Fourier series representation of the coupler curves of spatial linkages

In this paper, the three-dimensional Fourier series which can describe the coupler curves of spatial linkages are derived. Based on the geometric properties of one-dimensional and two-dimensional Fourier series, a geometric constraint conditions are proposed to establish the three-dimensional Fourier series. The geometric meaning of mechanism for the three-dimensional Fourier series is put forward. Finally, an example is given to show the feasibility and the validity of this approach.     

On the computation of spherical designs by a new optimization approach based on fast spherical Fourier transforms

Spherical t-designs are point sets \(X_{M}=\{x_{1},\ldots, x_{M}\} \subset {\mathbb{S}^{2}}\) which provide quadrature rules with equal weights for the sphere which are exact for polynomials up to degree t. In this paper we consider the problem of finding numerical spherical t-designs on the sphere \({\mathbb{S}^{2}}\) for high polynomial degree \({t \in {\mathbb{N}}}\) . That is, we compute numerically local minimizers of a certain quadrature error A _t (X _M ). The quadrature error A _t was also used for a variational characterization of spherical t-designs by Sloan and Womersley (J Approx Theory 159:308–318, 2009). For the minimization problem we regard several nonlinear optimization methods on manifolds, like Newton and conjugate gradient methods. We show that by means of the nonequispaced fast spherical Fourier transforms we perform gradient and Hessian evaluations in \({\mathcal {O}(t^{2}\log t + M \log^{2}(1/\varepsilon))}\) arithmetic operations, where \({\varepsilon >0 }\) is a prescribed accuracy. Using these methods we present numerical spherical t-designs for t ≤  1,000, even in the case \({M\approx \frac{1}{2} t^{2}}\) .     

Evolutes of dual spherical curves for ruled surfaces

E. Study found that there is a one‐to‐one correspondence between the oriented lines in Euclidean three space and the dual points of the dual unit sphere in dual three space, and it has wide applications in Engineering. In this paper, we investigate a ruled surface as a curve on the dual unit sphere by using E. Study's theory. Then we define the notion of evolutes of dual spherical curves for ruled surfaces and establish the relationships between singularities of these subjects and geometric invariants of dual spherical curves. Finally, we give an example to illustrate our findings. Copyright © 2015 John Wiley & Sons, Ltd.     

Reducible spectral curves and the hyperkahler geometry of adjoint orbits

We study the hyperkähler geometry of a regular semisimpleadjoint orbit of SL(k, ) via the algebraic geometry of the correspondingreducible spectral curve.     

Computer modelling and geometric construction for four-point synthesis of 4R spherical linkages

In this paper, computer modelling and geometric construction of Burmester curve for synthesis of spherical mechanisms is presented. Rigid shell guidance in four specified positions on a sphere is performed by a 4R spherical linkage. The synthesis of such a linkage requires obtaining the associated Burmester curves on the reference sphere. Based on the Burmester theory and beside the computational and modelling abilities of the symbolic mathematical software namely Maple, an accurate as well as fast procedure for geometric construction of Burmester curve is developed. In the first part, the concepts of orientation and position on the sphere, pole of the motion and the related parameters are extended for modelling by the Maple. In the second part, the concepts of complementary axis quadrilateral and its imaginary motion, the center and circle axis cones and Burmester curve are derived. In the final part, using the prepared procedure and through a numerical example, a 4R spherical linkage for guiding an antenna to meet four specified postures in a three-dimensional working space is synthesized.     

Inexactly constrained discrete adjoint approach for steepest descent-based optimization algorithms

The problem of constrained optimization via the gradient-based discrete adjoint steepest descent method is studied under the assumption that the constraint equations are solved inexactly. Error propagation from the constraint equations to the gradient is studied analytically, as is the convergence rate of the inexactly constrained algorithm as it relates to the exact algorithm. A method is developed for adapting the residual tolerance to which the constraint equations are solved. The adaptive tolerance method is applied to two simple test cases to demonstrate the potential gains in computational efficiency.     

A general procedure for solving boundary-value problems of elastostatics for a spherical geometry based on Love's approach

We develop a general procedure for solving the first and secondfundamental problems of the theory of elasticity for cases whereboundary conditions are prescribed on a spherical surface, usingLove's general solution of the elastostatic equilibrium equationsin terms of three scalar harmonic functions. It is shown thatthis general solution combined with a methodology by Brennerpaves an elegant way to determine the three harmonic functionsin terms of the boundary data. Thus, with this general scheme,solution of any such boundary-value problem is reducible toa routine exercise thereby providing some `economy of effort'.Furthermore, we develop a similar general scheme for thermoelasticproblems for cases when temperature type boundary conditionsare prescribed on a spherical surface. We then illustrate theapplication of the procedure by solving a number of problemsconcerning rigid spherical inclusions and spherical cavities.In particular, apart from furnishing alternative solutions tothe known problems, we demonstrate the use of this general procedurein solving the problem of interaction of a rigid spherical inclusionwith a concentrated moment and that of a concentrated heat sourcesituated at an arbitrary point outside the inclusion. We alsoderive closed-form expressions for the net force and the nettorque acting on a rigid spherical inclusion embedded into aninfinite elastic solid under an ambient displacement field characterizedby an arbitrary-order polynomial in the Cartesian coordinates.To the best of our knowledge, these results are new.     

Eulerian-Lagrangian method of cells based on localized adjoint method

The localized adjoint method, when applied using an Eulerian-Lagrangian frame, has been quite successful in treating advection-dominated transport. The resulting methodology is known as ELLAM. In previous work, bilinear functions were used as test functions. In this paper, local constant functions are used instead, leading to procedures which are appealing because, in addition to other advantages of ELLAM methods, they ensure local mass conservation, are easy to apply and can be combined without difficulty with existing solute-transport codes which are based on finite volumes. In addition, the procedures for deriving the algorithms presented here are used as an illustration of a general methodology for treating numerically partial differential equations, which is advocated by the authors. Such methodology consists in identifying the information about the sought solution which is contained in the approximate one and then using this insight to choose the interpolation procedure to be applied. © 1994 John Wiley & Sons, Inc.     

A new method for estimating derivatives based on a distribution approach

In many applications, the estimation of derivatives has to be done from noisy measured signal. In this paper, an original method based on a distribution approach is presented. Its interest is to report the derivatives on infinitely differentiable functions. Thus, the estimation of the derivatives is done only from the signal. Besides, this method gives some explicit formulae leading to fast calculus. For all these reasons, it is an efficient method in the case of noisy signals as it will be confirmed in several examples.     

Pseudo‐spherical normal Darboux images of curves on a lightlike surface

We introduce the notions of the pseudospherical normal Darboux images for the curve on a lightlike surface in Minkowski 3‐space and study these Darboux images by using technics of the singularity theory. Furthermore, we give a relation between these Darboux images and Darboux frame from the viewpoint of Legendrian dualities.     

On the characterization of spherical curves in 3-dimensional Sasakian spaces

In this paper, we give the spherical characterization of a regular curve in 3-dimensional Sasakian space. Furthermore the differential equation which expresses the mentioned characterization is solved.     

A new data envelopment analysis based approach for fixed cost allocation

Annals of Operations Research - In many real applications, there exist situations where some independent and decentralized entities will construct a common platform for production processes. A...     

Construction of spherical spline quasi-interpolants based on blossoming

A general theory of quasi-interpolants based on quadratic spherical Powell-Sabin splines on spherical triangulations of a sphere-like surface S is developed by using polar forms. As application, various families of discrete and differential quasi-interpolants reproducing quadratic spherical Bézier-Bernstein polynomials or the whole space of the spherical Powell-Sabin quadratic splines of class C¹ are presented.     

Semilinear fractional differential equations based on a new integral operator approach

Semilinear fractional differential equations (SFDEs) often arise in some dynamical systems. In most of the existing literature, the fixed point theorems are used to prove the existence and uniqueness of the solutions of SFDEs. In this paper, we give a new way to prove the existence and uniqueness of the solutions by introducing a new integral operator associated with the Mittag–Leffler function.     

A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.     

An approach to solving axially symmetric problems for spherical orthotropic bodies

A series expansion method is developed in which the small parameter is the deviation of the spherically orthotropic properties of deformable bodies from their transversally isotropic properties. The problem is reduced to a rigorous analytic solution of inhomogeneous boundary value problems. The efficiency of the approximation technique developed here and its practical convergence are examined in a centrally symmetric problem for an orthotropic sphere which permits an exact analytic solution. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 17–24, 1999.     

A theorem on the adjoint system for vector bundles

In this paper, we study the classification theory of uniruled varieties by means of the adjoint system for vector bundles on the varieties. We prove that ifE is an ample vector bundle on a smooth projective varietyX with rank(E)=dimX-2, thenK _X +C ₁ (E) is numerically effective except in a few cases. In all of the exceptional cases,X is a uniruled variety. As consequences, we generalized a result of Fujita [Fu3] and Ionescu [Io] and improve upon a theorem of Wiśniewski [Wi1].     

Locally supported spherical wavelets based on block grids

This paper presents a method for approximating spherical functions from discrete data of a block–wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotation–invariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from classical tensor product wavelet theory. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new flow on starlike curves in

In this note we find a new evolution equation for starlike curves in . We study the evolution of the subaffine curvature and subaffine torsion under the flow and show that it is completely integrable. The solutions to the evolution which move without changing affine shape are subaffine elastic curves. We integrate the subaffine elastica by quadratures.