一类四次有理插值样条的点控制

构造了一种有理四次插值样条,其分子为四次多项式分母为二次多项式.该有理插值样条是有界的、保单调且C~2连续的,仅带有一个调节参数δ_i.研究了有理四次插值样条的性质,同时给出了相应的函数值控制、导数值控制方法,这种方法的优点在于能够根据实际设计需要简单地选取适宜的参数,达到对曲线的形状进行局部调控的目的.     

The logarithmic energy of zeros and poles of a rational function

On assuming that certain lemniscates of a rational function are connected, we establish some sharp inequality that involves the logarithmic energy of a discrete charge concentrated at the zeros and poles of this function and the absolute values of its derivatives at these points. The equality in this estimate is attained for specially arranged zeros and poles of a suitable Zolotarev fraction and for special distributions of charge.     

一个几何概念──长度伸缩率

本文对一元函数在一点处导数绝对值的几何意义进行了讨论，指出在极限的意义下它实际上是映射在一点局部的一种长度的伸缩率。     

Neumann and Robin problems in a cracked domain with jump conditions on cracks

The boundary value problem for the Laplace equation is studied on a domain with smooth compact boundary and with smooth internal cracks. The Neumann or the Robin condition is given on the boundary of the domain. The jump of the function and the jump of its normal derivative is prescribed on the cracks. The solution is looked for in the form of the sum of a single layer potential and a double layer potential. The solvability of the corresponding integral equation is determined and the explicit solution of this equation is given in the form of the Neumann series. Estimates for the absolute value of the solution of the boundary value problem and for the absolute value of the gradient of the solution are presented.     

Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L ^p . This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L ^p -norm of its boundary values is attained at the radial direction. This confirms D. Khavinson’s conjecture for p = 1 and p = 2. Bibliography: 11 titles.     

A nonlocal problem with fractional derivatives for the mixed type equation

For an equation of the mixed elliptic-hyperbolic type we investigate the boundary-value problem, when on the elliptic part of domain boundary a co-normal derivative of solution is given, and in the hyperbolic part the generalized fractional derivatives of solution value on characteristics are pointwise connected with the solution value and its derivative on the line of parabolic degeneration. The unique resolvability of problem is proven.     

基于共轭几何变换程序的国债市场利率期限结构波动建模

本文变革已有的利率期限结构模型估计依赖于定价误差平方和最小化原则,引入几何双重变换程序解决非线性约束的误差绝对距离最小化问题,丰富国债市场利率波动和定价研究的理论体系和研究方法;运用负指数平滑立方L₁样条优化模型,克服B样条函数对节点数目与定位的过度敏感和放宽对贴现函数的二阶导数平滑要求,协同拟合误差绝对距离与贴现函数波动率最小化,保留B样条函数刻画中长期利率波动趋势的优势,增强对短期利率波动结构突变的估计、定价和预测能力,缓解B样条和NSS模型在利率期限结构拟合存在的过度波动问题。     

Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.     

Convergence rates of a family of barycentric osculatory rational interpolation

It is well-known that osculatory rational interpolation sometimes gives better approximation than Hermite interpolation, especially for large sequences of points. However, it is difficult to solve the problem of convergence and control the occurrence of poles. In this paper, we propose and study a family of barycentric osculatory rational interpolation function, the proposed function and its derivative function both have no real poles and arbitrarily high approximation orders on any real interval.     

CONSTRAINED RATIONAL CUBIC SPLINE AND ITS APPLICATION

1. IntroductionDesign of high quality, manufacturable surfaces, such as the outer shape of a ship, car oraeroplane, is an important yet challenging task in today's manufacturing industries. Althoughsignificam progress has been made in the last decade in developing and commercializing pro--duction quality CAD tools, demand for more effective tools is still high due to the ever increajsein model complexity and the needs to address and incorporate manufacturing requirements inthe early stage of …     

Harmonic Properties of the Logarithmic Potential and the Computability of Elliptic Fekete Points

We investigate the properties of the function sending each N-tuple of points to minus the logarithm of the product of their mutual distances. We prove that, as a function defined on the product of N spheres, this function is subharmonic, and indeed its (Riemannian) Laplacian is constant. We also prove a mean value equality and an upper bound on the derivative of the function. We use these results to get sharp upper bounds for the precision needed to describe an approximation to elliptic Fekete points (in the sense demanded by Smale’s 7th problem). We also conclude that Smale’s 7th problem has solutions given by rational spherical points of bounded (small) bit length, proving that there exists an exponential running time algorithm which solves it on the Turing machine model.     

Biorthogonal Laurent Polynomials,Toplitz Determinants,Minimal Toda Orbits and Isomonodromic Tau Functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulae. For all these classes of polynomials a 2 × 2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo-measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toplitz determinants of the moments of the pseudo-measure. In particular, the results imply that any (shifted) Toplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.     

Gears,pregears and related domains

We study conformal mappings from the unit disc to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or “gearlike”) domains fit into a more general category of domains we call “pregears” (images of gears under Möbius transformations), which aid in the study of the conformal mappings for gears and which we also describe in detail. Such domains being bounded by arcs of circles, the Schwarzian derivative of the Riemann mapping is known to be a rational function of a specific form. One accessory parameter of these mappings is naturally related to the conformal modulus of the gear (or pregear) and we prove several qualitative results relating it to the principal remaining accessory parameter. The corresponding region of univalence (parameters for which the rational function is the Schwarzian derivative of a conformal mapping) is determined precisely.     

On the Numerical Range of Rational Matrix Functions

In this paper we investigate bounds of the numerical range of the derivative of a rational matrix function. Moreover, some results on connectedness are presented.     

On the Numerical Range of Rational Matrix Functions

In this paper we investigate bounds of the numerical range of the derivative of a rational matrix function. Moreover, some results on connectedness are presented.     

A problem for the Helmholtz equation outside cuts on the plane with the dirichlet condition and the oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the Helmholtz equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition with pure imaginary coefficient of the tangential derivative is posed on the other side. We prove the uniqueness of the solution. The solvability of the problem is proved for the case in which the above-mentioned pure imaginary coefficient is less than unity in absolute value. In this case, we obtain an integral representation of the solution of the problem in the form of potentials. The densities of the potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. The boundary value problem considered here generalizes the mixed Dirichlet-Neumann problem.     

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.     

Sobolev space properties of superharmonic functions on metric spaces

Our objective is to study regularity of superharmonic functions of a nonlinear potential theory on metric measure spaces. In particular, we are interested in the local integrability properties of a superharmonic function and its derivative. We show that every superharmonic function has a weak upper gradient and provide sharp local integrability estimates. In addition, we study absolute continuity of a superharmonic function.     

Real and imaginary quadratic representations of hyperelliptic function fields

A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor's algorithm. We show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved. Furthermore, we show how the two representations are connected and compare the computational complexity.

     

Saddlepoint approximations to P-values for comparison of density estimates

Summary  This article is concerned with computing approximate p-values for the maximum of the absolute difference between kernel density estimates. The approximations are based on treating the process of local extrema of the differences as a nonhomogeneous Poisson Process and estimating the corresponding local intensity function. The process of local extrema is characterized by the intensity function, which determines the rate of local extrema above a given threshold. A key idea of this article is to provide methods for more accurate estimation of the intensity function by using saddlepoint approximations for the joint density of the difference between kernel density estimates and using the first and second derivative of the difference. In this article, saddlepoint approximations are compared to gaussian approximations. Simulation results from saddlepoint approximations show consistently better agreement between empirical p-value and predetermined value with various bandwidths of kernel density estimates.