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We study dense packings of a large number of congruent non-overlapping circles inside a square by looking for configurations which maximize the packing density, defined as the ratio between the area occupied by the disks and the area of the square container. The search for these configurations is carried out with the help of two algorithms that we have devised: a first algorithm is in charge of obtaining sufficiently dense configurations starting from a random guess, while a second algorithm improves the configurations obtained in the first stage. The algorithms can be used sequentially or independently. The performance of these algorithms is assessed by carrying out numerical tests for configurations with a large number of circles. [ABSTRACT FROM AUTHOR]

This paper proposes new methods to handle inverse problems. In particular, uncertain parameters have been taken here as fuzzy viz. Triangular Fuzzy Numbers (TFNs) involved in various physical models. Inverse approaches are very useful to approximate the unknown parameters that characterize the system when data from experimental measurements are already known. In this work we proposed different inverse approaches to approximate the unknown TFNs of various models of radon diffusion/dispersion problems with the help of obtained experimental data. [ABSTRACT FROM AUTHOR]

A mixed approximate method is proposed to simulate the generalized Hirota–Satsuma coupled Korteweg–de Vries equations. The equations are represented as linear dispersive equations for the material time derivative along the trajectory of physical particle movements described by the nonlinear Cauchy problem in the Lagrangian viewpoint. The proposed method employs the finite difference method and backward differentiation formula to solve the linear dispersive equations. To solve the nonlinear Cauchy problem, an error correction technique is applied. Moreover, the fifth-order weighted essentially non-oscillatory scheme interpolates solutions of non-grid points to control unnecessary oscillation due to the dispersion of the solution. From numerical experiments, we demonstrate the accuracy and efficiency of the proposed method. The proposed method numerically has the qth-order temporal and 4th-order spatial accuracies in the sense of the error norms L ∞ and L 2 . Consequently, the errors obtained using the proposed method are more accurate even with a larger time step size than those of the compared methods. [ABSTRACT FROM AUTHOR]

This study is aimed to relate nonlinear infinite-horizon optimal control problems (NLIHOCPs) with open-loop information. The difficulties of solving the two-point boundary value problems (TPBVPs) arising from NLIHOCPs can be assigned to the nonlinearity of differential equations, the combination of split boundary conditions, and how the transversality conditions in infinite-horizon are treated. In this paper, we propose a combined quasilinearization method (QLM) with the exponential Bernstein functions (EBFs) for solving nonlinear TPBVPs on the semi-infinite domain. First, the QLM is used to reduce the nonlinear TPBVP to a sequence of linear differential equations. Then, a collocation method based on the EBFs is utilized to find the approximate solution of the resulting linear differential equations. By applying the EBFs, the transversality conditions for TPBVP on the semi-infinite domain are satisfied. The convergence of the QLM + EBFs is proved. Some numerical experiments are performed to confirm the validity of the proposed computational scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a rational RBF interpolation method to approximate multivariate functions with poles or other singularities on or near the domain of approximation. The method is based on scattered point layouts and is flexible with respect to the geometry of the problem’s domain. Despite the existing rational RBF-based techniques, the new method allows the use of conditionally positive definite kernels as basis functions. In particular, we use polyharmonic kernels and prove that the rational polyharmonic interpolation is scalable. The scaling property results in a stable algorithm provided that the method be implemented in a localized form. To this aim, we combine the rational polyharmonic interpolation with the partition of unity method. Sufficient number of numerical examples in one, two and three dimensions are given to show the efficiency and the accuracy of the method. [ABSTRACT FROM AUTHOR]

In this study, an efficient method is developed for solving systems of partial differential equations governing seismic analysis of 2D dam reservoir interaction, in the frequency domain. Using Chebyshev higher-order polynomials as mapping function, special shape function, integration method of Clenshaw–Curtis and the integral form used to weighted residual method, coefficient matrices of the system of governing equations become diagonal. This means that the governing partial differential equation for each degree of freedom becomes independent from others. This feature and discretizing only boundaries of domain significantly reduce computational costs in comparison with other methods. In this regard, various problems such as dynamic analysis of empty gravity dam, calculating the hydrodynamic pressure on rigid dam, and dam–reservoir interaction analysis due to the horizontal motion of foundation are examined. Comparing the results of this method with other analytical/numerical methods shows high capability and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In previous studies, the nonlinear problem of electrohydrodynamic (EHD) ion drag flows in a circular cylindrical conduit has been studied by several authors. However, those studies seldom involve the computation for large physical parameters such as the electrical Hartmann number and the magnitude parameter for the strength of the nonlinearity due to the existence of strong nonlinearity in these extreme cases. To overcome this faultiness, the newly-developed homotopy Coiflets wavelet method is extended to solve this EHD flow problem with strong nonlinearity. The validity and reliability of the proposed technique are verified. Particularly, the highly accurate homotopy-wavelet solution is obtained for extreme large parameters, which seems to be overlooked before. Discussion about the effects of related physical parameters on the axial velocity field is presented. [ABSTRACT FROM AUTHOR]

In this paper, an analytical structural global sensitivity analysis (GSA) method based on direct integral is proposed to efficiently and precisely analyse the sensitivity of model variables. Sobol' sensitivity analysis method can comprehensively analyse the influences of model variables and their interaction on model responses, but this method is usually involved in a series of high-dimensional and high-order integrals which are generally difficult to calculate directly. Therefore, a kind of optimal polynomial response surface is established to replace the original model based on the polynomial structure-selection of error reduction ratio. This optimal polynomial has the advantages of simple structure, high precision and strong anti-noise ability. So the multiple integrals can be conveniently and analytically operated to obtain the accurate first-order and interactive sensitivity indices through the established optimal polynomial model. Compared with Sobol' sensitivity analysis method, this proposed method is more adaptable for the complex model and practical engineering problems as well as can meliorate the accuracy of GSA. Two numerical examples and two engineering applications are studied to illustrate the applicability and effectiveness of this method. [ABSTRACT FROM AUTHOR]

We construct a q-analog of the blossom for analytic functions, the analytic q-blossom. This q-analog also extends the notion of q-blossoming from polynomials to analytic functions. We then apply this analytic q-blossom to derive identities for analytic functions represented in terms of the q-Poisson basis, including q-versions of the Marsden identity and the de Boor-Fix formula for analytic functions. [ABSTRACT FROM AUTHOR]

This paper proposes a new technique based on interval Legendre polynomials in the collocation method for solving n-the order uncertain boundary value problems. Firstly, the proposed method is illustrated by different example problems. Further, the developed technique has been used to solve the steady state radon diffusion-advection equation with uncertain (interval) boundary conditions. The results obtained by the proposed method are compared with their exact solutions (in crisp case) in order to demonstrate the validity and applicability of the method. [ABSTRACT FROM AUTHOR]

The min-max symbols generalize the kneading symbols in that they contain also information about the minimum or maximum character of the critical values and their iterates. Interestingly enough, this additional information can be obtained from the kneading symbols without further computation. In this paper we review some interesting applications of the min-max symbols. The applications chosen concern new expressions for the topological entropy of multimodal maps, as well as a numerical algorithm to compute it.

We solve the non-linearized and linearized obstacle problems efficiently using a primal-dual hybrid gradients method involving projection and/or $$L^1$$ penalty. Since this method requires no matrix inversions or explicit identification of the contact set, we find that this method, on a variety of test problems, achieves the precision of previous methods with a speed up of 1-2 orders of magnitude. The derivation of this method is disciplined, relying on a saddle point formulation of the convex problem, and can be adapted to a wide range of other constrained convex optimization problems. [ABSTRACT FROM AUTHOR]

We introduce the G-blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G= G( t). By invoking the G-blossom, we construct G-Bernstein bases and G-Bézier curves and study their algebraic and geometric properties. We show that the G-blossom provides the dual functionals for the G-Bernstein basis functions and we use this dual functional property to prove that G-Bernstein basis functions form a partition of unity and satisfy a Marsden identity. We also show that G-Bézier curves share several other properties with classical Bézier curves, including affine invariance, interpolation of end points, and recursive algorithms for evaluation and subdivision. We investigate the effect of the linear functions G on the shape of the corresponding G-Bézier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G-Bézier curves are of Pólya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G-Bézier curve, and we derive the geometric rate of convergence of this algorithm. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the most general quantum Bernstein bases and quantum Bézier curves. Classical Bernstein bases satisfy a two-term formula for their classical derivatives; quantum Bernstein bases satisfy a two-term formula for their quantum derivatives. To study the properties of these general quantum polynomial schemes, a new variant of the blossom, the quantum blossom, is introduced by altering the diagonal property of the classical blossom. The significance of the quantum blossom is that the quantum blossom provides the dual functionals for quantum Bézier curves over arbitrary intervals. Using the quantum blossom, several fundamental identities involving the quantum Bernstein bases are developed, including a quantum variant of the Marsden identity and the partition of unity property. Based on these properties of quantum Bernstein bases, quantum Bézier curves are shown to be affine invariant, and under certain conditions lie in the convex hull of their control points. In addition, for each quantum Bézier curve of degree n , a collection of n ! , affine invariant, recursive evaluation algorithms are derived. Using two of these recursive evaluation algorithms, a recursive subdivision procedure for quantum Bézier curves is constructed. This subdivision procedure generates a sequence of control polygons that converges rapidly to the original quantum Bézier curve. [ABSTRACT FROM AUTHOR]

This paper makes three original contributions: (1) Explicit closed-form parametric formulas for the boundary of the Minkowski sum and difference of two arbitrarily oriented solid ellipsoids in n-dimensional Euclidean space are presented; (2) Based on this, new closed-form lower and upper bounds for the volume contained in these Minkowski sums and differences are derived in the 2D and 3D cases and these bounds are shown to be better than those in the existing literature; (3) A demonstration of how these ideas can be applied to problems in computational geometry and robotics is provided, and a relationship to the Principal Kinematic Formula from the fields of integral geometry and geometric probability is uncovered. [ABSTRACT FROM AUTHOR]

This work presents a new adaptive multilevel approximation of the gradient operator on a recursively refined spherical geodesic grid. The multilevel structure provides a simple way to adapt the computation to the local structure of the gradient operator so that high resolution computations are performed only in regions where singularities or sharp transitions occur. This multilevel approximation of the gradient operator is used to solve the linear spherical advection equation for both time-independent and time-dependent wind field geophysical test cases. In contrast with other approximation schemes, this approach can be extended easily to other curved manifolds by choosing an appropriate coarse approximation and using recursive surface subdivision. The results indicate that the adaptive gradient calculation and the solution of spherical advection equation accurate, efficient and free of numerical dispersion. [ABSTRACT FROM AUTHOR]

In this article, we consider a sample point (tj,sj) including a valuesj = f(tj) at heightsjand abscissa (time or location)tj. We apply wavelet decomposition by using shifts and dilations of the basic Häar transform and obtain an algorithm to analyze a signal or functionf. We use this algorithm in practical to approximating function by numerical example. Some relationships between wavelets coefficients and asymptotic distribution of wavelet coefficients are investigated. At the end, we illustrate the results on simulated data by using MATLAB and R software. [ABSTRACT FROM AUTHOR]

Blossoming is a useful technique to study bases and curve representations in computer-aided geometric design. Recently Simeonov et al. (Comput Aided Geom Des 28:549-565, ) have used a blossom generalization, namely the h-blossom, to derive new results about the h-Bernstein basis and h-Bézier curves that have previously been studied in approximation theory and computer-aided geometric design. This paper introduces a basis related to the h-Bernstein basis. There is a close relationship between this new basis and the h-Bernstein basis, between the new basis and the h-blossom, and between the new basis and 'progressive' curves. This paper explores these relationships and uses them to derive properties both of the new basis itself, and of curves represented in terms of the new basis. [ABSTRACT FROM AUTHOR]

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q ( quantum)- blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden's identity. [ABSTRACT FROM AUTHOR]

Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bézier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C 2 cubic Bézier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper. [ABSTRACT FROM AUTHOR]

We propose an algorithm to compute an approximation of capillary surfaces in a gravitational field. This algorithm is based on a decomposition-coordination method by augmented lagrangians and the discretization is done using the finite element method. We study the convergence of the algorithm and the error of discretization for the axisymmetric case; some numerical results are given. This method can be generalized to a two-dimensional space. [ABSTRACT FROM AUTHOR]