Journal articles: 'Pole assignment Mathematical models'

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Lam, J., and H. K. Tam. "Pole assignment with minimum eigenvalue differential sensitivity." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 211, no. 1 (February 1, 1997): 63–74. http://dx.doi.org/10.1243/0959651971539696.

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This paper introduces a set of mathematical formulae for calculating the eigenvalue differential sensitivities of the closed-loop state matrix with respect to the open-loop state matrix, input matrix and state feedback matrix. It provides a computational procedure for a robust pole assignment problem. The algorithm is based on a gradient flow minimization of a differentiate objective function which measures the sensitivity for all closed-loop poles. Two numerical examples are employed to illustrate the technique. Comparisons to other existing methods are made as well.

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Araki, Mituhiko, Ken’ichi Fukumitsu, and Tomomichi Hagiwara. "Simultaneous Stabilization and Pole Assignment by Two Level Controllers Consisting of a Gain Feedback and a Multirate Input Controller." Journal of Dynamic Systems, Measurement, and Control 121, no. 2 (June 1, 1999): 302–5. http://dx.doi.org/10.1115/1.2802470.

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A method of simultaneous pole assignment of a finite number of plant models is studied. Extending the idea of multi-structured multirate sampled-data controllers proposed in our previous work, this paper proposes a new method where a continuous-time gain feedback controller instead of a multirate sampled-data controller is used as the “internal structure.” By the proposed method, the plants precompensated by the internal structure can generally be made to satisfy an existing sufficient condition for simultaneous pole-assignability, and hence the class of the plant models for which simultaneous pole assignment can be attained becomes wider than that by the previous results. In particular, simultaneous pole assignment becomes possible for any finite number of plant models, provided that each model is controllable and observable.

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Halpern, M. E. "Modified pole-assignment controller for plant models with exact or near pole-zero cancellation." IEE Proceedings D Control Theory and Applications 135, no. 3 (1988): 189. http://dx.doi.org/10.1049/ip-d.1988.0026.

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Mclnnis, B. C., Deng L-Z, and R. Vogt. "Adaptive Pole Assignment Control of Blood Pressures using Bilinear Models." IFAC Proceedings Volumes 18, no. 5 (July 1985): 1209–12. http://dx.doi.org/10.1016/s1474-6670(17)60727-4.

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Tam, Hei Ka, and James Lam. "An Optimal Differential Sensitivity Approach to Pole Assignment for Structurally Perturbed Systems." Journal of Dynamic Systems, Measurement, and Control 121, no. 3 (September 1, 1999): 410–17. http://dx.doi.org/10.1115/1.2802489.

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This paper introduces a mathematical formula for calculating the eigenvalue differential sensitivities of the closed-loop state matrix with respect to a disturbance matrix when the closed-loop system is subjected to an additive structured perturbation. The robust pole assignment problem is then formulated as an optimization task in which the differential sensitivities of the closed-loop poles are minimized. The case of “multi-perturbations” can be considered naturally in our problem formulation. The gradient of the objective function to be minimized is then derived which permits the minimization to be proceeded effectively. Numerical examples are then employed to illustrate the design technique.

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Vikharev, D. Yu, and N. A. Rodin. "Model of implicit pole electric machine based on mathematical formulation of magnetic field in air gap." Vestnik IGEU, no. 6 (December 28, 2021): 27–37. http://dx.doi.org/10.17588/2072-2672.2021.6.027-037.

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When modeling transient processes in electric power systems, the issue of designing reliable models of electrical machines is of great interest. The most difficult task is to design the models of synchronous and asynchronous machines. The purpose of the project is to develop mathematical models of synchronous and asynchronous machines based on a generalized model of an implicit-pole machine that considers the change of rotor speed, geometric shapes, and the type of rotor and stator windings. Methods of mathematical modeling of electromagnetic fields in the air gap based on Maxwell equations and methods of the theory of electrical circuits are applied. A system of nonlinear differential equations is considered as a mathematical model of a generalized implicit-pole machine. The key assumption made is the high magnetic permeability of the stator and rotor cores. Technical data of real electric machines have been used for verification. Mathematical models of an implicit pole synchronous machine and an asynchronous machine with a phase rotor have been developed. These models can be used to analyze transient processes in the aggregate without dividing them into electromagnetic and electromechanical ones. The novelty of the models is the fact of non-sinusoidal distribution of the magnetic field in the air gap, various types of multiphase AC windings and rejecting the principle of constant rotation frequency at the time step of the simulation. The proposed mathematical models can be used to solve design problems, to analyze the modes of electric power systems, the operation of relay protection and automation of electric power facilities, and emergency automation. In the future, the equations that consider the influence of the damper circuits and the peculiarities of the windings will be added to the developed mathematical models. The purpose of further research is to develop models of salient-pole synchronous machines and asynchronous machines with a squirrel-cage rotor.

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Obredor-Baldovino, Thalía, Evis Barcasnegras-Moreno, Nohora Mercado-Caruso, Katherinne Salas-Navarro, and Shib Sankar Sana. "Coverage Reduction: A Mathematical Model." Journal of Advanced Manufacturing Systems 17, no. 03 (July 30, 2018): 317–31. http://dx.doi.org/10.1142/s0219686718500191.

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This paper deals with a mathematical model for reduction of the lack of coverage (LC) involving multiple coverage in presence of partial covering. The model proposes a new structure of assignment of facilities in a facility location system to cover in greater proportion of the demand territory, avoiding assignment of several facilities in the same space of the territory. A comparison between the engendered solution and its representation is carried out through known indicators to measure the improvement of the solution. The results of our proposed model are contrast and better compared to defined referred models in order to evaluate the reduction of LC.

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V. R. Sheelavant, R. L. Chakrasali, Manjunath B. Ranadev,. "Modeling of Three-Phase Written-Pole Motor and Performance Predetermination using Numerical Methods." Mathematical Statistician and Engineering Applications 71, no. 3s (July 19, 2022): 08–18. http://dx.doi.org/10.17762/msea.v71i3s.3.

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Modeling has attracted increasing attention from researchers. A physical system transformation to a mathematical base is important to analyse the behaviour of the system. The solution of various engineering problems depends on appropriate mathematical models. Differential equations are the basic tool for the mathematical modeling of electric machines. A model that describes the machine performance is used to analyse the dynamic behaviour of a 3-phase written-pole motor. Mathematical modeling of a 3-phase written-pole motor will help us to do the numerical analysis. It provides insight and useful guidance for originating the applications. A method for studying the behaviour of a 3-phase written-pole motor under steady state condition is described. The numerical tools such as Gauss-Seidel method and Successive Over-Relaxation method are used. This study forms the foundation for further research and development.

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Al-Yakoob, Salem M., and Hanif D. Sherali. "Mathematical programming models and algorithms for a class–faculty assignment problem." European Journal of Operational Research 173, no. 2 (September 2006): 488–507. http://dx.doi.org/10.1016/j.ejor.2005.01.052.

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Batiha, Khaled. "Development of Mathematical Models to Estimate Optimum Capacity Assignment Network Connections." International Journal of Computer Applications 129, no. 16 (November 17, 2015): 12–14. http://dx.doi.org/10.5120/ijca2015906964.

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Santos, Daryl. "A Mathematical Model for In-Person Office Assignment During COVID-19." Industrial and Systems Engineering Review 9, no. 1 (April 9, 2021): 68–74. http://dx.doi.org/10.37266/iser.2021v9i1.pp68-74.

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The global pandemic that is COVID-19 has altered our world as we know it. Educational systems have been seriously affected, businesses – whether “essential” or not – have been severely stressed, how people socialize has, perhaps forever, been changed, and telecommuting is the new norm. The purpose of this paper is to discuss a situation wherein a company was having difficulty in scheduling their in-person office staffing in a move to allow rotational schedules for their employees to decongest their office when an entirely telecommuting operation was not possible. Due to the different types of employees, their various work-related constraints, and the need to minimize the number of bodies in the office while addressing minimum coverage necessary to perform the company’s daily activities, a quick solution to their scheduling/assignment needs was not obvious or trivially obtained. As such, mathematical models, specifically, integer programming assignment models, were developed and ultimately solved using the Python/Gurobi solver to address their scheduling needs. This paper will describe the constraints faced by the company and the models developed to solve their tricky assignment problem.

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Anosova, Natalia, Alexandra Filippova, Sergej Krylov, and Vadim Perepelkin. "Prediction of Earth polar motion in short time interval." MATEC Web of Conferences 362 (2022): 01001. http://dx.doi.org/10.1051/matecconf/202236201001.

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The accuracy characteristics of mathematical models of the Earth pole motion in a short time intervals have been investigated. The modeling was carried out using two approaches: a numerical-analytical model of the Earth’s rotational motion relative to the center of mass, based on the weighted least squares method and a neural network model for predicting the Earth pole motion. A comparison is made between the standard deviations of short-term forecasts of the Earth pole motion for 1–2 days according to the developed models and the standard deviations of forecasts published by the International Earth Rotation Service in Bulletin A. The models under consideration did not use data smoothing. The model based on neural network approach showed a higher accuracy of short-term forecasting.

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González, Gilberto A., Noe G. Barrera, Gerardo Ayala, J. Aaron Padilla, and David Z. Alvarado. "Quasi-Steady-State Models of Three Timescale Systems: A Bond Graph Approach." Mathematical Problems in Engineering 2019 (October 20, 2019): 1–20. http://dx.doi.org/10.1155/2019/9783740.

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Modelling in bond graph to obtain reduced models of systems with singular perturbations is applied. This singularly perturbed system is characterized by having three timescales, i.e., slow, medium, and fast dynamics. From a bond graph whose storage elements have an integral causality assignment (BGI), the mathematical model of the complete system can be determined. By assigning a derivative causality assignment to the storage elements for the fast dynamics and maintaining an integral causality assignment for the slow and medium dynamics on the bond graph, reduced models for the slow and medium dynamics are obtained. When a derivative causality to the storage elements for the fast and medium dynamics is assigned and an integral causality assignment to the slow dynamics is applied, the most reduced model is determined. Finally, the proposed methodology to the Ward Leonard system is applied.

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Dominguez, Victor H., Octavio Garcia-Salazar, Luis Amezquita-Brooks, Luis A. Reyes-Osorio, Carlos Santana-Delgado, and Erik G. Rojo-Rodriguez. "Micro Coaxial Drone: Flight Dynamics, Simulation and Ground Testing." Aerospace 9, no. 5 (May 1, 2022): 245. http://dx.doi.org/10.3390/aerospace9050245.

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This paper describes the conceptual design of a micro coaxial unmanned aerial vehicle (MCR UAV v3.0) based on its flight dynamics and a simple aerodynamic analysis using computational fluid dynamics (CFD). In addition, a simple linear control is proposed with the pole assignment technique. The methodology proposed in this paper involves a standardized path for designing the novel micro coaxial UAV. This begins by selecting the avionics to create a primary dimensional design for a later transient and stationary CFD analysis. In effect, the mathematical model is obtained using the Newton–Euler formulation and is linearized to obtain the dynamical requirements of the vehicle. The requirements allow us to design the control scheme with a linear control technique. This process is iterative and uses a combination of flight dynamics and CFD. The control technique is based on pole assignment, ensuring a specific phase condition is used in the controller gain for the stabilization of the proposed aerial vehicle. The control scheme is analyzed once the CFD analysis is correctly performed; in this sense, the methodology proposed in this paper is capable of converging as a result of the dimensional design. This design ensures a suitable vehicle performance according to the dynamical requirements. Thus, the micro coaxial UAV is completely designed based on its flight dynamics along with a CFD analysis, generating a robust methodology.

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Trunina, E. S., and A. V. Zotov. "Multi-pole extension of the elliptic models of interacting integrable tops." Theoretical and Mathematical Physics 209, no. 1 (October 2021): 1331–56. http://dx.doi.org/10.1134/s0040577921100020.

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Goher, Khaled M. "Mathematical Modelling and Control of a Two-Wheeled PUMA-LikeVehicle." Mechanical Engineering Research 6, no. 2 (November 22, 2016): 11. http://dx.doi.org/10.5539/mer.v6n2p11.

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<p class="1Body">This paper presents mathematical modelling and control of a two-wheeled single-seat vehicle. The design of the vehicle is inspired by the Personal Urban Mobility and Accessibility (PUMA) vehicle developed by General Motors® in collaboration with Segway®. The body of the vehicle is designed to have two main parts. The vehicle is activated using three motors; a linear motor to activate the upper part in a sliding mode and two DC motors activating the vehicle while moving forward/backward and/or manoeuvring. Two stages proportional-integral-derivative (PID) control schemes are designed and implemented on the system models. The state space model of the vehicle is derived from the linearized equations. Controller based on the Linear Quadratic Regulator (LQR) and the pole placement techniques are developed and implemented. Further investigation of the robustness of the developed LQR and the pole placement techniques is emphasized through various experiments using an applied impact load on the vehicle.</p>

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Golovkov, Alexander A., and Vladimir A. Golovkov. "Algorithm for parametric synthesis of cascade-connected matching mixed quadripoles according to the criterion of ensuring stationary generation mode." Physics of Wave Processes and Radio Systems 25, no. 1 (March 29, 2022): 45–54. http://dx.doi.org/10.18469/1810-3189.2022.25.1.45-54.

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Introduction: the analysis of the well-known literature shows that the use of various types of four-pole devices (reactive, resistive, complex) allows to increase the area of physical feasibility of the stationary generation mode. The purpose of the work: increasing the area of physical feasibility of the stationary mode of generation by optimizing the parameters of the matching mixed four-poles. One part of such four-pole devices consists only of resistive elements, and the second part consists only of reactive elements. Materials and methods: four-pole theory, matrix algebra, decomposition method, method of synthesis of microwave control devices, immitance stability criterion. Results: mathematical models of matching mixed four-pole devices are obtained in the form of relationships between the elements of their transmission matrix and the dependencies of the resistances of their two-pole devices on the frequency, optimal according to the criterion of ensuring a stationary generation mode. Conclusion: a comparative analysis of the theoretical results (frequency response and frequency response of the autogenerator in the amplification mode) obtained by mathematical modeling in the MathCad system, and the experimental results obtained by circuit modeling in the OrCad (in the amplification mode) and MicroCap (in the generation mode) systems shows their satisfactory coincidence. The frequency response and frequency response in the amplification mode are similar to the amplitude and phase spectra of the generated oscillations in the generation mode.

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Ridkokasha, I. "Confronting dual models of the strong interaction." Modern Physics Letters A 36, no. 05 (February 1, 2021): 2150031. http://dx.doi.org/10.1142/s0217732321500310.

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Studies of the mathematical properties of Regge-pole and dual amplitudes are important both for their applications in high energy phenomenology and in their generalizations to strings, superstrings, branes, and other theoretical developments. In this paper, we investigate the similarities and differences between two classes of dual amplitudes: one with Mandelstam analyticity (DAMA) and another one with logarithmic trajectories (Dual-log). By using quantum (q-) deformations, new features of Dual-log amplitude are unveiled, in particular those concerning its asymptotic behavior and the spectrum of resonances. The two classes of dual amplitudes are compared in various kinematic regions: at fixed transferred momenta asymptotic, fixed angle asymptotic, and in the resonance region.

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Bhoi, Sunil B., and Jayesh M. Dhodiya. "Multi-Objective Faculty Course Assignment Problem with Result and Feedback Based Uncertain Preferences." International Journal of Mathematical, Engineering and Management Sciences 6, no. 4 (July 18, 2021): 1055–75. http://dx.doi.org/10.33889/ijmems.2021.6.4.062.

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In this paper, a multi-objective faculty course allocation problem with result analysis and feedback analysis based on uncertain preferences mathematical model is presented. To deal with an uncertain model, three different ranking criteria are being used to develop: a) Expected value, b) Optimistic value, c) Dependent optimistic value criterion. These mathematical models are transformed into their corresponding deterministic forms using the basic concepts of uncertainty theory. The deterministic model of DOCM consists of fractional objectives which are converted into their linear form using Charnes and Cooper’s transformation. These deterministic formulations MOFCAP are converted into a single objective problem by using the fuzzy programming technique with linear and exponential membership functions. Further, the single objective problem for all the defined models is solved in the Lingo 18.0 software to derive the Pareto-optimal solution. The sensitivity of the models is also performed to examine the variation in the objective function due to the variation in parameters. Finally, a numerical example is given to exhibit the application and algorithm of the models.

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Ismayilova, Nigar. "CHALLENGES OF USING DIFFERENT MATHEMATICAL MODELS FOR LOAD BALANCING OPTIMIZATION IN MULTI-CORE COMPUTING SYSTEMS." Azerbaijan Journal of High Performance Computing 3, no. 2 (December 29, 2020): 190–95. http://dx.doi.org/10.32010/26166127.2020.3.2.190.195.

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This paper examines the role of applying different artificial intelligence techniques for the implementation of load balancing in the dynamic environment of distributed multi-core computing systems. Were investigated several methods to optimize the assignment process between computing nodes and executing tasks after the occurrence of a dynamic and interactive event, when traditional discrete load balancing techniques are ineffective.

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Elwakil, A. S., and A. M. Soliman. "Two Modified for Chaos Negative Impedance Converter Op Amp Oscillators with Symmetrical and Antisymmetrical Nonlinearities." International Journal of Bifurcation and Chaos 08, no. 06 (June 1998): 1335–46. http://dx.doi.org/10.1142/s0218127498001030.

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Two sinusoidal oscillator circuits, that employ an operational amplifier (op amp) as a current negative impedance converter (CNIC), are modified for chaos using a nonlinear resistor of anti-symmetrical current-voltage characteristics formed by a junction field effect transistor (JFET) operating in the triode region. The internal op amp dominant pole is found to contribute significantly to the chaotic nature of one circuit while the other circuit develops different chaotic attractors when cubic and fifth power odd symmetrical nonlinearities are used. Mathematical models of the two generators are presented. Experimental laboratory results, circuit simulations and numerical simulations of the mathematical models well agree and are included.

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Tajani, Francesco, and Pierluigi Morano. "A model for the elaboration of fair divisional projects in inheritance disputes." Property Management 36, no. 2 (April 16, 2018): 186–202. http://dx.doi.org/10.1108/pm-07-2016-0033.

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Purpose The purpose of this paper is to develop a method to support the definition of efficient and fair divisional projects in particularly complex cases concerning inheritance disputes. Design/methodology/approach First, the approach involves an appraisal of the market value of the assets, along with an analysis of the respective conditions of concrete divisibility; then, two mathematical models have been developed for the assignment of the assets to the subjects involved in the divisional projects. The logic underlying of both models has been translated into mathematical algorithms that allow for the minimization of the monetary compensations resulting from the differences between the legal right shares and the actual portions to be attributed to them. Findings Both models have been developed through mathematical formulas that can be easily implemented by using an appropriate calculation software. They can be used in particularly complex inheritance divisions, in which the deceased’s assets are numerous and there are several heirs with similar or different legal right shares. Originality/value The methodology is useful in the disputes that could arise in hereditary successions. The fundamental value is that the models could support the definition of the best solution in particularly complex situations, characterized by a large number of assets to be assigned and/or the existence of “preferential” constraints for the assignment of the assets.

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BURDETT, ROBERT L., and ERHAN KOZAN. "THE ASSIGNMENT OF INDIVIDUAL RENEWABLE RESOURCES IN SCHEDULING." Asia-Pacific Journal of Operational Research 21, no. 03 (September 2004): 355–77. http://dx.doi.org/10.1142/s021759590400028x.

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Resource constrained scheduling problems are concerned with the allocation of limited resources to tasks over time. The solution to these problems is often a sequence, resource allocation, and schedule. When human workers are incorporated as a renewable resource, the allocation is defined as the number of workers assigned to perform each task. In practice, however, this solution does not adequately address how individual workers are to be assigned to tasks. This paper, therefore, provides mathematical models and heuristic techniques for solving this multi-period precedence constrained assignment problem. Results of a significant numerical investigation are also presented.

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Ding, Haoyang, Yu Bao, Sida Luo, Hanxia Shen, Wei Wang, and Man Long. "A Capacity-Restraint Transit Assignment Model When a Predetermination Method Indicates the Invalidity of Time Independence." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/821574.

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The statistical independence of time of every two adjacent bus links plays a crucial role in deciding the feasibility of using many mathematical models to analyze urban transit networks. Traditional research generally ignores the time independence that acts as the ground of their models. Assumption is usually made that time independence of every two adjacent links is sound. This is, however, actually groundless and probably causes problematic conclusions reached by corresponding models. Many transit assignment models such as multinomial probit-based models lose their effects when the time independence is not valid. In this paper, a simple method to predetermine the time independence is proposed. Based on the predetermination method, a modified capacity-restraint transit assignment method aimed at engineering practice is put forward and tested through a small contrived network and a case study in Nanjing city, China, respectively. It is found that the slope of regression equation between the mean and standard deviation of normal distribution acts as the indicator of time independence at the same time. Besides, our modified assignment method performs better than the traditional one with more reasonable results while keeping the property of simplicity well.

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Song, Qiang, and Augustine O. Esogbue. "Mathematical models for optimal usage of tributary cards in wavelength assignment for DWDM ring networks." Mathematical and Computer Modelling 50, no. 3-4 (August 2009): 509–15. http://dx.doi.org/10.1016/j.mcm.2008.10.022.

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Parra, Enrique. "Transfer prices assignment with integrated production and marketing optimization models." Journal of Industrial Engineering and Management 11, no. 2 (April 6, 2018): 262. http://dx.doi.org/10.3926/jiem.2512.

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Purpose: In decentralized organizations (today a great majority of the large multinational groups), much of the decision-making power is in its individual business units-BUs-. In these cases, the management control system (MCS) uses transfer prices to coordinate actions of the BUs and to evaluate their performance with the goal of guaranteeing the whole corporation optimum. The purpose of the investigation is to design transfer prices that suit this goal.Design/methodology/approach: Considering the results of the whole company supply chain optimization models (in the presence of seasonality of demand) the question is to design a mechanism that creates optimal incentives for the managers of each business unit to drive the corporation to the optimal performance. Mathematical programming models are used as a start point.Findings: Different transfer prices computation methods are introduced in this paper for decentralised organizations with two divisions (production and marketing). The methods take into account the results of the solution of the whole company supply chain optimization model, if exists, and can be adapted to the type of information available in the company. It is mainly focused on transport costs assignment.Practical implications: Using the methods proposed in this paper a decentralized corporation can implement more accurate transfer prices to drive the whole organization to the global optimum performance.Originality/value: The methods proposed are a new contribution to the literature on transfer prices with special emphasis on the practical and easy implementation in a modern corporation with several business units and with high seasonality of demand. Also, the methods proposed are very flexible and can be tuned depending on the type of information available in the company.

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Mei, Liu, Huaguan Li, Yunlai Zhou, Dawang Li, Wujian Long, and Feng Xing. "Output-Only Damage Detection of Shear Building Structures Using an Autoregressive Model-Enhanced Optimal Subpattern Assignment Metric." Sensors 20, no. 7 (April 6, 2020): 2050. http://dx.doi.org/10.3390/s20072050.

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This paper proposes a novel output-only structural damage indicator by incorporating the pole-based optimal subpattern assignment distance with autoregressive models to localize and relatively assess the severity of damages for sheared structures. Autoregressive models can model dynamic systems well, while their model poles can represent the state of the dynamic systems. Structural damage generally causes changes in the dynamic characteristics (especially the natural frequency, mode shapes and damping ratio) of structures. Since the poles of the autoregressive models can solve the modal parameters of the structure, the poles have a close relationship with the modal parameters so that the changes in the poles of its autoregressive model reflect structural damages. Therefore, we can identify the damage by tracking the shifts in the dynamic system poles. The optimal subpattern assignment distance, which is the performance evaluator in multi-target tracking algorithms to measure the metric between true and estimated tracks, enables the construction of damage sensitive indicator from system poles using the Hungarian algorithm. The proposed approach has been validated with a five-story shear-building using numerical simulations and experimental verifications, which are subjected to excitations of white noise, El Centro earthquake and sinusoidal wave with frequencies sweeping, respectively; the results indicate that this approach can localize and quantify structural damages effectively in an output-only and data-driven way.

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Khadka, Mukesh, Alexander Paz, Cristian Arteaga, and David K. Hale. "Simultaneous Generation of Optimum Pavement Clusters and Associated Performance Models." Mathematical Problems in Engineering 2018 (December 12, 2018): 1–17. http://dx.doi.org/10.1155/2018/2159865.

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With regard to developing pavement performance models (PPMs), the existing state-of-the-art proposes Clusterwise Linear Regression (CLR) to determine the pavement clusters and associated PPMs simultaneously. However, the approach does not determine optimal clustering to minimize error; that is, the number of clusters and explanatory variables are prespecified to determine the corresponding coefficients of the PPMs. In addition, existing formulations do no address issues associated with overfitting as there is no limit to include parameters in the model. In order to address this limitation, this paper proposes a mathematical program within the CLR approach to determine simultaneously (1) an optimal number of clusters, (2) assignment of segments into clusters, and (3) regression coefficients for all prespecified explanatory variables required to minimize the estimation error. The Bayesian Information Criteria is proposed to limit the number of optimal clusters. A simulated annealing coupled with ordinary least squares was used to solve the mathematical program.

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Fukuyama, Takeshi, Yuuko Kakigi, and Takashi Okamura. "Multipole Expansion Model in Gravitional Lensing." International Journal of Modern Physics D 06, no. 04 (August 1997): 425–47. http://dx.doi.org/10.1142/s021827189700025x.

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Nontransparent models of the multipole expansion model and the two point-mass model are analyzed from the catastrophe theory. Singularity behaviours of 2n-pole moments are discussed. We apply these models to the triple quasar PG1115+080 and compare with the typical transparent model, softened power law spheroids. The multipole expansion model gives the best fit among them.

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Sawik, Bartosz. "A single and triple-objective mathematical programming models for assignment of services in a healthcare institution." International Journal of Logistics Systems and Management 15, no. 2/3 (2013): 249. http://dx.doi.org/10.1504/ijlsm.2013.053770.

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Bunge, H. J. "Physical Versus Mathematical Aspects in Texture Analysis." Textures and Microstructures 25, no. 2-4 (January 1, 1996): 71–108. http://dx.doi.org/10.1155/tsm.25.71.

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Texture deals with the orientational aspects of the crystal lattice in polycrystalline aggregates. This comprises the classical orientation distribution function ODF as well as higher-order textural quantities. The quantitative treatment of these quantities requires a good deal of mathematical methods. This concerns particularly the representation of orientation data including all kinds of symmetries, the transformation of experimentally measured raw data into the required distribution functions, as well as mathematical models of texture formation by physical processes and of the texture-property relationship.When physical facts are idealized in terms of a mathematical description or by mathematical models then the physical limits, within which these are valid, must be known. Such physical limits are, for instance, definition of crystal orientation by the crystal lattice which leads to an unsharpness relationship between location and orientation resolution as well as a relationship between statistical relevance and angular resolving power. Other physical limits are given by the “fuzzyness” of sample symmetry. A central problem is pole figure inversion i.e. the inversion of the projection equation. This problem may have a “physical” solution even if it does not have a mathematical one. Finally, in the problem of rationalizing orientation distribution functions in terms of a low number of “components”, mathematical aspects may be quite different from physical ones.In all these problems it is thus necessary to keep the mathematical aspects apart from physical aspects.

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Lujak, Marin, Stefano Giordani, Andrea Omicini, and Sascha Ossowski. "Decentralizing Coordination in Open Vehicle Fleets for Scalable and Dynamic Task Allocation." Complexity 2020 (July 16, 2020): 1–21. http://dx.doi.org/10.1155/2020/1047369.

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One of the major challenges in the coordination of large, open, collaborative, and commercial vehicle fleets is dynamic task allocation. Self-concerned individually rational vehicle drivers have both local and global objectives, which require coordination using some fair and efficient task allocation method. In this paper, we review the literature on scalable and dynamic task allocation focusing on deterministic and dynamic two-dimensional linear assignment problems. We focus on multiagent system representation of open vehicle fleets where dynamically appearing vehicles are represented by software agents that should be allocated to a set of dynamically appearing tasks. We give a comparison and critical analysis of recent research results focusing on centralized, distributed, and decentralized solution approaches. Moreover, we propose mathematical models for dynamic versions of the following assignment problems well known in combinatorial optimization: the assignment problem, bottleneck assignment problem, fair matching problem, dynamic minimum deviation assignment problem, Σk-assignment problem, the semiassignment problem, the assignment problem with side constraints, and the assignment problem while recognizing agent qualification; all while considering the main aspect of open vehicle fleets: random arrival of tasks and vehicles (agents) that may become available after assisting previous tasks or by participating in the fleet at times based on individual interest.

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Xu, F., Ke Zhang, and Zheng Xing Cui. "Research on Spindle Induction Motor Stator Flux Observer Algorithm." Key Engineering Materials 455 (December 2010): 507–10. http://dx.doi.org/10.4028/www.scientific.net/kem.455.507.

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In order to enhance observation accuracy of the spindle induction stator flux and decrease observation error which is influenced by temperature, frequency and magnetic path in operation process, a full order observer was used to observe stator flux. The change of stator flux observer which adopted u-i model and low pass filter estimation model observation error was analyzed when the parameters of motor changed through the mathematical model in the stationary coordinates. A method to determine the gain matrix of full order observer was presented. The results showed that the error changed with the changes of motor speed and torque. On the premise of assuring stability of the observer, according to different speed ranges, choosing the proper observer pole assignment, the method can be used to enhance the observation accuracy of stator flux at high speed, and it is also validated in the Matlab/Simulink condition. It is shown that it can improve observer accuracy of the induction motor stator flux effectively.

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Weinstein, Alan M. "Mathematical models of renal fluid and electrolyte transport: acknowledging our uncertainty." American Journal of Physiology-Renal Physiology 284, no. 5 (May 1, 2003): F871—F884. http://dx.doi.org/10.1152/ajprenal.00330.2002.

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Mathematical models of renal tubular function, with detail at the cellular level, have been developed for most nephron segments, and these have generally been successful at capturing the overall bookkeeping of solute and water transport. Nevertheless, considerable uncertainty remains about important transport events along the nephron. The examples presented include the role of proximal tubule tight junctions in water transport and in regulation of Na+ transport, the mechanism by which axial flow in proximal tubule modulates solute reabsorption, the effect of formate on proximal Cl− transport, the assessment of potassium transport along collecting duct segments inaccessible to micropuncture, the assignment of pathways for peritubular Cl− exit in outer medullary collecting duct, and the interaction of carbonic anhydrase-sensitive and -insensitive pathways for base exit from inner medullary collecting duct. Some of these uncertainties have had intense experimental interest well before they were cast as modeling problems. Indeed, many of the renal tubular models have been developed based on data acquired over two or three decades. Nevertheless, some uncertainties have been delineated as the result of model exploration and represent communications from the modelers back to the experimental community that certain issues should not be considered closed. With respect to model refinement, incorporating more biophysical detail about individual transporters will certainly enhance model reliability, but ultimate confidence in tubular models will still be contingent on experimental development of critical information at the tubular level.

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Szeto, W., and S. Wong. "Dynamic traffic assignment: model classifications and recent advances in travel choice principles." Open Engineering 2, no. 1 (January 1, 2012): 1–18. http://dx.doi.org/10.2478/s13531-011-0057-y.

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AbstractDynamic Traffic Assignment (DTA) has been studied for more than four decades and numerous reviews of this research area have been conducted. This review focuses on the travel choice principle and the classification of DTA models, and is supplementary to the existing reviews. The implications of the travel choice principle for the existence and uniqueness of DTA solutions are discussed, and the interrelation between the travel choice principle and the traffic flow component is explained using the nonlinear complementarity problem, the variational inequality problem, the mathematical programming problem, and the fixed point problem formulations. This paper also points out that all of the reviewed travel choice principles are extended from those used in static traffic assignment. There are also many classifications of DTA models, in which each classification addresses one aspect of DTA modeling. Finally, some future research directions are identified.

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Chang, Wen-Jer, Yan-Horng Lin, Jialu Du, and Chin-Ming Chang. "Fuzzy Control with Pole Assignment and Variance Constraints for Continuous-time Perturbed Takagi-Sugeno Fuzzy Models: Application to Ship Steering Systems." International Journal of Control, Automation and Systems 17, no. 10 (July 26, 2019): 2677–92. http://dx.doi.org/10.1007/s12555-018-0917-9.

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Suarez-Rivera, Guiovanny, Nelson David Muñoz-Ceballos, and Henry Mauricio Vásquez-Carvajal. "Development of an Adaptive Trajectory Tracking Control of Wheeled Mobile Robot." Revista Facultad de Ingeniería 30, no. 55 (February 13, 2021): e12022. http://dx.doi.org/10.19053/01211129.v30.n55.2021.12022.

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Classical modeling and control methods applied to differential locomotion mobile robots generate mathematical equations that approximate the dynamics of the system and work relatively well when the system is linear in a specific range. However, they may have low accuracy when there are many variations of the dynamics over time or disturbances occur. To solve this problem, we used a recursive least squares (RLS) method that uses a discrete-time structure first-order autoregressive model with exogenous variable (ARX). We design and modify PID adaptive self-adjusting controllers in phase margin and pole allocation. The main contribution of this methodology is that it allows the permanent and online update of the robot model and the parameters of the adaptive self-adjusting PID controllers. In addition, a Lyapunov stability analysis technique was implemented for path and trajectory tracking control, this makes the errors generated in the positioning and orientation of the robot when performing a given task tend asymptotically to zero. The performance of the PID adaptive self-adjusting controllers is measured through the implementation of the criteria of the integral of the error, which allows to determine the controller of best performance, being in this case, the PID adaptive self-adjusting type in pole assignment, allowing the mobile robot greater precision in tracking the trajectories and paths assigned, as well as less mechanical and energy wear, due to its smooth and precise movements.

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Meng, Leilei, Biao Zhang, Yaping Ren, Hongyan Sang, Kaizhou Gao, and Chaoyong Zhang. "Mathematical Formulations for Asynchronous Parallel Disassembly Planning of End-of-Life Products." Mathematics 10, no. 20 (October 18, 2022): 3854. http://dx.doi.org/10.3390/math10203854.

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Disassembly is one of the most time-consuming and labor-intensive activities during the value recovery of end-of-life (EOL) products. The completion time (makespan) of disassembling EOL products is highly associated with the allocation of operators, especially in parallel disassembly. In this paper, asynchronous parallel disassembly planning (APDP), which avoids the necessity to synchronize disassembly tasks of manipulators during the parallel disassembly process, is studied to optimize the task assignment of manipulators for minimal makespan. We utilize four mixed integer linear programming (MILP) formulations to identify the optimal solutions. A set of different-sized instances are used to test and compare the performance of the proposed models, including some real-world cases. Finally, the proposed exact algorithm is further compared with the existing approach to solving APDP. Results indicate that a significant difference exists in terms of the computational efficiency of the MILP models, while three of four MILP formulations can efficiently achieve better solutions than that of the existing approach.

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Kneller, James, Rafael J. Ramirez, Denis Chartier, Marc Courtemanche, and Stanley Nattel. "Time-dependent transients in an ionically based mathematical model of the canine atrial action potential." American Journal of Physiology-Heart and Circulatory Physiology 282, no. 4 (April 1, 2002): H1437—H1451. http://dx.doi.org/10.1152/ajpheart.00489.2001.

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Ionically based cardiac action potential (AP) models are based on equations with singular Jacobians and display time-dependent AP and ionic changes (transients), which may be due to this mathematical limitation. The present study evaluated transients during long-term simulated activity in a mathematical model of the canine atrial AP. Stimulus current assignment to a specific ionic species contributed to stability. Ionic concentrations were least disturbed with the K+ stimulus current. All parameters stabilized within 6–7 h. Inward rectifier, Na+/Ca2+ exchanger, L-type Ca2+, and Na+-Cl− cotransporter currents made the greatest contributions to stabilization of intracellular [K+], [Na+], [Ca2+], and [Cl−], respectively. Time-dependent AP shortening was largely due to the outward shift of Na+/Ca2+exchange related to intracellular Na+(Na[Formula: see text]) accumulation. AP duration (APD) reached a steady state after ∼40 min. AP transients also occurred in canine atrial preparations, with the APD decreasing by ∼10 ms over 35 min, compared with ∼27 ms in the model. We conclude that model APD and ionic transients stabilize with the appropriate stimulus current assignment and that the mathematical limitation of equation singularity does not preclude meaningful long-term simulations. The model agrees qualitatively with experimental observations, but quantitative discrepancies highlight limitations of long-term model simulations.

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Vovsha, Peter, and Shlomo Bekhor. "Link-Nested Logit Model of Route Choice: Overcoming Route Overlapping Problem." Transportation Research Record: Journal of the Transportation Research Board 1645, no. 1 (January 1998): 133–42. http://dx.doi.org/10.3141/1645-17.

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A new link-nested logit model of route choice is presented. The model is derived as a particular case of the generalized-extreme-value class of discrete choice models. The model has a flexible correlation structure that allows for overcoming the route overlapping problem. The corresponding stochastic user equilibrium is formulated in two equivalent mathematical programming forms: as a particular case of the general Sheffi formulation and as a generalization of the logit-based Fisk formulation. A stochastic network loading procedure is proposed that obviates route enumeration. The proposed model is then compared with alternative assignment models by using numerical examples.

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Adamou-Mitiche, Amel BH, and Lahcène Mitiche. "Extended approaches based on Genetic Algorithms (GAs) for MIMO singular systems approximation." Transactions of the Institute of Measurement and Control 40, no. 1 (July 21, 2016): 148–54. http://dx.doi.org/10.1177/0142331216657869.

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In this paper we present an extension of three important model reduction techniques: namely, the stability equation, the modified pole clustering and the dominant modes methods for conventional (regular) systems to reduce complexity relating to high dimensionality of mathematical models representing physical, generalized (also called singular) systems. Combining these methods to Genetic Algorithms’ tools and exploiting a special representation base where a full order singular system is deflating into proper and improper subsystems, different natures of stable, optimal low order models are obtained. To show the effectiveness of the proposed algorithms, a numerical example is given, where six approximants are derived from a multi-input multi-output singular system. By the use of two optimal norms, the MOR errors are quantified and permits to conclude to the quality of the proposed reduced order models.

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Aguedo, Reinel Beltran, Ania Lussón Cervantes, José Ricardo Núñez Alvarez, and Yolanda Llosas Albuerne. "Speed control in DC and AC drives." International Journal of Power Electronics and Drive Systems (IJPEDS) 12, no. 4 (December 1, 2021): 2006. http://dx.doi.org/10.11591/ijpeds.v12.i4.pp2006-2017.

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<div><span lang="EN-US">Three speed-control strategies for DC and AC drives are presented in this study: a proportional<strong> </strong>integral derivative (PID) control strategy; an internal model control (IMC); and a state-space control by pole assignment with full state observer (ESSO). The three strategies are applied to a case study, demonstrating the potential of each one. Experimental identification was used to obtain the drive models used for the synthesis of the controllers. The three strategies showed satisfactory results when compared with the requirements imposed on the system, in addition to the good rejection of disturbances. However, the IMC strategy showed itself to be a little softer and with no maximum overshoot, which in some cases and some applications is usually a restriction.</span></div>

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LEE, TZU-YING, PO-CHUAN CHEN, and DER-SHIN JUANG. "SLIDING MODE CONTROL ON ISOLATED BRIDGES WITH COLUMNS OF IRREGULAR HEIGHTS USING POLE ASSIGNMENT AND PSO-SA HYBRID ALGORITHM." International Journal of Structural Stability and Dynamics 12, no. 03 (May 2012): 1250014. http://dx.doi.org/10.1142/s0219455412500149.

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The effectiveness of sliding mode control on the seismic response of an isolated bridge with columns of irregular heights, which exhibit hysteretic behaviors at both the columns and isolators, is studied. The bridge of concern consists of a two-span continuous deck and three columns of irregular heights, adjoining two single-span approaches each at the two ends. The irregular isolated bridge is idealized by an equivalent model to reduce the number of degrees of freedom involved. Compared with typical isolated bridges, the irregular isolated bridge has more poles of sliding surface, which dominates the dynamic characteristics of the controlled system and should be determined for the sliding mode control. The particle swarm optimization-simulated annealing (PSO-SA) hybrid searching algorithm is thus employed and shown to outperform the PSO algorithm and a parametric approach in finding the best sliding surface. Numerical simulations reveal that the sliding mode control together with the PSO-SA hybrid searching algorithm provides a simple and powerful technique for controlling the nonlinear seismic responses of irregular isolated bridges. Such a technique combining the control and optimization technology can be applied to practical bridges or structures, which are generally complicated and should be idealized by sophisticated numerical models.

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Fang, Jian Cheng, Wen Ji Xu, Zhi Yu Zhao, and H. Y. Li. "Numerical Simulation of Magnetic Field Distribution in MAF." Key Engineering Materials 304-305 (February 2006): 565–69. http://dx.doi.org/10.4028/www.scientific.net/kem.304-305.565.

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Magnetic field distribution (MFD) plays an important role in magnetic abrasive finishing (MAF). The mathematical models of the MFD in MAF have been established, and forced behavior of magnetic abrasive (MA) in the magnetic field has been analyzed in this paper. The MFD characteristics of the grooved poles have been numerically simulated, and the distribution law of magnetic force, interaction force and interface pressure have been investigated based on a single particle. The calculated results show that the MFD in the finishing zone is directly affected by the pole structure and working gap. The mapping function of interaction force and interface pressure corresponds to corner-effect in the peak point of the pole, and the magnetic brush stiffness is changed by the diameter and magnetization characteristics of MA, and the magnetic flux intensity (MFI). The numerical simulations are in a good agreement with the experiments. All of results mentioned above will offer an important method to study further on the finishing mechanism.

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Alonso-Mora, Javier, Samitha Samaranayake, Alex Wallar, Emilio Frazzoli, and Daniela Rus. "On-demand high-capacity ride-sharing via dynamic trip-vehicle assignment." Proceedings of the National Academy of Sciences 114, no. 3 (January 3, 2017): 462–67. http://dx.doi.org/10.1073/pnas.1611675114.

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Ride-sharing services are transforming urban mobility by providing timely and convenient transportation to anybody, anywhere, and anytime. These services present enormous potential for positive societal impacts with respect to pollution, energy consumption, congestion, etc. Current mathematical models, however, do not fully address the potential of ride-sharing. Recently, a large-scale study highlighted some of the benefits of car pooling but was limited to static routes with two riders per vehicle (optimally) or three (with heuristics). We present a more general mathematical model for real-time high-capacity ride-sharing that (i) scales to large numbers of passengers and trips and (ii) dynamically generates optimal routes with respect to online demand and vehicle locations. The algorithm starts from a greedy assignment and improves it through a constrained optimization, quickly returning solutions of good quality and converging to the optimal assignment over time. We quantify experimentally the tradeoff between fleet size, capacity, waiting time, travel delay, and operational costs for low- to medium-capacity vehicles, such as taxis and van shuttles. The algorithm is validated with ∼3 million rides extracted from the New York City taxicab public dataset. Our experimental study considers ride-sharing with rider capacity of up to 10 simultaneous passengers per vehicle. The algorithm applies to fleets of autonomous vehicles and also incorporates rebalancing of idling vehicles to areas of high demand. This framework is general and can be used for many real-time multivehicle, multitask assignment problems.

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Pinarbasi, Mehmet, Hacı Mehmet Alakas, and Mustafa Yuzukirmizi. "A constraint programming approach to type-2 assembly line balancing problem with assignment restrictions." Assembly Automation 39, no. 5 (November 4, 2019): 813–26. http://dx.doi.org/10.1108/aa-12-2018-0262.

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Purpose Main constraints for an assembly line balancing problem (ALBP) are cycle time/number of stations and task precedence relations. However, due to the technological and organizational limitations, several other restrictions can be encountered in real production systems. These restrictions are called as assignment restrictions and can be task assignment, station, resource and distance limitations. The purpose of the study is to evaluate the effects of these restrictions on ALBP using constraint programming (CP) model. Design/methodology/approach A novel CP model is proposed and compared to mixed-integer programming (MIP) as a benchmark. The objective is to minimize the cycle time for a given number of stations. The authors also provide explicit anthology of the assignment restriction effects on line efficiency, the solution quality and the computation time. Findings The proposed approach is verified with the literature test instances and a real-life problem from a furniture manufacturing company. Computational experiments show that, despite the fact that additional assignment restrictions are problematic in mathematical solutions, CP is a versatile exact solution alternative in modelling and the solution quality. Practical implications Assembly line is a popular manufacturing system in the making of standardized high volume products. The problem of assembly line balancing is a crucial challenge in these settings and consists of assigning tasks to the stations by optimizing one or more objectives. Type-2 AR-ALBP is a specific case with the objective function of minimizing the cycle time for a given number of stations. It further assumes assignment restrictions that can be confronted due to the technological limitations or the strategic decisions of the company management. This is especially encountered in rebalancing lines. Originality/value Several solution approaches such as mathematical modelling, heuristic and meta-heuristic are proposed to solve the ALBP in the literature. In this study, a new approach has been presented using CP. Efficient models are developed for Type-2 ALBP with several assignment restrictions. Previous studies have not considered the problem to the presented extent. Furthermore, to the best of the authors’ knowledge, the paper is the first study that solves ALBP with assignment restrictions using CP.

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Bolaños Zuñiga, Johanna, Jania Astrid Saucedo Martínez, Tomas Eloy Salais Fierro, and José Antonio Marmolejo Saucedo. "Optimization of the Storage Location Assignment and the Picker-Routing Problem by Using Mathematical Programming." Applied Sciences 10, no. 2 (January 10, 2020): 534. http://dx.doi.org/10.3390/app10020534.

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The order picking process involves a series of activities in response to customer needs, such as the selection or programming of orders (batches), and the selection of different items from their storage location to shipment. These activities are accomplished by a routing policy that determines the picker sequence for retrieving the items from the storage location. Therefore, the order picking problem has been plenty investigated; however, in previous research, the proposed models were based on demand fulfilling, putting aside factors such as the product weight—which is an important criterion—at the time of establishing routes. In this article, a mathematical model is proposed; it takes into account the product’s weight derived from a case study. This model is relevant, as no similar work was found in the literature that improves the order picking by making simultaneous decisions on the storage location assignment and the picker-routing problem, considering precedence constraints based on the product weight and the characteristics of the case study, as the only location for each product in a warehouse with a general layout.

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Dahite, Lamiaa, Abdeslam Kadrani, and Rachid Benmansour. "Optimization models for train load and transport planning problems." MATEC Web of Conferences 200 (2018): 00010. http://dx.doi.org/10.1051/matecconf/201820000010.

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Optimization of rail transport is complex because of the industry’s multiple constraints. The transport of containers is very particular since it is characterized in addition to its specificities as a product to be loaded and transported by a strong instability of the demand. So far research in this area has dealt only with the separate treatment of the train load and transport problem. The present study focuses on optimizing resources facing unstable demand for the combined problem. Mathematical models are proposed to assign customers demands to wagons and for railcars allowance per axis depending on the available park, the locomotive capacity and the train length. An algorithm for the train load problem is also suggested. The models have been tested to measure their efficiency by comparing them to an existing train planning model and to manual assignment adopted in the rail industry. Some test results are finally reported to show how a novel formulation can simplify the resolution of a complex problem.

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Chiel, Hillel J., Jeffrey M. McManus, and Kendrick M. Shaw. "From Biology to Mathematical Models and Back: Teaching Modeling to Biology Students, and Biology to Math and Engineering Students." CBE—Life Sciences Education 9, no. 3 (September 2010): 248–65. http://dx.doi.org/10.1187/cbe.10-03-0022.

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We describe the development of a course to teach modeling and mathematical analysis skills to students of biology and to teach biology to students with strong backgrounds in mathematics, physics, or engineering. The two groups of students have different ways of learning material and often have strong negative feelings toward the area of knowledge that they find difficult. To give students a sense of mastery in each area, several complementary approaches are used in the course: 1) a “live” textbook that allows students to explore models and mathematical processes interactively; 2) benchmark problems providing key skills on which students make continuous progress; 3) assignment of students to teams of two throughout the semester; 4) regular one-on-one interactions with instructors throughout the semester; and 5) a term project in which students reconstruct, analyze, extend, and then write in detail about a recently published biological model. Based on student evaluations and comments, an attitude survey, and the quality of the students' term papers, the course has significantly increased the ability and willingness of biology students to use mathematical concepts and modeling tools to understand biological systems, and it has significantly enhanced engineering students' appreciation of biology.

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Hasan, Mohamad K., Mohammad Saoud, and Raed Al-Husain. "Supernetwork Representation Formulation of a Multiclass Simultaneous Transportation Equilibrium Model as a Fixed Demand User Equilibrium Problem." International Journal of Operations Research and Information Systems 12, no. 3 (July 2021): 18–33. http://dx.doi.org/10.4018/ijoris.20210701.oa2.

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A multiclass simultaneous transportation equilibrium model (MSTEM) explicitly distinguishes between different user classes in terms of socioeconomic attributes, trip purpose, pure and combined transportation modes, as well as departure time, all interacting over a physically unique multimodal network. It enhances the prediction process behaviorally by combining the trip generation and departure time choices to trip distribution, modal split, and trip assignment choices in a unified and flexible framework that has many advantages from both supply and demand sides. However, the development of this concept of multiple classes increases the mathematical complexity of travel forecasting models. In this research, the authors reduce this mathematical complexity by using the supernetwork representation formulation of the diagonalized MSTEM as a fixed demand user equilibrium (FDUE) problem.

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Journal articles on the topic 'Pole assignment Mathematical models'

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