Nonlinear Equations

2001 ◽  
Author(s):  
Krzysztof A. Sikorski
Keyword(s):  
Nonlinear Equations ◽  
Nonlinear Function ◽  
Multivariate Problems ◽  
Univariate Function

In this chapter we address the problem of approximationg zeros ∝ of nonlinear function f, f (∝ ) = 0, where f ϵ F ⊂ {f : D ⊂ Rd →Rl}. In order to define our solution operators, we first review several error criteria that are commonly used to measure the quality of approximations to zeros of nonlinear equations. This is done for univariate function f : [a, b] → R. Straightforward generalizations to the mulivariate case are based on replacing the magnitude function by a specific norm. These are considered in section 2.2 when we review multivariate problems. A number of error criteria are used in practice for approximation of a zero ∝ of f.

Model Order Identification of Combustion Instability Using Lipschitz Indices

2019 ◽  
Author(s):  
Salil Harris ◽  
Aniruddha Sinha ◽  
Sudarshan Kumar
Keyword(s):  
Active Control ◽  
Combustion Instability ◽  
Nonlinear Function ◽  
Model Structure ◽  
Control Methods ◽  
Nonlinear Functions ◽  
Lean Premixed Combustion ◽  
Input And Output ◽  
Inputs And Outputs

Abstract Gas turbine combustors employing lean premixed combustion are prone to combustion instability. Combustion instability, if unchecked, will have deleterious effects to the combustor and hence needs to be controlled. Active control methods are preferred to obtain better off-design performance. The effectiveness of active control methods is dependent on the quality of controller which in-turn depends on the quality of model. In the present work, an input-output model structure, where the output of the system at the current instant is modelled as a nonlinear function of delayed inputs and outputs is chosen. As there are infinite possibilities for representation of nonlinear functions, all parameters in the model structure like time delay between input and output, number of delayed input and output terms and the appropriate form of nonlinear function can be obtained only iteratively. However, prior knowledge of delay and number of delayed inputs and outputs reduces the computational intensity. To this end, the present work utilizes the method of Lipschitz indices to obtain the number of delayed inputs and outputs.

Profiling Rotors for Limaçon-to-Limaçon Compression-Expansion Machines

2005 ◽  
Vol 128 (4) ◽  
pp. 787-793 ◽  
Author(s):  
Ibrahim A. Sultan
Keyword(s):  
Nonlinear Equations ◽  
Volumetric Efficiency ◽  
Radial Clearance ◽  
A Value ◽  
First Case ◽  
Rotor Profile ◽  
Mathematical Expressions ◽  
Second Mode ◽  
Extreme Care

Limaçon-to-limaçon compression-expansion machines have housings and rotors whose profiles are manufactured of limaçon curves. For these machines to perform satisfactorily, extreme care should be given to the geometric particulars of their rotor profile. The main characteristics that govern the quality of the rotor profile are the volumetric efficiency and the prevention of interference. In this work, the interference problem is handled from two different mathematical standpoints: the slope of tangents to both the rotor and housing curves at the apices; and the value of the minimum radial clearance that separates the two limaçon curves. In the first case, mathematical expressions, relating the radii of the limaçon base circles is presented to ensure that interference would not take place during normal operations of the limaçon-to-limaçon machine. The second mode of analysis produces a set of nonlinear equations that can be solved to obtain a value of the radial clearance. This value has to be machined off the rotor profile to prevent interference. A numerical example is given at the end of the paper to prove the validity of the models proposed and graphs are produced to support the claims presented.

scholarly journals An Efficient and Straightforward Numerical Technique Coupled to Classical Newton’s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Grégory Antoni
Keyword(s):  
Approximate Solution ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Input Data ◽  
Numerical Technique ◽  
Nonlinear Function ◽  
Numerical Procedure ◽  
Approximate Solutions ◽  
First Order

This study concerns the development of a straightforward numerical technique associated with Classical Newton’s Method for providing a more accurate approximate solution of scalar nonlinear equations. The proposed procedure is based on some practical geometric rules and requires the knowledge of the local slope of the curve representing the considered nonlinear function. Therefore, this new technique uses, only as input data, the first-order derivative of the nonlinear equation in question. The relevance of this numerical procedure is tested, evaluated, and discussed through some examples.

scholarly journals Nonlinear Inertia Classification Model and Application

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mei Wang ◽  
Pai Wang ◽  
Jzau-Sheng Lin ◽  
Xiaowei Li ◽  
Xuebin Qin
Keyword(s):  
Search Algorithm ◽  
Great Influence ◽  
Nonlinear Function ◽  
Classification Model ◽  
Support Vector ◽  
Power Cable ◽  
Optimal Position ◽  
Kernel Parameter ◽  
Svm Model

Classification model of support vector machine (SVM) overcomes the problem of a big number of samples. But the kernel parameter and the punishment factor have great influence on the quality of SVM model. Particle swarm optimization (PSO) is an evolutionary search algorithm based on the swarm intelligence, which is suitable for parameter optimization. Accordingly, a nonlinear inertia convergence classification model (NICCM) is proposed after the nonlinear inertia convergence (NICPSO) is developed in this paper. The velocity of NICPSO is firstly defined as the weighted velocity of the inertia PSO, and the inertia factor is selected to be a nonlinear function. NICPSO is used to optimize the kernel parameter and a punishment factor of SVM. Then, NICCM classifier is trained by using the optical punishment factor and the optical kernel parameter that comes from the optimal particle. Finally, NICCM is applied to the classification of the normal state and fault states of online power cable. It is experimentally proved that the iteration number for the proposed NICPSO to reach the optimal position decreases from 15 to 5 compared with PSO; the training duration is decreased by 0.0052 s and the recognition precision is increased by 4.12% compared with SVM.

Some Analytical Approximations and Numerical Solutions of EHL Problems for Non-Newtonian Fluids Fatigue

2005 ◽  
Author(s):  
Ilya I. Kudish
Keyword(s):  
Shear Stress ◽  
Shear Strain ◽  
Nonlinear Equations ◽  
Numerical Solutions ◽  
Nonlinear Function ◽  
Shear Stresses ◽  
Rolling Shear ◽  
Analytical Approximations ◽  
Problem Formulations ◽  
Generalized Reynolds Equation

Plane steady isothermal and non-isothermal EHL problems for non-Newtonian lubricants in a line contact are considered. Two cases of lubricant rheologies are studied. Namely, lubricants for which (a) the shear stress is any explicitly given nonlinear function of the shear strain and (b) the shear strain is any explicitly given nonlinear function of the shear stress are analyzed. The isothermal EHL problem is reduced to solution of a nonlinear equation for the sliding shear stress f, generalized Reynolds equation for pressure p, and the equation for gap h as well as the classic balance and boundary conditions. The non-isothermal EHL problem is reduced to solution of the above mentioned nonlinear equations and the nonlinear equations for temperatures of the lubricant T and of the contact surfaces Tw1 and Tw2. The EHL problems are considered in the case of heavily loaded contact when the rolling shear stress in lubricant is much smaller than the lubricant sliding shear stress [1–7]. Therefore, the problems contain a small parameter represented by the ratio of the characteristic rolling and sliding shear stresses. That leads to the opportunity to use the perturbation methods for simplifying the EHL problem formulations.

scholarly journals Application of Traditional Cultural Symbols in Art Design under the Background of Artificial Intelligence

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Cuifang Lin
Keyword(s):  
Neural Network ◽  
Artificial Intelligence ◽  
Nonlinear Function ◽  
Fractal Model ◽  
Nonlinear Transformations ◽  
Cultural Symbols ◽  
Fractal Models ◽  
Neural Network Algorithm ◽  
Art Design

In order to solve the declining influence of traditional cultural symbols, the research on traditional cultural symbols has become more meaningful. This article aims to study the application of traditional cultural symbols in art design under the background of artificial intelligence. In this paper, a fractal model with self-combined nonlinear function changes is constructed. By combining nonlinear transformations and multiparameter adjustments, various types of fractal models can be automatically rendered. The convolutional neural network algorithm is used to extract the characteristics of the style picture, and it is compared with the trained picture many times to avoid the problem of excessive tendency of the image with improper weight. The improved L-BFGS algorithm is also used to optimize the loss of the traditional L-BFGS, which improves the quality of the generated pictures and reduces the noise of the chessboard. The experimental results in this paper show that the improved L-BFGS algorithm has the least loss and the shortest time in the time used for more than 500 s. Compared with the traditional AdaGrad method, its loss is reduced by about 62%; compared with the traditional AdaDelta method, its loss is reduced by 46%. Its loss is reduced by about 8% compared with the newly optimized Adam method, which is a great improvement.

scholarly journals Mathematical modeling in the Urumari micro-watershed using Streeter-Phelps mathematical models and the enhanced Do-Bod model

2020 ◽  
Vol 11 (6) ◽  
pp. 332-345
Author(s):  
Lucinewton Silva de Moura ◽  
Ruy Bessa Lopes ◽  
Joseph Simões Ribeiro ◽  
Graciene Do Socorro Taveira Fernandes ◽  
Rodolfo Maduro Almeida ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Experimental Data ◽  
Water Quality ◽  
Mathematical Models ◽  
Surface Waters ◽  
Nonlinear Function ◽  
Absolute Deviation ◽  
Recovery Mechanism ◽  
Photosynthesis And Respiration

When a certain load of pollutants is dumped in urban surface waters, the natural hydrological parameters are altered and a consequent dispersion of a contaminant occurs. Water self-purification is a natural recovery mechanism that aims to restore the natural parameters of water quality. In the municipality of Santarém, the Urumari micro-watershed, one of the main urban streams and a tributary of the Amazon river, suffers from increasing degradation. Thus, it is necessary to systematically monitor the water quality of the micro-watershed, through mathematical modeling, in order to identify environmental impairment and propose mitigating tools that facilitate self-purification capacity. The aim of this work was to evaluate the degree of disturbance of the Urumari micro-watershed using Streeter-Phelps mathematical models and the Enhanced DO-BOD model. The software Curve Expert 1.4 was used to adjust the coefficients of the mathematical models, using the Levenberg-Marquardt nonlinear function adjustment method. The complete mathematical model of DO and BOD considered the parcels of punctual deficit, microbial BOD, nitrification, sediment demand, photosynthesis and respiration. Three coefficients of the Streeter-Phelps model and six coefficients of the Enhanced DO-BOD model were adjusted according to the experimental data of dissolved oxygen and water travel time. The mathematical models satisfactorily correlated the experimental data. The values of the correlation coefficient, mean absolute deviations and quadratic deviations are for the Streeter-Phelps model and the Enhanced DO-BOD model, respectively. The Streeter-Phelps model presented the best correlation to the experimental data with the values of R2 equal to 0.83; mean absolute deviation equal to 0.56 and quadratic deviations equal to 4.98.

scholarly journals A Backstepping Approach to Ship Course Control

2007 ◽  
Vol 17 (1) ◽  
pp. 73-85 ◽  
Author(s):  
Anna Witkowska ◽  
Miroslaw Tomera ◽  
Roman Śmierzchalski
Keyword(s):  
Control System ◽  
Nonlinear Function ◽  
Control Algorithms ◽  
Backstepping Method ◽  
Control Structures ◽  
Nonlinear Controllers ◽  
Backstepping Approach ◽  
Course Control ◽  
The Mathematical Model

A Backstepping Approach to Ship Course ControlAs an object of course control, the ship is characterised by a nonlinear function describing static manoeuvring characteristics that reflect the steady-state relation between the rudder deflection and the rate of turn of the hull. One of the methods which can be used for designing a nonlinear ship course controller is the backstepping method. It is used here for designing two configurations of nonlinear controllers, which are then applied to ship course control. The parameters of the obtained nonlinear control structures are tuned to optimise the operation of the control system. The optimisation is performed using genetic algorithms. The quality of operation of the designed control algorithms is checked in simulation tests performed on the mathematical model of a tanker. In order to obtain reference results to be used for comparison with those recorded for nonlinear controllers designed using the backstepping method, a control system with the PD controller is examined as well.

Immunoferritin localization of intracellular antigens in positively stained frozen sections

1978 ◽  
Vol 36 (2) ◽  
pp. 168-169
Author(s):  
K. T. Tokuyasu
Keyword(s):  
Surface Tension ◽  
Water Surface ◽  
Thin Layers ◽  
The Other ◽  
Positive Staining ◽  
Frozen Sections ◽  
The Past ◽  
Ultrathin Frozen Sections ◽  
Definition Of

During the past investigations of immunoferritin localization of intracellular antigens in ultrathin frozen sections, we found that the degree of negative staining required to delineate u1trastructural details was often too dense for the recognition of ferritin particles. The quality of positive staining of ultrathin frozen sections, on the other hand, has generally been far inferior to that attainable in conventional plastic embedded sections, particularly in the definition of membranes. As we discussed before, a main cause of this difficulty seemed to be the vulnerability of frozen sections to the damaging effects of air-water surface tension at the time of drying of the sections.Indeed, we found that the quality of positive staining is greatly improved when positively stained frozen sections are protected against the effects of surface tension by embedding them in thin layers of mechanically stable materials at the time of drying (unpublished).

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