Projection Equations Research Articles

Computer Program for the Inverse Transformation of the Winkel Projection

The map projection problem involves transforming the graticule of meridians and parallels of a sphere onto a plane using a specified mathematical method according to certain conditions. Map projection transformations are a research field dealing with the method of transforming one kind of map projection coordinates to another. The conversion from geographical to plane coordinates is the normal practice in cartography, which is called forward transformation. The inverse transformation, which yields geographical coordinates from map coordinates, is a more recent development due to the need for transformation between different map projections, especially in Geographic Information Systems (GIS). The direct inverse equations for most of the map projections are already in existence, but for the projections, which have complex functions for forward transformation, defining the inverse projection is not easy. This paper describes an iteration algorithm to derive the inverse equations of the Winkel tripel projection using the Newton–Raphson iteration method.

A method for constructing exact solutions and application to Benjamin Ono equation

By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.

Solving semidefinite programming problems via alternating direction methods

In this paper, we consider an alternating direction algorithm for the solution of semidefinite programming problems (SDP). The main idea of our algorithm is that we reformulate the complementary conditions in the primal–dual optimality conditions as a projection equation. By using this reformulation, we only need to make one projection and solve a linear system of equation with reduced dimension in each iterate. We prove that the generated sequence converges to the solution of the SDP under weak conditions.

EXACT SOLITON SOLUTIONS FOR THE HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION

Based on the complex envelope ansatz method, the projective Riccati equation method and q-deformed hyperbolic functions, a method is developed for constructing a series of exact analytical solutions for higher-order nonlinear Schrödinger (HNLS) equation, which describes propagation of femtosecond light pulse in optical fiber under certain parametric conditions. With the help of symbolic computation, six families of new solitary wave solutions are obtained. The solitary wave solutions obtained by Li et al.18 are special cases of our solutions. The novel soliton solutions can describe W-shaped, bright and dark soliton properties in the same expression and their amplitude may approach nonzero when the time variable approaches infinity. Furthermore, the soliton propagation and solitons interaction scenario are discussed and simulated by computer.

IDEALS OF MINORS DEFINING GENERIC SINGULARITIES AND THEIR GRöBNER BASES

In this article, using the local parametric equations of a generic projection π of a smooth projective variety X, at an analytically irreducible singular point y of X′ = π(X), the defining ideals J and J′ of X′ and its singular locus at y are expressed as ideals of maximal and sub-maximal minors of certain Sylvester matrix @. The proof is obtained by a convenient reduction of @ to a “generic pluri-circulant matrix” P and the construction of minimal Gröbner bases for the ideal of t-minors of P and for the ideals J and J′. The depth of local rings of X′ and Sing (X′) at y are also computed in terms of the multiplicity at y.

A Delayed Neural Network for Solving Linear Projection Equations and its Analysis

In this paper, we present a delayed neural network approach to solve linear projection equations. The Lyapunov-Krasovskii theory for functional differential equations and the linear matrix inequality (LMI) approach are employed to analyze the global asymptotic stability and global exponential stability of the delayed neural network. Compared with the existing linear projection neural network, theoretical results and illustrative examples show that the delayed neural network can effectively solve a class of linear projection equations and some quadratic programming problems.

New expansion algorithm of three Riccati equations and its applications in nonlinear mathematical physics equations

Based on the study of tanh function method and the coupled projective Riccati equation method, we propose a new algorithm to search for explicit exact solutions of nonlinear evolution equations. We use the higher-order Schrodinger equation and mKdV equation to illustrate this algorithm. As a result, more new solutions are obtained, which include new solitary solutions, periodic solutions and singular solutions. Some new solutions are illustrated in figures.

Relationship Between Soliton-like Solutions and Soliton Solutionsto a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series ofdoubly periodic solutions (Jacobi elliptic function solution) for a classof nonlinear partial differential equations are obtained in aunified way. When the module m→1, these solutions exactlydegenerate to the soliton solutions of the equations. Then wereveal the relationship between the soliton-like solutionsobtained by other authors and these soliton solutions of theequations.

Basal Area and Diameter Distribution Prediction Equations for Young Douglas-Fir Plantations with Hardwood Competition: Coast Ranges

Abstract A stand-level modeling system for young Douglas-fir plantations applicable to the Coast Ranges of the Pacific Northwest was developed by using data from 84 growth-monitoring plots. Specific components of the system for predicting current size distributions of planted Douglas-fir include: dominant height and survival of planted Douglas-fir; a basal area projection equation for hardwoods; and diameter distribution percentile prediction functions for planted Douglas-fir. Prediction equations for height, crown width, and height to crown base of individual trees describe stand structure of the planted Douglas-fir. The stand- and tree-level equations include both direct and indirect effects of hardwood competition and site preparation, competition release, and precommercial thinning treatments. With the exception of minimum dbh, stand-level equations for planted Douglas-fir explained at least 92% of the observed variation. Basal area dynamics were simulated through plantation age 20 years for site preparation and competition release treatments of different effectiveness by varying hardwood basal area at plantation age 5 years. The simulations indicate that site preparation must reduce hardwood basal area at plantation age 5 years by 75% or more to realize an increase in Douglas-fir basal area and shift of the diameter distribution toward larger trees. This is due to increased growth of hardwoods that survive site preparation. In contrast, competition release treatments exhibited positive Douglas-fir response with 50% efficacy or more. Therefore, ineffective site preparation treatments need to be followed with competition release treatments having at least 50% efficacy to produce similar Douglas-fir basal area and diameter distributions at age 20 as in stands with no site preparation. West. J. Appl. For. 20(2):77–93.

New exact solutions to the generalized coupled Hirota–Satsuma Kdv system☆

Abstract In this paper, the projective Riccati equations method is improved. The improved method can be applied to find more and new exact solutions to the nonlinear evolution equations with the aid of symbolic computation system like Maple. Some of the solutions are found for the first time.

Exact travelling wave solutions of nonlinear evolution equations in [formula omitted] and [formula omitted] dimensions

Based upon the symbolic computation system Maple, the projective Riccati equation method is improved to seek for travelling wave solutions in ( 1 + 1 ) and ( 2 + 1 ) -dimensional nonlinear evolution equations by introducing two new elementary bell-shaped and kink-shaped functions. The validity and reliability of the method is tested by applying it to the ( 2 + 1 ) -dimensional generalized of the shallow water wave equation, potential KdV equation and Burgers–KdV equation.

Exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres

First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed.

New soliton-like and periodic-like solutions for the KdV equation

In this paper based on a system of Riccati equations, a newly generally projective Riccati equation expansion method is presented. It is direct and more powerful than the tanh-function method, generally projective Riccati equation method and can be used to construct more new exact solutions of nonlinear differential equations in mathematical physics. The well-known KdV equation is chosen to illustrate the algorithm such that more families of new explicit solutions are obtained, which contain soliton-like and periodic-like solutions. This algorithm can also be applied to many other nonlinear differential equations.

Extended projective Riccati equations method and its application

The projective Riccati equations method is extended and applied to find new exac t solutions to the Whitham-Broer-Kaup(WBK) equation with the aid of symbolic com putation system Maple.

The stochastic soliton-like solutions of stochastic mKdV equations

In this paper, the generally projective Riccati equations method is improved by means of a generalized transformation. The improved method can be applied to find not only some exact travelling wave solutions but also some soliton-like solutions with the aid of symbolic computation system — Maple. We choose Wick-type stochastic mKdV equations to illustrate the method. As a result, some stochastic soliton-like solutions are obtained.

Exact Soliton Solutions To An Averaged Dispersion-Managed Fiber System Equation

By introducing a set of ordinary differential equations which possess q-deformed hyperbolic function solutions, and a new ansatz, a method is developed for constructing a series of exact analytical solutions of some nonlinear evolution equations. The proposed method is more powerful than various tanh methods, the secq-tanhq-method, generalized hyperbolic-function method, generalized Riccati equation expansion method, generalized projective Riccati equations method and other sophisticated methods. As an application of the method, an averaged dispersion-managed (DM) fiber system equation, which governs the dynamics of the core of the DM soliton, is chosen to illustrate the method. With the help of symbolic computation, rich new soliton solutions are obtained. From these solutions, some previously known solutions obtained by some authors can be recovered by means of some suitable choices of the arbitrary functions and arbitrary constants. Further, the soliton propagation and solitons interaction scenario are discussed and simulated by computer.

New Soliton-like Solutions to Variable Coefficients MKdV Equation**The project supported by the State Key Basic Research Development Program of China under Grant No. 1998030600 and National Natural Science Foundation of China under Grant No. 10072013

A further improved projective Riccati equation method is proposed. By applying it to solve the variable coefficients MKdV equation, we obtain many new-type soliton-like solutions to this equation.

Multiresolution Modeling and Estimation of Multisensor Data

This paper presents a multiresolution multisensor data fusion scheme for dynamic systems to be observed by several sensors of different resolutions. A state projection equation is introduced to associate the states of a system at each resolution with others. This projection equation together with the state transition equation and the measurement equations at each of the resolutions construct a continuous-time model of the system. The model meets the requirements of Kalman filtering. In discrete time, the state transition is described at the finest resolution and the sampling frequencies of sensors decrease successively by a factor of two in resolution. We can build a discrete model of the system by using a linear projection operator to approximate the state space projection. This discrete model satisfies the requirements of discrete Kalman filtering, which actually offers an optimal estimation algorithm of the system. In time-invariant case, the stability of the Kalman filter is analyzed and a sufficient condition for the filtering stability is given. A Markov-process-based example is given to illustrate and evaluate the proposed scheme of multiresolution modeling and estimation with multiple sensors.

Using symbolic computation to construct travelling wave solutions to nonlinear partial differential equations

Based upon the symbolic computation and the coupled projective Riccati equation, the tanh function method is further improved. As its applications, Wu–Zhang equation (which describes a (2+1)-dimensional dispersive long wave) and the (1+1)-dimensional dispersive long wave equation obtained from Wu–Zhang equation by scaling transformation and symmetry reduction are chosen to illustrate the validity of the proposed approach.

New families of non-travelling wave solutions to the (2+1)-dimensional modified dispersive water-wave system

In this paper, we introduce a further generalized projective Riccati equation method and apply it to solve the (2+1)-dimensional modified dispersive water-wave system. Many new types of non-travelling wave solutions are obtained for this system.