Refinements of Gurland’s formula for pi

The aim of this paper is to refine Gurland’s formula for approximating pi. We prove the complete monotonicity of a class of functions related to Gurland’s formula, and then we state some sharp inequalities.     

Optimal solution of linear inequalities with applications to pattern recognition

An algorithm for the optimal solution of consistent and inconsistent linear inequalities is presented, where the optimality criterion is the maximization of the number of constraints satisfied. In the terminology of pattern recognition, the algorithm finds a linear decision function which minimizes the number of patterns misclassified. The algorithm is developed as a nonenumerative search procedure based on several new results established in this paper. Bounds on the search are also developed and the method is experimentally evaluated and shown to be computationally superior to other techniques for finding minimum-error solutions.     

A source of inequalities for circular functions

In this work, a basic theorem is established and it is found to be a source of inequalities for circular functions, such as Cusa’s, Huygens’, Wilker’s, the Sandor–Bencze, Carlson’s, the Shafer–Fink inequality, and the one in the problem of Oppenheim. Furthermore, these inequalities described above will be extended by this basic theorem.     

Lyapunov functions for quadratic differential equations with applications to adaptive control

Conditions are derived for the existence of quadratic Lyapunov functions for vector differential equations with quadratic terms. These conditions are used to establish a set of nonlinear equations for which the null solution is asymptotically stable in the whole. Many of the recent results in the stability of model reference adaptive systems are particular cases of these general conditions.     

Useful formula for moment computation of normal random variables with nonzero means

It is well known that higher order moments of normal random variables (RV) with zero means can be expressed by terms of second-order moments. In this correspondence an extension will be given for moments of normal RV with nonzero means.     

A complete ranking of trapezoidal fuzzy numbers and its applications to multi-criteria decision making

The problem (or scenario) involving qualitative or imprecise information is not solvable by classical set theory. To overcome the shortcoming of classical set theory, Zadeh (Inf Control 8(3):338–356, 26) introduced the concept of fuzzy sets that generalizes the concept of classical sets. Fuzzy set theory allows modelling and handling of imprecise information in an effective way. As a special class of fuzzy sets, fuzzy numbers (FN) which are very much important in decision making was introduced by Dubois and Prade (Int J Syst Sci 9:631–626, 12). The available methods for solving multi-criteria decision making problems (MCDM) are problem dependent in nature due to the partial ordering on the class of FN. Total ordering on the class of FN by countable number of real-valued parameters was achieved by Wang and Wang (Fuzzy Sets Syst 243:131–141, 21). A complete ranking on the class of trapezoidal fuzzy numbers (TrFNs) using finite number of score functions is achieved in this paper. In this paper, a new ranking procedure (complete) on the class of TrFNs using the concepts of mid-point, radius, left and right fuzziness of TrFN is proposed and further we introduce a method for solving fuzzy multi-criteria decision making (Fuzzy MCDM) problem. Finally, comparisons of our proposed method with familiar existing methods are listed.

     

On coefficient inequalities of functions associated with conic domains

In this paper, the concepts of Janowski functions and the conic regions are combined to define a new domain which represents the conic type regions. Different views of this modified conic domain for specific values are shown graphically for better understanding of the behavior of this domain. The class of such functions which map the open unit disk E onto this modified conic domain is defined. Also the classes of k-uniformly Janowski convex and k-Janowski starlike functions are defined and their coefficient inequalities are formulated. The coefficient bound for a certain class of analytic functions, proved by Owa et al. (2006) in [16], has also been improved.     

Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration

In this paper numerical approximation of history-dependent hemivariational inequalities with constraint is considered, and corresponding Céa’s type inequality is derived for error estimate. For a viscoelastic contact problem with normal penetration, an optimal order error estimate is obtained for the linear element method. A numerical experiment for the contact problem is reported which provides numerical evidence of the convergence order predicted by the theoretical analysis.     

Two general integral inequalities and their applications to stability analysis for systems with time‐varying delay

Integral inequalities have been widely used in stability analysis for systems with time‐varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger‐based inequality, the Bessel–Legendre inequality, the Wirtinger‐based double integral inequality, and the auxiliary function‐based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time‐varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

The applications of harmonic functions to robotics

Harmonic functions are solutions to Laplace's equation. Such functions can be used to advantage for potential-field path planning because they do not exhibit spurious local minima. Harmonic functions are shown here to have a number of properties that are essential to robotics applications. Paths derived from harmonic functions are generally smooth. Harmonic functions also offer a complete path-planning algorithm. We show how a harmonic function can be used as the basis for a reactive admittance control. Such schemes allow incremental updating of the environment model. Methods for computing harmonic functions respond well to sensed changes in the environment, and can be used for control while the environment model is being updated.     

Minimizing binary functions with simulated annealing algorithm with applications to binary tomography

The minimization of binary functions finds many applications in practice, and can be solved by the simulated annealing (SA) algorithm. However, the SA algorithm is designed for general combinatorial problems, not specifically for binary problems. Consequently, a direct application of the SA algorithm might not provide optimal performance and efficiency. Therefore, this study specifically investigated the performance of various implementations of the SA algorithm when applied to binary functions. Results obtained in this investigation demonstrated that 1) the SA algorithm can reliably minimize difficult binary functions, 2) a simple technique, analogous to the local search technique used in minimizing continuous functions, can exploit the special structure of binary problems and significantly improve the solution with negligible computational cost, and 3) this technique effectively reduces computational cost while maintaining reconstruction fidelity in binary tomography problems. This study also developed two classes of binary functions to represent the typical challenges encountered in minimization.     

Coefficient formula and matrix of nonuniform B-spline functions

The paper derives a coefficient formula of nonuniform B-spline functions of arbitrary degree from the Coxde Boor recursive algorithm. An efficient numerical algorithm for the coefficient matrix of nonuniform B-spline functions is also presented that is based on this formula. The results in the paper are useful for the evaluation and conversion of NURBS curves and surfaces in CAD/CAM systems. They will promote the application of product data-exchange standards in industry.     

Neural network approximation for periodically disturbed functions and applications to control design

This paper addresses the approximation problem of functions affected by unknown periodically time-varying disturbances. By combining Fourier series expansion into multilayer neural network or radial basis function neural network, we successfully construct two kinds of novel approximators, and prove that over a compact set, the new approximators can approximate a continuously and periodically disturbed function to arbitrary accuracy. Then, we apply the proposed approximators to disturbance rejection in the first-order nonlinear control systems with periodically time-varying disturbances, but it is straightforward to extend the proposed design methods to higher-order systems by using adaptive backstepping technique. A simulation example is provided to illustrate the effectiveness of control schemes designed in this paper.