Notes on pedal and contrapedal curves of fronts in the Euclidean plane

A pedal curve (a contrapedal curve) of a regular plane curve is the locus of the feet of the perpendiculars from a point to the tangents (normals) to the curve. These curves can be parametrized by using the Frenet frame of the curve. Yet provided that the curve has some singular points, the Frenet frame at these singular points is not well‐defined. Thus, we cannot use the Frenet frame to examine pedal or contrapedal curves. In this paper, pedal and contrapedal curves of plane curves, which have singular points, are investigated. By using the Legendrian Frenet frame along a front, the pedal and contrapedal curves of a front are introduced and properties of these curves are given. Then, the condition for a pedal (and a contrapedal) curve of a front to be a frontal is obtained. Furthermore, by considering the definitions of the evolute, the involute, and the offset of a front, some relationships are given. Finally, some illustrated examples are presented.     

On derivatives of hypergeometric functions and classical polynomials with respect to parameters

Closed expressions are obtained for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial.     

A Legendre‐spectral element method for flow and heat transfer about an accelerating droplet

The problem of flow and heat transfer associated with a spherical droplet accelerated from rest under gravitational force is studied using a Legendre‐spectral element method in conjunction with a mixed time integration procedure to advance the solution in time. An influence matrix technique that exploits the superposition principle is adapted to resolve the lack of vorticity boundary conditions and to decouple the equations from the interfacial couplings. The computed flow and temperature fields, the drag coefficient, the Nusselt number, and the interfacial velocity and vorticity are presented for a drop moving vertically in a quiescent gas of infinite extent to illustrate the evolution of the flow and temperature fields. Comparison of the predicted drag coefficient and the Nusselt number against previous numerical and experimental results indicate good agreement. Copyright © 2000 John Wiley & Sons, Ltd.     

带非局部边界条件的热方程的LEGENDRE配置法

对带非局部边界条件的热方程的初边值问题提出了LEGENDRE配置法,并给出其半离散逼近和全离散逼近的稳定性和收敛性分析.数值试验验证了方法的有效性.     

Legendre-type optimality conditions for a variational problem with inequality state constraints

, they differ from the Legendre-Clebsch condition. They give information about the Hesse matrix of the integrand at not only inactive points but also active points. On the other hand, since the inequality state constraints can be regarded as an infinite number of inequality constraints, they sometimes form an envelope. According to a general theory [9], one has to take the envelope into consideration when one consider second-order necessary optimality conditions for an abstract optimization problem with a generalized inequality constraint. However, we show that we do not need to take it into account when we consider Legendre-type conditions. Finally, we show that any inequality state constraint forms envelopes except two trivial cases. We prove it by presenting an envelope in a visible form. Received April 18, 1995 / Revised version received January 5, 1998 Published online August 18, 1998     

Approximate solution of the linear quadratic control problem

In this paper the Legendre polynomials method is applied in order to find the approximate solution of the linear quadratic optimal control problem with imposed inequality constraints. The computational procedure based on the obtained results, is presented     

Pedal curves of frontals in the Euclidean plane

In this paper, we will give the definition of the pedal curves of frontals and investigate the geometric properties of these curves in the Euclidean plane. We obtain that pedal curves of frontals in the Euclidean plane are also frontals. We further discuss the connections between singular points of the pedal curves and inflexion points of frontals in the Euclidean plane.     

关于完全平方数的一个性质

运用不定方程的理论讨论了完全平方数的一个基本性质,得到了关于完全平方数的几个重要定理.     

Asymptotic Results for Semigroups of Order-Preserving Partial Transformations

Let 𝒫  _n be the semigroup of all decreasing and order-preserving partial transformations of an n-element chain, and let E(𝒫  _n ) be its set of idempotents. Among other results, asymptotic formulae for |𝒫  _n | and |E(𝒫  _n )|/|𝒫  _n | are obtained. Similar results for 𝒫  _n the (larger) semigroup of all order-preserving partial transformations of an n-element chain are also obtained.     

Adaptive order polynomial algorithm in a multiwavelet representation scheme

We have developed a new strategy to reduce the storage requirements of a multivariate function in a multiwavelet framework. We propose that alongside the commonly used adaptivity in the grid refinement one can also vary the order of the representation k as a function of the scale n. In particular the order is decreased with increasing refinement scale. The consequences of this choice, in particular with respect to the nesting of scaling spaces, are discussed and the error of the approximation introduced is analyzed. The application of this method to some examples of mono- and multivariate functions shows that our algorithm is able to yield a storage reduction up to almost 60%. In general, values between 30 and 40% can be expected for multivariate functions. Monovariate functions are less affected but are also much less critical in view of the so called “curse of dimensionality”.