yufei, Zhitong Chen, Qiwei Zhang, Xiaoya Fan, and Yanzhao Ma. "A Hybrid Four-Switch Inductor-First Quadratic Step-Down Converter With Internal Ripple Compensated COT Control." In 2024 IEEE Asian Solid-State Circuits Conference (A-SSCC). IEEE, 2024. https://doi.org/10.1109/a-sscc60305.2024.10848668.

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Dattaroy, Arkabrata, and Avik Bhattacharya. "A Novel Three-Output Port Isolated DC-DC Converter with Four Switches Providing High Gain Quadratic Non-Inverting Buck-Boost or Derived Boost Characteristics Along with Continuous Input Battery Current Operation." In 2024 IEEE International Conference on Power Electronics, Drives and Energy Systems (PEDES). IEEE, 2024. https://doi.org/10.1109/pedes61459.2024.10961088.

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Adiga, Shreemathi. "On bi-quadratic equation with four unknowns 7xy+3z2 = 3w4." In PROCEEDINGS OF INTERNATIONAL CONFERENCE ON ADVANCES IN MATERIALS RESEARCH (ICAMR - 2019). AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0016866.

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Tahmasebi, Farhad. "Closed Form Dimensional Synthesis of a High-Accuracy Four-Position Tilt Mechanism." In ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49294.

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The conceptual and closed-form dimensional synthesis of a high-accuracy four-position tilt mechanism is discussed in detail. The mechanism, which consists of a customized five-bar linkage, is capable of accurately rotating an optical bench, which supports an object such as a space instrument, or another type of similar platform to four required and discrete angular postures. The mechanism is driven by two stepper motors. Due to the special characteristics of the five-bar linkage, even relatively large stepper motor errors produce very minor errors in the four desired angular postures of the optical bench. The dimensional synthesis of the mechanism involves solving a system of four non-linear equations in four unknowns. A methodology is introduced for reducing this system of non-linear equations to a closed-form quadratic equation in one unknown. A numerical example of the closed-form dimensional synthesis methodology is also presented.

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Huang, Chintien, Chenning Hung, and Kuenming Tien. "Numerical Solutions of Polynomial Equations for the Eight-Position Synthesis of the Cylindrical-Spherical Dyad." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70587.

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This paper investigates the numerical solutions of equations for the eight-position rigid-body guidance of the cylindrical-spherical (C-S) dyad. We seek to determine the number of finite solutions by using the numerical continuation method. We derive the design equations using the geometric constraints of the C-S dyad and obtain seven quartic polynomial equations and one quadratic equation. We then solve the system of equations by using the software package Bertini. After examining various specifications, including those with random complex numbers, we conclude that there are 804 finite solutions of the C-S dyad for guiding a body through eight prescribed positions. When designing spatial dyads for rigid-body guidance, the C-S dyad is one of the four dyads that result in systems of equal numbers of equations and unknowns if the maximum number of allowable positions is specified. The numbers of finite solutions in the syntheses of the other three dyads have been obtained previously, and this paper provides the computational kinematic result of the last unsolved problem, the eight-position synthesis of the C-S dyad.

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Liu, Zongmin, Lifu Liang, Tao Fan, and Haiyan Song. "Gurtin-Type Quasi-Variational Principle for Large Deformation Elasto-Dynamics Based on Base Forces Theory." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-12488.

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Considering that “gap function” is a nuisance term that contains both the stress and the quadratic function of displacement, it is difficult to establish Gurtin-type variational principles for large deformation elasto-dynamics. Luo En, Finlayson B. A. and Scriven L. established Gurtin-type variational principles by restricted variational idea. Based on base forces theory framework, the basic equations for large deformation elasto-dynamics are defined. Using the concept of base forces, Gurtin-type quasi-variational principle for large deformation elasto-dynamics can be established by variational integral method directly. When the base forces and the conjugate variable “displacement gradients” are taken as the basic unknowns, there is no “gap function” in the functional of Gurtin-type quasi-variational principle. Then quasi-stationary value conditions of Gurtin-type quasi-variational principle are derived. One of the quasi-stationary value conditions is the famous wave equation. Finally, some correlative problems are discussed.

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Mummolo, Carlotta, Luigi Mangialardi, and Joo H. Kim. "Concurrent Contact Planning and Trajectory Optimization in One Step Walking Motion." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-47745.

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Bipeds’ trajectories and control during walking are closely coupled with the contact force distribution in time and space as an indeterminate problem. Therefore, generating the motion of redundant bipeds in presence of unilateral contact is usually formulated as a nonlinear constrained optimization problem. The optimal walking motion must be solved in terms of trajectories, control, contact status (i.e., when, where, and whether a foot is in contact), and contact response (i.e., ground reaction forces). The solution for this problem requires predictive methods within the general optimal motion planning framework. However, there is a lack of fully predictive methods that can concurrently solve for all the above mentioned unknowns. This represents an important challenge in the simulation, design, analysis, and control of general robotic systems. A novel approach for the optimal motion planning of multibody systems with contacts is developed, based on a Sequential Quadratic Programming (SQP) algorithm for Nonlinear Programming (NLP). The complete formulation is presented and demonstrated with numerical experiments on a simple planar biped with the assigned task of one complete step motion in forward progression.

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Kwon, Hyun Jung, Yujiang Xiang, Salam Rahmatalla, R. Timothy Marler, Karim Abdel-Malek, and Jasbir S. Arora. "Optimization-Based Digital Human Dynamics: Santos™ Walking Backwards." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35616.

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An objective of this study is to simulate the backward walking motion of a full-body digital human model. The model consists of 55 degree of freedom – 6 degrees of freedom for global translation and rotation and 49 degrees of freedom representing the kinematics of the entire body. The resultant action of all the muscles at a joint is represented by the torque for each degree of freedom. The torques and angles at a joint are treated as unknowns in the optimization problem. The B-spline interpolation is used to represent the time histories of the joint angles and the well-established robotics formulation of the Denavit-Hartenberg method is used for kinematics analysis of the mechanical system. The recursive Lagrangian formulation is used to develop the equations of motion, and was chosen because of its known computational efficiency. The backwards walking problem is formulated as a nonlinear optimization problem. The control points of the B-splines for the joint angle profiles are treated as the design variables. For the performance measure, total dynamic effort that is represented as the integral of the sum of the squares of all the joint torques is minimized using a sequential quadratic programming algorithm. The solution is simulated in the Santos™ environment. Results of the optimization problem are the torque and joint angle profiles. The torques at the key joints and the ground reaction forces are compared to those for the forward walk in order to study the differences between the two walking patterns. Simulation results are approximately validated with the experimental data which is motion captured in the VSR Lab at the University of Iowa.

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Reports on the topic "Quadratic with four unknowns"

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de Dieu Niyigena, Jean, Innocent Ngaruye, Joseph Nzabanita, and Martin Singull. Approximation of misclassification probabilities using quadratic classifier for repeated measurements with known covariance matrices. Linköping University Electronic Press, 2024. http://dx.doi.org/10.3384/lith-mat-r-2024-02.

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Quadratic discriminant analysis is a well-established supervised classification method, which extends the linear the linear discriminant analysis by relaxing the assumption of equal variances across classes. In this study, quadratic discriminant analysis is used to develop a quadratic classification rule based on repeated measurements. We employ a bilinear regression model to assign new observations to predefined populations and approximate the misclassification probability. Through weighted estimators, we estimate unknown mean parameters and derive moments of the quadratic classifier. We then conduct numerical simulations to compare misclassification probabilities using true and estimated mean parameters, as well as probabilities computed through simulation. Our findings suggest that as the distance between groups widens, the misclassification probability curve decreases, indicating that classifying observations is easier in widely separated groups compared to closely clustered ones.

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Casella, Alessandra, and Luis Sanchez. Storable Votes and Quadratic Voting. An Experiment on Four California Propositions. National Bureau of Economic Research, 2019. http://dx.doi.org/10.3386/w25510.

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Riveros, Guillermo, Felipe Acosta, Reena Patel, and Wayne Hodo. Computational mechanics of the paddlefish rostrum. Engineer Research and Development Center (U.S.), 2021. http://dx.doi.org/10.21079/11681/41860.

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Purpose – The rostrum of a paddlefish provides hydrodynamic stability during feeding process in addition to detect the food using receptors that are randomly distributed in the rostrum. The exterior tissue of the rostrum covers the cartilage that surrounds the bones forming interlocking star shaped bones. Design/methodology/approach – The aim of this work is to assess the mechanical behavior of four finite element models varying the type of formulation as follows: linear-reduced integration, linear-full integration, quadratic-reduced integration and quadratic-full integration. Also presented is the load transfer mechanisms of the bone structure of the rostrum. Findings – Conclusions are based on comparison among the four models. There is no significant difference between integration orders for similar type of elements. Quadratic-reduced integration formulation resulted in lower structural stiffness compared with linear formulation as seen by higher displacements and stresses than using linearly formulated elements. It is concluded that second-order elements with reduced integration and can model accurately stress concentrations and distributions without over stiffening their general response. Originality/value – The use of advanced computational mechanics techniques to analyze the complex geometry and components of the paddlefish rostrum provides a viable avenue to gain fundamental understanding of the proper finite element formulation needed to successfully obtain the system behavior and hot spot locations.

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A simple four-moment approximation to the distribution of a positive definite quadratic form, with applications to testing. Cemmap, 2022. http://dx.doi.org/10.47004/wp.cem.2022.0222.

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