Complex Algebraic Equations Research Articles

An inventory model for deteriorating products under preservation technology with time-dependent quality demand

In many real-world inventory deterioration concerns, the rate of deterioration of individual commodities is based on the deteriorating items during the storage period. Inventory models for degrading commodities have become more important in the food production sectors due to the unpredictability of product life spans. This deterioration can be overcome by involving a preservation factor to reduce the effects of decaying objects. This model is applicable for food production, in particular, dairy product (ice cream) suppliers to make inventory decisions in the inventory system under the effect of optimal values. In this paper, a preservative-based inventory model for deteriorating products with quality demand is discussed. The proposed model with cost functions yields the best operating plan and total sales. The policy combinations for dairy products are generally designed to reduce the deterioration rate, the expenditures, and the total cost. To illustrate the value of the proposed model numerical examples involving three flavours of ice creams have been studied. The model was tested with the real-time data obtained from the ice cream factory. Also, the proposed model’s numerical results are compared with the existing model. Further, the sensitivity analysis is performed to study the effect of changes in different parameters of the model. The complex algebraic equations are solved using MATLAB R2013a.

Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees.

Dynamic behavior of earth dams under different kinematic impacts

The paper provides a detailed analysis of the current state of the problem. A mathematical model is presented to determine the dynamic behavior of earth dams, considering the viscoelastic properties of soil, using the hereditary Boltzmann-Volterra theory with the A.R. Rzhanitsyn kernel under periodic kinematic impacts. To solve the problems considered, the finite element method and complex arithmetics were used to reduce integrodifferential equations to a high-order complex algebraic equation. The accuracy of the methods was verified by solving test problems. Steadystate forced vibrations of the Pskem earth dam 195 m high are studied considering the real geometry and soil properties under resonant vibration modes. It was stated that the largest stress amplitudes in the body of the dam occur not only under the first resonance, but they can occur under other dense spectra of the eigenfrequencies of the dam, due to the interaction between close natural modes of vibration. The strength of various sections of the dam body was tested under kinematic impact using the Coulomb- Mohr theory of strength; the most dangerous sections of the dam were identified in terms of the highest stress.

A parametric frequency domain approach to analysis and design of critical design parameters of nonlinear energy harvesting systems: Parametric output spectrum and power generation functions

Nonlinear energy harvesting systems have been extensively studied which usually involve quite a few critical design parameters to determine. However, a straightforward parametric design method for such a purpose is still far from developed. Traditional methods with harmonic balance or numerical simulation usually cannot reveal a straightforward relationship between the energy harvesting performance and system parameters, and in most cases face relatively large computational costs during parametric studies. To this aim, a novel Volterra series-based frequency domain method is established in this study with respect to a typical nonlinear harvesting system. A concise parametric characteristic approach, namely the linear and nonlinear characteristic output spectrum (lnCOS) method, is thus proposed to achieve parametric characteristic expression for the nonlinear output frequency response and power generation function (PGF) of the system, referred to as the lnCOS function and the Parametric-PGF respectively. Compared with the harmonic balance method, the vibration responses/power generation can be obtained in a more straightforward way without solving complex algebraic equations, and the lnCOS method directly links the power outputs to critical system parameters in a concise parametric way. It is for the first time to provide explicit expressions for vibration responses and power generations with respect to structural parameters of a nonlinear energy harvesting system, which cannot be achieved in a traditional way. Based on the obtained explicit expressions of the lnCOS function/Parametric-PGF, the sensitivity analysis of vibration response and power generation with respect to structural parameters can be directly performed. Our work provides an efficient way to characterize both the linear and nonlinear effects on a nonlinear energy harvesting system and demonstrates an efficient approach to the analysis and design of nonlinear energy harvesting systems.

Generalized phasor modeling of dynamic gas flow for integrated electricity-gas dispatch

In this paper, a generalized phasor method (GPM) for modeling gas flow dynamics is proposed and applied in integrated electricity-gas dispatch (IEGD). An accurate dynamic model of natural gas systems is described by partial differential equations (PDEs). In existing studies, PDEs are often approximated by using the finite difference method (FDM), but many variables and constraints are required in that approach. The proposed GPM transforms PDEs into ordinary differential equations (ODEs) by using the Fourier method and then eliminates spatial variables by analytically solving the transformed ODEs. Since spatial differences are avoided, the GPM has fewer constraints and variables than the traditional FDM. Moreover, the GPM contains only linear complex algebraic equations, so IEGD problems based on the GPM can be efficiently solved. In case studies, IEGD models based on the GPM and FDM are compared for two integrated electricity-gas systems. The results illustrate that the proposed GPM and FDM yield similar accuracies, but the GPM has a shorter calculation time than the FDM.

Optimization-based primary and secondary control of microgrids

Abstract This article discusses how to use optimization-based methods to efficiently operate microgrids with a large share of renewables. We discuss how to apply a frequency-based method to tune the droop parameters in order to stabilize the grid and improve oscillation damping after disturbances. Moreover, we propose a centralized real-time feasible nonlinear model predictive control (NMPC) scheme to achieve efficient frequency and voltage control while considering economic dispatch results. Centralized NMPC for secondary control is a computationaly challenging task. We demonstrate how to reduce the computational burden using the Advanced Step Real-Time Iteration with nonuniform discretization grids. This reduces the computational burden up to 60 % compared to a standard uniform approach, while having only a minor performance loss. All methods are validated on the example of a 9-bus microgrid, which is modeled with a complex differential algebraic equation.

Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept

In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by $ \alpha [\varphi(z) \varphi (z) +(\varphi' (z))^2]+ a_m \varphi^m(z)+a_{m-1} \varphi^{m-1}(z)+...+ a_1 \varphi(z)+ a_0 = 0. $ The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of $e^z.$ Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.

Integrated Heat and Electricity Dispatch for District Heating Networks With Constant Mass Flow: A Generalized Phasor Method

Using thermal inertia in district heating systems (DHSs) to improve the dispatch flexibility and economy of integrated heat and electricity systems (IHESs) is a research hotspot and difficulty. In most existing studies, the partial differential equations (PDEs) of thermal inertia are approximated by discrete-time models, making it difficult to accurately describe the continuous dynamic processes. In this paper, we propose a novel generalized phasor method (GPM) for thermal inertia in DHSs with constant mass flow. Based on the analytical solution of the PDEs and the Fourier transform, the intractable PDEs are transformed into a series of complex algebraic equations represented by phasors. The GPM has higher accuracy compared to traditional discrete models because it is essentially a continuous model in the time domain. Then, we present a different representation of an integrated heat and electricity dispatch (IHED) model combining a DHS model in phasor form and a traditional electrical power system model. The IHED model is a convex programming problem and can be easily solved. The effectiveness of the proposed GPM and dispatch model is verified in three test systems. Compared with traditional methods for modeling the thermal inertia, the proposed GPM is more accurate.

Modular and parallel VLSI architecture of multi-dimensional quad-core GA co-processor for real time image/video processing

In modern days multi-dimensional signal processing has appeared as a trendy research area, including hyper-spectral image/video processing. It involves computationally complex algebraic equations, thereby considering conventional algebra incompetent. To alleviate this problem, Geometric Algebra (GA) establishes its competency uniquely due to its compact and less complex equations for operating in higher dimensional frameworks in contrast to the conventional one. Hardware realization is indispensible for any real time signal processing algorithm. As any GA based algorithm usually results in a significantly complex, outsized and sluggish digital circuit, it lags behind the conventional algebra in case of hardware implementation. Hence, in this article, a parallel-pipelined VLSI architecture of GA co-processor has been proposed to mitigate this intricacy, thereby becoming proficiently competent for computing a variety of GA operations. 4 channel parallelisms and 8-stage pipelining is the main attraction of our proposed design which does not necessitate any wait-state even in the situation of concurrent accessing of Memory. Supremacy of our design has been established over other state-of-the-art GA architectures in terms of number of processing cycle, latency and throughput. Superiority of our proposed design has also been proved in case of exemplary real time applications.

Exact results relating spin–orbit interactions in two-dimensional strongly correlated systems

ABSTRACT A 2D square, two-bands, strongly correlated and non-integrable system is analysed exactly in the presence of many-body spin–orbit interactions via the method of Positive Semidefinite Operators. The deduced exact ground states in the high concentration limit are strongly entangled, and given by the spin–orbit coupling are ferromagnetic and present an enhanced carrier mobility, which substantially differs for different spin projections. The described state emerges in a restricted parameter space region, which however is clearly accessible experimentally. The exact solutions are provided via the solution of a matching system of equations containing 74 coupled, non-linear and complex algebraic equations. In our knowledge, other exact results for 2D interacting systems with spin–orbit interactions are not present in the literature.

Quasi optimum PI controller tuning rules for a grid-connected three phase AC to DC PWM rectifier

Controller tuning is deemed to be critical when dealing with the voltage oriented control (VOC) as it ultimately defines the system performance. This paper contributes to the tuning issue associated with VOC in three key points. Firstly, an algebraic solution (i.e. PI-parameters) based on the first-order-plus-time-delay (FOPTD) approximations of the plant dynamics is developed, along with approximating relationships and methodology. Secondly, a non-linear multi-criteria optimization algorithm, with PI-controller parameters being the decision variables, is developed in order to achieve the optimum disturbance rejection. The cost function employs a weighted-sum criterion to account for two sources of disturbances: grid voltage sag/dip and load current disturbances, which affect the regulated dc voltage. Simulation results suggest a quasi-optimum system performance, in terms of minimum integral time absolute of the error (ITAE) performance index of the dc-bus voltage, using the FOPTD tuning parameters. Hence, the developed FOPTD approximations demonstrate an accurate linear approximation. Thirdly and consequently, analytic tuning rules for the PI-controllers are proposed. The proposed rules provide quasi-optimum PI-parameters by direct substitution of system parameters in the tuning formulae, rather than solving complex algebraic equations as in the FOPTD tuning approach. The obtained results are verified experimentally.

Investigation of modal interactions and their effects on the nonlinear dynamics of a periodic coupled pendulums chain

The nonlinear dynamics of a weakly coupled pendulums chain is investigated under primary resonance. The coupled equations governing the nonlinear vibrations are normalized and transformed into a set of coupled complex algebraic equations using the multiple scales method coupled with standing wave decomposition. A model reduction method is proposed to calculate the dominant dynamics without significant loss of accuracy compared to the full model. The validity of the proposed semi-analytical method is verified, and its role in identifying the type of the solution branches is highlighted. The modal interactions and their effects on the nonlinear dynamics are studied in the frequency domain in order to emphasize the large number of multimode solution branches and the bifurcation topology transfer between the modal intensities. Basins of attraction analysis have been performed, showing that the distribution of the multimodal solution branches generated by all modes collectively increases by increasing the number of coupled pendulums.

Synthesis of Linear Coherent Quantum Control Systems Using A Differential Evolution Algorithm

We propose a new method to construct an optimal linear coherent quantum controller based on an evolutionary optimization method, namely a differential evolution algorithm. The aim is to provide a straightforward approach to deal with both nonlinear and nonconvex constraints arising in the coherent quantum controller synthesis. The solution to this control problem involves a complex algebraic Riccati equation, which corresponds to a physical realizability condition for the coherent quantum controller. The proposed method is demonstrated through an example of an entanglement control problem for a quantum network comprising two cascaded optical parametric amplifiers.

Using a Spreadsheet Scroll Bar to Solve Equilibrium Concentrations

A simple, conceptual method is described for using the spreadsheet scroll bar to find the composition of a system at chemical equilibrium. Simulation of any kind of chemical equilibrium can be carried out using this method, and the effects of different disturbances can be predicted. This simulation, which can be used in general chemistry university courses (and also in the higher years of secondary schools due to its simplicity), is carried out by the students, avoids the solving of complex algebraic equations, and allows many of the common misconceptions and difficulties in this subject to be dealt with.

Geometrical interpretation of acoustic holography: Adaptation of the propagator and minimum quality guaranteed in the presence of errors

A large number of inverse problems in acoustics consist of a reverse propagation of the acoustic pressure measured with an array of microphones. The goal is usually to identify the acoustic source location and strength or the surface velocity of a vibrating structure. The quality of the results obtained depends on the propagation model, on the accuracy of the pressure measurements and, finally, on the inverse problem conditioning. How to quantify this quality is the issue addressed in this paper. For this purpose, a geometrical interpretation of the inverse acoustic problem is proposed. The main application will, eventually, be near-field acoustic holography (NAH), but it is expected that the proposed approach will also apply to other types of inverse acoustic problems. First, the geometrical representation of the inverse problem is proposed. The inverse problem is stated from a direct linear problem in the frequency domain. For each frequency, an overdetermined system of linear complex algebraic equations must be inverted. The concept of quality is discussed and a quality index is proposed based upon the residue of the inverse problem, solved in a mean square sense. Then, a simple one-dimensional (plane wave) acoustic example consisting of a source and two pressure measurements is used to illustrate the proposed geometrical representation of the inverse problem and the quality criterion inspired by it. In the simple example, the propagation model can be improved by searching for a reflection coefficient at the origin of the simulated hologram. This reflection coefficient is used to simulate the presence of a hidden source placed behind the source. An artificial attenuation is introduced to simulate the effect of geometrical attenuation present in real NAH problems. Again, using the geometrical representation, it is shown how, from an improved propagation model together with a given measurement noise level in the hologram, one can guarantee a certain quality level of the inverse procedure. Finally, numerical results show, in a preliminary way, how the identified source strength converges towards the exact velocity when the estimated propagation model tends to the exact propagation model.

Importance of verifying queue model assumptions before planning with simulation software

This case study uses empirical data gathered at an Australian refinery to verify the assumptions for queue distributions before using special-purpose software to plan the off-road-truck hauling of titanium dioxide to a refinery ( n = 773). Easy-to-use spreadsheet software is utilized to verify assumptions for queue models. Managers are able to make decisions based on economic implications of queue models to avoid making costly planning mistakes. Analysts can use nonparametric hypothesis-testing techniques to verify distribution assumptions for optimization without having to write hard-to-maintain and complex algebraic linear equations or nonlinear search routines.

Coherent $H^{\infty }$ Control for a Class of Annihilation Operator Linear Quantum Systems

This paper considers a coherent H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> control problem for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only. For this class of quantum systems, a solution to the H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> control problem can be obtained in terms of a pair of complex algebraic Riccati equations. In addition, the question of physical realizability of the resulting quantum controllers is related to a bounded real property.

A result of systems of nonlinear complex algebraic differential equations

In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gackstatter and Laine's result concerning complex differential equations to the systems of algebraic differential equations.

Multiharmonic Response Analysis of Systems With Local Nonlinearities Based on Describing Functions and Linear Receptance

Direct time integration methods are usually applied to determine the dynamic response of systems with local nonlinearities. Nevertheless, these methods are computationally expensive to predict the steady state response. To significantly reduce the computational effort, a new approach is proposed for the multiharmonic response analysis of dynamical systems with local nonlinearities. The approach is based on the describing function (DF) method and linear receptance data. With the DF method, the kinetic equations are converted into a set of complex algebraic equations. By using the linear receptance data, the dimension of the complex algebraic equations, which should be solved iteratively, are only related to nonlinear degrees of freedom (DOFs). A cantilever beam with a local nonlinear element is presented to show the procedure and performance of the proposed approach. The approach can greatly reduce the size and computational cost of the problem. Thus, it can be applicable to large-scale systems with local nonlinearities.

Nonlinear vibration analysis of spacecraft with local nonlinearity

Most spacecraft are usually assembled from some simple substructures by different kinds of connectors, which include various kinds of joints and hinges. Most of the connectors have properties of nonlinearity, and can strongly affect the dynamic characteristics of spacecraft. Mathematical models of such spacecraft usually have a large number of degrees of freedom (DOFs), but their nonlinear connectors are generally spatially localized. In general, it is impractical and time-consuming to directly calculate the frequency response of the spacecraft using current methods. To enhance the calculation efficiency of the frequency response, an improved approach is proposed in the present paper. With describing functions (DFs) and linear receptance data, the kinetic equations are firstly converted into a set of complex algebraic equations whose dimension is only associated with nonlinear DOFs and interested DOFs. Subsequently the number of iterative equations is reduced and only related to nonlinear DOFs. Hence the improved approach can be applicable to large-scale and complicated finite element (FE) models. An FE model of a satellite with some nonlinear joints is used to show and demonstrate the usefulness of the proposed method. Besides, the effects induced by nonlinear joints on payloads’ vibration of the satellite are discussed.