Projection Equations Research Articles

Design and Optimisation of a Simple Filter Group for Reactive Power Distribution

Basic methods are presented to design a simple filter group and a method of shaping the resultant of the filter group’s impedance characteristics (distribution of the characteristics’ extremes) and then project equations were transformed into a uniform, common form that addresses issues of the reactive power distribution between component filters. The analysis also takes into account the filters’ detuning from the reduced harmonics and quality factors of passive elements. Another important factor of the analysis considered was the power grid equivalent impedance affecting the filtration effectiveness. A criterion for the filter group’s filtration effectiveness evaluation was proposed and optimisation was completed for the reactive power distribution between separate filters in the function of the power grid’s equivalent inductance.

Group classification of projective type second-order ordinary differential equations

Group classification with respect to admitted point transformation groups is carried out for second-order ordinary differential equations with cubic nonlinearity of the first-order derivative. The result is obtained with use of the invariants of the equivalence transformation group of the family of equations under consideration. The corresponding Riemannian metric is found for the equations that are the projection of the system of geodesics to a two-dimensional surface.

A Novel Neural Network for Generally Constrained Variational Inequalities.

This paper presents a novel neural network for solving generally constrained variational inequality problems by constructing a system of double projection equations. By defining proper convex energy functions, the proposed neural network is proved to be stable in the sense of Lyapunov and converges to an exact solution of the original problem for any starting point under the weaker cocoercivity condition or the monotonicity condition of the gradient mapping on the linear equation set. Furthermore, two sufficient conditions are provided to ensure the stability of the proposed neural network for a special case. The proposed model overcomes some shortcomings of existing continuous-time neural networks for constrained variational inequality, and its stability only requires some monotonicity conditions of the underlying mapping and the concavity of nonlinear inequality constraints on the equation set. The validity and transient behavior of the proposed neural network are demonstrated by some simulation results.

Approach for home appliance recognition using vector projection length and Stockwell transform

Recognition of residential electrical appliances has been a highlighted area of research in recent years due to increasing energy consumption in residential sectors around the world. The use of instant current from 17 commonly presented electrical appliances in homes is proposed. The Stockwell transform, a time–frequency-domain transform, is applied to current collected data to be used as a load signature. Then, a novel method for identification of electrical appliances was employed to analyse the data. This method is an image pattern recognition technique, called vector projection length (VPC). In this case, the idea is to recognise patterns in appliances transformed into a 2D image. An extensive analysis is presented, analysing the trade-off between the training set database, the number of current cycles used and the variable k from the VPC projection equation, searching for optimal parameters for this application. Results have shown successful identification employing VPC, close to 90% of devices being identified.

Caution with respect to “new” variable separation solutions and their corresponding localized structures

Caution is advised regarding so-called “new” variable separation solutions obtained by the generalizing Riccati equation mapping method and the improved projective Riccati equation method because many seemingly independent variable separation solutions actually depend on each other. To illustrate this point, we employ the (2+1)-dimensional Bogoyavlenskii–Schiff model as an example and we derive 10 different variable separation solutions. Based on a detailed investigation, we show that many of the so-called “new” solutions are equivalent to each other. Thus, when we discuss localized structures based on variable separation solution, we should consider all of the field components to avoid the appearance of false unphysical related structures, where seemingly abundant structures are obtained for a special component, while unphysical (non-localized) or even divergent structures may appear for other corresponding components of the same equation.

Sigma-Model Solitons on Noncommutative Spaces

We use results from time-frequency analysis and Gabor analysis to construct new classes of sigma-model solitons over the Moyal plane and over noncommutative tori, taken as source spaces, with a target space made of two points. A natural action functional leads to self-duality equations for projections in the source algebra. Solutions, having non-trivial topological content, are constructed via suitable Morita duality bimodules.

Quasi indicatrix approach for distortion visualization and analysis for map projections

Tissot’s indicatrix or ellipse of distortion is a diagram that is the projection of an infinitesimal circle on the original surface. It is normally an ellipse of which elongation depends on the amount of distortion caused by map projection. It provides a medium for analyzing existing projections and developing new ones. The ellipse can be scaled and depicted on the map for visualization purposes. This paper presents an alternative approach, in which the projection of a finite small circle on the sphere is used. Its projection is normally an ellipse that can be very close to Tissot’s indicatrix, and is called quasi indicatrix, here. Its parameters can be derived from the forward projection equations without using partial derivatives. Therefore, it is a useful and practical approach from a programmer’s point of view. The quasi indicatrix approach is also numerically tested on Aitoff–Hammer projection with a set of points. The indicatrix parameters obtained by using this approach deviate 0.5% from the ground truths at most, being the average less than 0.2%.

English

Microtubules (MTs) are major cytoskeletal proteins. They are hollow cylinders formed by protofilaments (PFs) representing series of proteins known as tubulin dimers. Each dimer is an electric dipole. These diamers are in a straight position within PFs or in radially displaced positions pointing out of cylindrical surface. In this paper, the authors demonstrate how the generalized projective Riccati equations method can be used in the study of the nonlinear dynamics of MTs. To this end, the authors apply this method to construct the exact solutions with parameters for two nonlinear PDEs describing MTs. The first equation describes the model of microtubules as nanobioelectronics transmission lines. The second equation describes the dynamics of radial dislocations in microtubules. As a result, hyperbolic, trigonometric and rational function solutions are obtained. When these parameters are taken as special values, solitary wave solutions are derived from the exact solutions. Comparison between our recent results and the well-known results is given. Key words: Generalized projective Riccati equations method, models of microtubules (MTs), exact solutions, solitary solutions, trigonometric solutions rational solutions.

On fuzzy linear projection equation and applications

This paper introduces the concept of fuzzy projection of a fuzzy number on a set of fuzzy numbers based on r-cut approach. It is proved that the projection of a fuzzy number on the set of all fuzzy...

Self-calibration of vision parameters via genetic algorithms with simulated binary crossover and laser line projection

A self-calibration technique based on genetic algorithms (GAs) with simulated binary crossover (SBX) and laser line imaging is presented. In this technique, the GA determines the vision parameters based on perspective projection geometry. The GA is constructed by means of an objective function, which is deduced from the equations of the laser line projection. To minimize the objective function, the GA performs a recombination of chromosomes through the SBX. This procedure provides the vision parameters, which are represented as chromosomes. The approach of the proposed GA is to achieve calibration and recalibration without external references and physical measurements. Thus, limitations caused by the missing of references are overcome to make self-calibration and three-dimensional (3-D) vision. Therefore, the proposed technique improves the self-calibration obtained by GAs with references. Additionally, 3-D vision is carried out via laser line position and vision parameters. The contribution of the proposed method is elucidated based on the accuracy of the self-calibration, which is performed with GAs.

An efficient projection neural network for solving bilinear programming problems

In this paper the application of projection neural network for solving bilinear programming problems (BLPs) is obtained. So far as we know, no study has yet been attempted for these problems via neural network. In fact, some interesting reformulations of BLP and mixed-integer bilinear programming problem (MIBLP) with a binary vector to linear complementarity problem (LCP) are given. Additionally, we show that the special type of MIBLP with a binary vector is equal to a quadratic program and on the other hand, it is equal to a mixed-integer linear program (MILP). Finally, we use a neural network to solve projection equation which has the same solution with LCP. Then, by presenting a Lyapunov function, we show that the proposed neural network is globally asymptotically stable. Illustrative examples are given to show the effectiveness and efficiency of our method.

Research and application of forward imaging technology with systemic blur

Forward imaging technique is the base of the combined method for density reconstruction with forward calculation and inverse problem solution. This paper proposes the idea of girding the areal source as many point sources. Using the blurring window as the affecting factor of the detector blur, the projecting matrix from any point source to any detector pixel with X-ray trace technique is obtained, the projection equation for the radiographic system with areal source blur and detector blur is introduced. The projection equation is used to gain the same deviation information about the object edge as the experimental image. The forward projection equation is combined with the constrained conjugate gradient method to form a full procedure of density reconstruction, which is applied on a simulated image of French Test Object and an experimental image. The results show that using the projection equation, the affecting range of the blur is decreased and can be controlled in one or two pixels.

Lie algebraic similarity transformed Hamiltonians for lattice model systems

We present a class of Lie algebraic similarity transformations generated by exponentials of two-body on-site hermitian operators whose Hausdorff series can be summed exactly without truncation. The correlators are defined over the entire lattice and include the Gutzwiller factor $n_{i\uparrow}n_{i\downarrow}$, and two-site products of density $(n_{i\uparrow} + n_{i\downarrow})$ and spin $(n_{i\uparrow}-n_{i\downarrow})$ operators. The resulting non-hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with polynomial computational cost. The proposed similarity transformation generates locally weighted orbital transformations of the reference determinant. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the 1D and 2D repulsive Hubbard model where we find accurate results across all interaction strengths.

Dynamic Modeling and Force Control of a Redundant Robot for Polishing Applications

AbstractFor many robotic applications with tasks such as cutting, assembly or polishing, it is necessary to get in contact with the surrounding. In this paper a redundant robot with seven degrees of freedom in a metal polishing task is considered. For simulation as well as for the controller design a dynamic model of the robot and a contact model are required. The equations of motion of the robot are calculated with the Projection Equation in subsystem representation and the contact model contains linear tool elasticities and work piece elasticities. In the case of a polishing task, a constant contact force during the process is required even if the robot moves along a trajectory. Thus some degrees of freedom of the robot tool center point have to be position controlled while the other ones have to be force controlled. The redundant robot offers the possibility to avoid singular positions or to maximize the available end‐effector forces within the inverse kinematics and is therefore best suited for polishing large objects. The actual process forces are measured with a six axis force‐torque‐sensor mounted at the tool center point. These forces are used in a parallel force/position control law to achieve the desired behavior. Results from measurements of a test arrangement are presented. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Projective Stochastic Equations and Nonlinear Long Memory

A projective moving average {Xt, t ∈ ℤ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of Xt on ‘intermediate’ lagged innovation subspaces with given coefficients αi and βi,j. The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution Xt. We show that, under certain conditions on Q, αi, and βi,j, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

Projective Stochastic Equations and Nonlinear Long Memory

A projective moving average {X t , t ∈ ℤ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of X t on ‘intermediate’ lagged innovation subspaces with given coefficients α i and β i,j . The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution X t . We show that, under certain conditions on Q, α i , and β i,j , this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

Acceleration-Level Inequality-Based MAN Scheme for Obstacle Avoidance of Redundant Robot Manipulators

In this paper, a new inequality-based criterion is proposed and investigated for obstacle avoidance of redundant robot manipulators at the joint-acceleration level. By incorporating such a dynamically updated inequality criterion and the joint physical constraints (i.e., joint-angle limits, joint-velocity limits, and joint-acceleration limits), a novel minimum-acceleration-norm (MAN) scheme is presented and investigated for robots' redundancy resolution. In addition, the resultant obstacle-avoidance MAN scheme resolved at the joint-acceleration level is reformulated as a general quadratic program (QP). Moreover, two important “Bridge” theorems are established, which show that such a QP problem can be equivalent to linear variational inequality (LVI) and then to piecewise-linear projection equation (PLPE). An LVI-based numerical method is thus developed and applied for online solution of the QP problem and the inequality-based obstacle-avoidance MAN scheme. Simulative results based on the PA10 robot manipulator in the presence of window-shaped and point obstacles further demonstrate the efficacy and superiority of the proposed acceleration-level inequality-based MAN scheme for obstacle avoidance of redundant robot manipulators. In addition, experimental verification conducted on a practical six-link planar robot manipulator substantiates the effectiveness and physical realizability of the proposed obstacle-avoidance scheme.

Time Optimal Path Planning for Industrial Robots: A Dynamic Programming Approach Considering Torque Derivative and Jerk Constraints

AbstractThis paper presents a dynamic programming approach for calculating time optimal trajectories for industrial robots, subject to various physical constraints. In addition to path velocity, motor torque, joint velocity and acceleration constraints, the present contribution also shows how to deal with torque derivative and joint jerk limitations. First a Cartesian path for the endeffector is defined by splines using Bernstein polynomials as basis functions and is parameterized via a scalar path parameter. In order to compute the belonging quantities in configuration space, inverse kinematics is solved numerically. Using this and in combination with the dynamical model, joint torques as well as their derivatives can be constrained. For that purpose the equations of motion are calculated with the help of the Projection Equation. As a consequence of the used optimization problem formulation, the dynamical model as well as the restrictions have to be transformed to path parameter space. Due to the additional consideration of jerk and torque derivative constraints, the phase plane is expanded to a phase space. The parameterized restrictions lead to feasible regions in this space, in which the optimal solution is sought. Result of the optimization is the time behavior of the path parameter and subsequently the feed forward torques for the optimal motion on the spatial path defined by previously mentioned splines. Simulation results as well as experimental results for a three axes industrial robot are presented. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Posterior Model Reduction Method Based on Weighted Residue for Linear Partial Differential Equations

This paper is concerned with a new model reduced method based on optimal large truncated low-dimensional dynamical system, by which the solution of linear partial differential equation (PDE) is able to be approximate with highly accuracy. The method proposed is based on the weighted residue of PDE under consideration, and the weighted residue is used as an alternative optimal control condition (POT-WR) while solving the PDE. A set of bases is constructed to describe a dynamical system required in case. The Lagrangian multiplier is introduced to eliminate the constraints of the Galerkin projection equation, and the penalty function is used to remove the orthogonal constraint. According to the extreme principle, a set of the ordinary differential equations is obtained by taking the variational operation on generalized optimal function. A conjugate gradients algorithm on FORTRAN code is developed to solve these ordinary differential equations with Fourier polynomials as the initial bases for iterations. The heat transfer equation under a potential initial condition is used to verify the method proposed. Good agreement between the simulations and the analytical solutions of example was obtained, indicating that the POT-WR method presented in this paper provides the most effective posterior way of capturing the dominant characteristics of an infinite-dimensional dynamical system with only finitely few bases.

The projective translation equation and unramified 2-dimensional flows with rational vector fields

Let x = (x, y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1−z)ϕ(x) = ϕ(ϕ(xz)(1−z)/z); here ϕ(x) = (u(x,y),v(x,y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in \({\mathbb{C}^{2}\setminus\{\text{union of curves}\}}\)) projective flows whose vector field is still rational. If an orbit of the flow is given by a homogeneous rational function of degree N, then N is called the level of the flow. We prove that, up to conjugation with 1-homogenic birational plane transformation, unramified non-singular flows are of 6 types: (1) the identity flow; (2) one flow for each non-negative integer N—these flows are rational of level N; (3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; (4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; (5) the level 4 flow expressable in terms of lemniscatic elliptic functions; (6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Pólya–Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus. This study, though seemingly analytic, is in fact algebraic and our method reduces to algebraic transformations in various quotient rings of rational function fields.