Using variable-entered Karnaugh maps to produce compact parametric general solutions of Boolean equations

The variable-entered Karnaugh map (VEKM) is shown to be the natural map for representing and manipulating general ‘big’ Boolean functions that are not restricted to the switching or two-valued case. The VEKM is utilized herein in producing a compact general solution of a system of Boolean equations. It serves as a powerful manual tool for function inversion, implementation of the solution procedure, handling don't-care conditions and minimization of the final expressions. The rules of using the VEKM are semi-algebraic and collective in nature, and hence are much easier to state, remember and implement than are the tabular and per-cell rules of classical maps. As a result, the maps used are significantly smaller than those required by classical methods. As an offshoot, the paper contributes some pictorial insight into the representation of ‘big’ Boolean algebras and functions. It also predicts the correct number of particular solutions of a Boolean equation, and produces an exhaustive list of particular solutions. Details of the method are carefully explained and further demonstrated via an illustrative example.     

Boolean sets and most general solutions of Boolean equations

The aim of this paper is twofold. First we determine the most general form of the subsumptive general solution of a Boolean equation (Theorem 1 and Theorem 2). Then we discuss several characterizations of Boolean sets, meaning sets of zeros of Boolean functions, and prove that every Boolean transformation X=Φ(T) is the parametric general solution of a certain Boolean equation.     

Efficient iterative solutions to general coupled matrix equations

Linear matrix equations are encountered in many systems and control applications.In this paper,we consider the general coupled matrix equations（including the generalized coupled Sylvester matrix equations as a special case）l t=1EstYtFst = Gs,s = 1,2,···,l over the generalized reflexive matrix group（Y1,Y2,···,Yl）.We derive an efcient gradient-iterative（GI） algorithm for fnding the generalized reflexive solution group of the general coupled matrix equations.Convergence analysis indicates that the algorithm always converges to the generalized reflexive solution group for any initial generalized reflexive matrix group（Y1（1）,Y2（1）,···,Yl（1））.Finally,numerical results are presented to test and illustrate the performance of the algorithm in terms of convergence,accuracy as well as the efciency.     

Gradient based iterative solutions for general linear matrix equations

In this paper, we present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective.     

Parallel Solutions for Large-Scale General Sparse Nonlinear Systems of Equations

In solving application problems,many large-scale nonlinear systems of equaions result in sparse Jacobian matrices.Such nonlinear systems are called sparse nonlinear systems.The irregularity of the locations of nonzrero elements of a general sparse matrix makes it very difficult to generally map sparse matrix computations to multiprocessors for parallel processing in a well balanced manner.To overcome this difficulty,we define a new storage scheme for general sparse matrices in this paper,With the new storage scheme,we develop parallel algorithms to solve large-scale general sparse systems of equations by interval Newton/Generalized bisection methods which reliably find all numerical solutions within a given domain.I n Section 1,we provide an introduction to the addressed problem and the interval Newton‘s methods.In Section 2,some currently used storage schemes for sparse systems are reviewed.In Section 3,new index schemes to store general sparse matrices are reported.In Section 4,we present a parallel algorithm to evaluate a general sparse Jacobian matrix.In Section 5,we present a parallel algorithm to solve the corresponding interval linear system by the all-row preconditioned scheme.Conclusions and future work are discussed in Section 6.     

Implicitization of rational parametric equations

Based on the Gröbner basis method, we present algorithms for a complete solution to the following problems in the implicitization of a set of rational parametric equations. (1) Find a basis of the implicit prime ideal determined by a set of rational parametric equations. (2) Decide whether the parameters of a set of rational parametric equations are independent. (3) If the parameters of a set of rational parametric equations are not independent, reparameterize the parametric equations so that the new parametric equations have independent parameters. (4) Compute the inversion maps of parametric equations, and as a consequence, give a method to decide whether a set of parametric equations is proper. (5) In the case of algebraic curves, find a proper reparameterization for a set of improper parametric equations.     

Computing all parametric solutions for blending parametric surfaces

In this paper we prove that, for a given set of parametric primary surfaces and parametric clipping curves, all parametric blending solutions can be expressed as the addition of a particular parametric solution and a generic linear combination of the basis of a free module of rank 3. As a consequence, we present an algorithm that outputs a generic expression for all the parametric solutions for the blending problem. In addition, we also prove that the set of all polynomial parametric solutions (i.e. solutions that have polynomial parametrizations) for a parametric blending problem can also be expressed in terms of the basis of a free module of rank 3, and we prove an algorithmic criterion to decide whether there exist parametric polynomial solutions. As a consequence we also present an algorithm that decides the existence of polynomial solutions, and that outputs (if this type of solution exists) a generic expression for all polynomial parametric solutions for the problem.     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

Iterative and approximate solutions of general parameter dependent nonlinear operator equations in Hilbert spaces

Iterative solutions of a general nonlinear operator equation of the form Ax + λTx = f, where A and T are in general nonlinear operators in an appropriate space, have not been developed so far. In this paper, the three well-known Banach, Mann, and Ishikawa iteration processes are used to find solutions of this equation in the Hilbert space case. An error estimate in each case is given. The established results generalize many of the known results in literature.     

A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization

A general algorithm is presented to approximately solve a great variety of linear and nonlinear ordinary differential equations (ODEs) independent of their form, order, and given conditions. The ODEs are formulated as optimization problem. Some basic fundamentals from different areas of mathematics are coupled with each other to effectively cope with the propounded problem. The Fourier series expansion, calculus of variation, and particle swarm optimization (PSO) are employed in the formulation of the problem. Both boundary value problems (BVPs) and initial value problems (IVPs) are treated in the same way. Boundary and initial conditions are both modeled as constraints of the optimization problem. The constraints are imposed through the penalty function strategy. The penalty function in cooperation with weighted-residual functional constitutes fitness function which is central concept in evolutionary algorithms. The robust metaheuristic optimization technique of the PSO is employed to find the solution of the extended variational problem. Finally, illustrative examples demonstrate practicality and efficiency of the presented algorithm as well as its wide operational domain.     

Periodic conjugate direction algorithm for symmetric periodic solutions of general coupled periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The conjugate direction (CD) method is a famous iterative algorithm to find the solution to nonsymmetric linear systems $Ax=b$. In this work, a new method based on the CD method is proposed for computing the symmetric periodic solutions $(X_1,X_2,\dots ,X_{\lambda })$ and $(Y_1,Y_2,\dots ,Y_{\lambda })$ of general coupled periodic matrix equations

for $i=1,2,\dots ,\lambda$. The key idea of the scheme is to extend the CD method by means of Kronecker product and vectorization operator. In order to assess the convergence properties of the method, some theoretical results are given. Finally two numerical examples are included to illustrate the efficiency and effectiveness of the method.     

Iterative algorithm to compute the maximal and stabilising solutions of a general class of discrete-time Riccati-type equations

In this article an iterative method to compute the maximal solution and the stabilising solution, respectively, of a wide class of discrete-time nonlinear equations on the linear space of symmetric matrices is proposed. The class of discrete-time nonlinear equations under consideration contains, as special cases, different types of discrete-time Riccati equations involved in various control problems for discrete-time stochastic systems. This article may be viewed as an addendum of the work of Dragan and Morozan (Dragan, V. and Morozan, T. (2009), ‘A Class of Discrete Time Generalized Riccati Equations’, Journal of Difference Equations and Applications, first published on 11 December 2009 (iFirst), doi: 10.1080/10236190802389381) where necessary and sufficient conditions for the existence of the maximal solution and stabilising solution of this kind of discrete-time nonlinear equations are given. The aim of this article is to provide a procedure for numerical computation of the maximal solution and the stabilising solution, respectively, simpler than the method based on the Newton–Kantorovich algorithm.     

Ground state solutions of Pohoz̆aev type for fractional Choquard equations with general nonlinearities

In this paper, we study the fractional Choquard equation

${(?\Delta )}^{s}u+u=\left(\right|x{|}^{?\mu }1F\left(u\right))f\left(u\right),\text{in}{\mathbb{R}}^N,$

where $N\ge 3$, $0<s<1$, $0<\mu <min\{N,4s\}$, and $f\in C(\mathbb{R},\mathbb{R})$ satisfies the general Berestycki–Lions conditions. Combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz?aev type for the above equation. The result improves some ones in Shen et al. (2016).     

Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations

In this study it is shown that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared.     

n-tuple complex helical geometry modeling using parametric equations

Complex helical structures are difficult to model in three-dimensional form to conduct finite element analysis. In general, parametric mathematical equations for single and double helical geometries are readily available in existing literature. However, more complex forms such as triple or in general n-tuple helical structures are still not widely studied. In this paper, at first, definitions of single and double helical structures are presented in parametric mathematical forms. Centerlines, curvatures, and torsions of these geometries are found, and these two helical geometries are visualized in three-dimensional structures. Next, one of the untouched helical models, the triple helical geometry, is investigated and a procedure to find the centerline of the triple helical geometry is presented. In addition, the first three-dimensional generated solid model of a triple helix geometry is presented. Finally, the steps used to create triple helix geometry are generalized to find parametric mathematical equations for n-tuple helical geometries.     

基于曲线参数方程的植物果实造型*

基于曲线参数方程的植物果实造型方法通过果实主轴线和截面曲线参数方程计算出果实表面的网格面以及各顶点的法线,从而实现了植物果实的造型;通过在曲线方程上叠加扰动函数解决了果实生长过程中的形变问题。实验证明该方法能够较逼真地模拟植物果实形态及其生长过程。     

Ray tracing general parametric surfaces using interval arithmetic

This paper describes an algorithm for ray tracing general parametric surfaces. After dividing the surface adaptively into small parts, a binary tree of these parts is built. For each part a bounding volume is calculated with interval arithmetic. From linear approximations and intervals for the partial derivatives it is possible to construct parallelepipds that adapt the orientation and shape of the surface parts very well and form very tight enclosures. Therefore we can develop an algorithm for rendering that is similar to that used with Bèzier and B-spline surfaces, where the bounding volumes are derived from the convex hull property. The tree of enclosures (generated once in a preprocessing step) guarantees that each ray that hits the surface leads to an iteration on a very small surface part; this iteration can be robustly (and very quickly) performed in real arithmetic.     

Scaling of general quadratic matrix equations

A scaling framework for general quadratic algebraic matrix equations is presented. All algebraic quadratic equations can be considered as special cases of a single generalized algebraic quadratic matrix equation (GQME). Hence, the paper is focused on the analysis and solution of the scaling problem of that GQME. The presented scaling method is based on the assignment of predetermined values of the coefficients and the unknown matrices of the GQME. The proposed framework is independent of any numerical method and therefore its use is general. Implementations are presented for the special case of matrix algebraic Riccati equations (AREs). Some new results of matrix algebraic identities considering Kronecker and Hadamard products are also reported.     

Nonoscillatory solutions for discrete equations

A nonoscillatory theory is presented for discrete equations. Our results rely on a nonlinear alternative of Leray-Schauder type for condensing operators.