Order Partial Derivatives Research Articles

Extension of a Riemannian metric with vanishing curvature

Let Ω be a connected and simply-connected open subset of R n such that the geodesic distance in Ω is equivalent to the Euclidean distance. Let there be given a Riemannian metric ( g ij ) of class C 2 and of vanishing curvature in Ω, such that the functions g ij and their partial derivatives of order ⩽ 2 have continuous extensions to Ω . Then there exists a connected open subset Ω of R n containing Ω and a Riemannian metric ( g ̃ ij) of class C 2 and of vanishing curvature in Ω that extends the metric ( g ij ). To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

APPROXIMATELY MULTIPLICATIVE FUNCTIONALS ON ALGEBRAS OF SMOOTH FUNCTIONS

Let φ be ab ounded linear functional on A ,w hereA is a commutative Banach algebra, then the bilinear functional ˇ φ is defined as ˇ φ(a, b )= φ(ab) − φ(a)φ(b )f or eacha and b in A .I f thenorm of ˇ φ is small then φ is approximately multiplicative, and it is of interest whether or not � ˇ φ� being small implies that φ is near to a multiplicative functional. If this property holds for a commutative Banach algebra then A is an AMNM algebra (approximately multiplicative functionals are near multiplicative functionals). The main result of the paper shows that C N [0, 1] M (the complex valued functions defined on [0, 1] M with all Nth order partial derivatives continuous) is AMNM. It is also shown that a similar proof can be applied to certain Lipschitz algebras.

An alternative approach for calculation of the first and higher order derivatives in the DRM-MD

The partial derivatives (PDs) in the classical dual reciprocity method (DRM) are represented through PDs of the DRM approximation function. This approach is very flexible and easy to implement but may introduce singularities depending on the approximation function used, especially for higher order PDs. A new scheme is proposed that reduces the order of the PD of the DRM approximation function by one in respect to the one that appears in the classical DRM representation of PDs. This scheme can be applied for a case when the proportion of corner nodes is high, that is, in the DRM-MD approach. The new approach is tested and the results show improvements in the accuracy of at least one order of magnitude for the case when the derivatives of the approximation function introduce singularities.

Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

Let ${\mit\Omega}$ be a bounded domain in $\mathbb R^N$ with smooth boundary. Consider the following elliptic system: $$\eqalign{ -{\mit\Delta} u&=\partial_vH(u,v,x)\quad\ \hbox{in ${\mit\Omega}$,}\cr -{\mit\Delta} v&=\partial_uH(u,v,x)\quad\ \hbox{in $

The quadratic approximation lemma and decompositions of superlative indexes

It was shown in 1976 that a difference in a quadratic function of N variables evaluated at two points is exactly equal to the sum of the arithmetic average of the first order partial derivatives of the function evaluated at the two points times the differences in the independent variables. In the present paper, this result is generalized and the resulting generalized quadratic approximation lemma is used to establish all of the superlative index number formulae that were derived in Diewert [4]. In addition, some new exact decompositions of the percentage change in the Fisher and Walsh superlative indexes into N components are derived. Each component in this decomposition represents the contribution of a change in a single independent variable to the overall percentage change in the index. Finally, these components are given economic interpretations.

ELECTRIC CIRCUIT DESIGN AND TESTING OF INTEGRATED DISTRIBUTED STRUCTRONIC SYSTEMS

Modelling distributed parameter systems (DPS) by electric circuits and fabricating the complicated equivalent circuits to evaluate system responses poses many challenging research issues for many years. Electrical modelling and analysis of distributed sensing/control of smart structures and distributed structronic systems are even scarce. This paper is to present a technique to model distributed structronic control systems with electric circuits and to evaluate control behaviors with the fabricated equivalent circuits. Electrical analogies and analysis of distributed structronic systems is proposed and dynamics and control of beam/sensor/actuator systems are investigated. To determine the equivalent circuits and system parameters, higher order partial derivatives are simplified using the finite difference method; partial differential equations (PDE) are transformed to finite difference equations and further represented by electronic components and circuits. To provide better signal management and stability, active electrical circuit systems are designed and fabricated. Electrical signals from the distributed system circuits (i.e., soft and hard) are compared with results obtained by the classical theoretical, finite element, and experimental techniques.

Option Pricing with Stochastic Volatility: A Closed-form Solution Using the Fourier Transform

The Black and Scholes (1973) option pricing model was developed starting from the hypothesis of constant volatility. However, many empirical studies, have argued that the mentioned hypothesis is subject to debate. A few authors, among who - Stein and Stein (1991), Heston (1993), Bates (1996) and Bakshi et al.(1997, 2000) - suggested the use of the Fourier transform for the density of the underlying return or for the risk-neutral probabilities, in order to evaluate the fair price of an option. In this paper we propose a stochastic valuation model using the Fourier transform for option price. This model can be used for the valuation of European options, characterized by two state variables: the price of the underlying asset and its volatility. We model the stochastic processes described by the two variables and we obtain a partial derivatives equation of which the solution is the price of the derivative. We propose a solution to this partial derivatives equation using the Fourier transform. When we apply the Fourier transform, we demonstrate that a second order partial derivatives equation is solved as an ordinary differential equation. We consider a correlation between the underlying asset price and its volatility and two sources of risk: return and volatility. The first part of the paper describes the hypotheses of the model. After describing the Fourier transforms, we propose a formula for the valuation of European options with stochastic volatility. In the second part, we present a few empirical results on the pricing of CAC 40 index call options.

Evaluation of 3D Operators for the Detection of Anatomical Point Landmarks in MR and CT Images

Point-based registration of images strongly depends on the extraction of suitable landmarks. Recently, different 3D operators have been proposed in the literature for the detection of anatomical point landmarks in 3D images. In this paper, we investigate nine 3D differential operators for the detection of point landmarks in 3D MR and CT images. These operators are based on either first, second, or first and second order partial derivatives of an image. In our investigation, we use measures which reflect different aspects of the detection performance of the operators. In the first part of the investigation, we analyze the number of corresponding detections under elastic deformations and noise, in the second part, we use statistical measures to determine the detection performance for landmarks within regions of interest (ROIs), and in the third part, we investigate the separability of the detections. It turns out that operators based on only first order partial derivatives of an image (i) yield a larger number of corresponding points than the other operators, (ii) that their performance on the basis of the statistical measures is better, and (iii) that the separability of the detections is better so that a suitably chosen threshold can significantly decrease the number of false detections.

Worst Case Complexity of Weighted Approximation and Integration over [formula omitted

We study the worst case complexity of weighted approximation and integration for functions defined over Rd. We assume that the functions have all partial derivatives of order up to r uniformly bounded in a weighted Lp-norm for a given weight function ψ. The integration and the error for approximation are defined in a weighted sense for another given weight ϱ. We present a necessary and sufficient condition on weight functions ϱ and ψ for the complexity of the problem to be finite. Under additional conditions, we show that the complexity of the weighted problem is proportional to the complexity of the corresponding classical problem defined over a unit cube and with ϱ=ψ=1. Similar results have been obtained recently for scalar functions (d=1) and for multivariate functions under restriction that ψ=1 and p=∞.

Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems

Consider an infinite dimensional diffusion process process on T Z d , where T is the circle, defined by the action of its generator L on C2 (T Z d ) local functions as . Assume that the coefficients, a i and b i are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that a i is only a function of and that . Suppose ν is an invariant product measure. Then, if ν is the Lebesgue measure or if d=1,2 , it is the unique invariant measure. Furthermore, if ν is translation invariant, then it is the unique invariant, translation invariant measure. Now, consider an infinite particle spin system, with state space {0,1}Z d , defined by the action of its generator on local functions f by , where is the configuration obtained from η altering only the coordinate at site x . Assume that are of finite range, bounded and that . Then, if ν is an invariant product measure for this process, ν is unique when d=1,2 . Furthermore, if ν is translation invariant, it is the unique invariant, translation invariant measure. The proofs of these results show how elementary methods can give interesting information for general processes.

An elementary proof of the uniqueness of invariant product measures for some infinite dimensional processes

Consider an infinite dimensional diffusion process with state space T Z d , where T is the circle, and defined by an infinitesimal generator L which acts on local functions f as Lf(η)=∑ i∈ Z d ( a i 2(η i) 2 ∂ 2f ∂η i 2 +b i(η) ∂f ∂η i ) . Suppose that the coefficients a i and b i are smooth, bounded, of finite range, have uniformly bounded second order partial derivatives, that a i are uniformly bounded from below by some strictly positive constant, and that a i is a function only of η i . Suppose that there is a product measure ν which is invariant. Then if ν is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if ν is translation invariant, it is the unique invariant, translation invariant measure. The proofs are elementary. Similar results can be proved in the context of an interacting particle system with state space {0,1} Z d , with uniformly positive bounded flip rates which are finite range. To cite this article: A.F. Ramı́rez, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 139–144

Complexity of Weighted Approximation over [formula omitted

We study approximation of multivariate functions defined over Rd. We assume that all rth order partial derivatives of the functions considered are continuous and uniformly bounded. Approximation algorithms U(f) only use the values of f or its partial derivatives up to order r. We want to recover the function f with small error measured in a weighted Lq norm with a weight function ρ. We study the worst case (information) complexity which is equal to the minimal number of function and derivative evaluations needed to obtain error ε. We provide necessary and sufficient conditions in terms of the weight ρ and the parameters q and r for the weighted approximation problem to have finite complexity. We also provide conditions guaranteeing that the complexity is of the same order as the complexity of the classical approximation problem over a finite domain. Since the complexity of the weighted integration problem is equivalent to the complexity of the weighted approximation problem with q=1, the results of this paper also hold for weighted integration. This paper is a continuation of [7], where weighted approximation over R was studied.

Bending of large curvature beams.: I. Stress method approach

The paper presents a coherent theory of the uniform bending problem in a circular curved beam, with multi-connected cross-section, having a large radius of curvature with respect to its width. The three-dimensional elastic problem is solved, in the case of linear homogeneous isotropic body, assuming the stress tensor as the unknown and by exactly satisfying the field compatibility equations. The mathematical structure of the governing boundary value problem (BVP), enlightened here for the first time, is unexpectedly complicated: a fourth-order elliptic (variable coefficients) partial differential equation with two degenerate unstable boundary conditions (i.e. involving second and third order partial derivatives in a direction that becomes tangent at several points of the boundary). Such a kind of BVP seems to be typical of the curved beam bending problem since it also appears in the displacement approach ( Mentrasti, 2001. part II, Int. J. Solids Struct. 38, 5727–5745). As a final point, it is reduced to a simpler problem by an ad hoc integral representation, assuming the ρ-convexity of the cross-section domain.

A SHARP ERROR ESTIMATE FOR THE NUMERICAL SOLUTION OF MULTIVARIATE DIRICHLET PROBLEM FOR THE HEAT EQUATION

For the multidimensional Dirichlet problem of the heat equation on a cylinder, this study examines convergence properties with rates of approximate solutions, obtained by a naturally arising difference scheme over inscribed uniform grids. Sharp quantitative estimates are given by the use of first and second moduli of continuity of some first and second order partial derivatives of the exact solution. This is accomplished by using the probabilistic method an appropriate random walk.

An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives

We present a code to compute the oscillatory cuspoid canonical integrals and their first order partial derivatives. The algorithm is based on the method of Connor and Curtis [J. Phys. A 15 (1982) 1179–1190], in which the integration path along the real axis is replaced by a more convenient contour in the complex plane, rendering the oscillatory integrand more amenable to numerical quadrature. Our code is a modern implementation of this method, presented in a modular fashion as a Fortran 90 module containing the relevant subroutines. The code is robust, with the novel feature that the algorithm implements an adaptive contour procedure, choosing contours that avoid the violent oscillatory and exponential natures of the integrand and modifying its choice as necessary. In many cases, the subroutines are efficient and provide results of high accuracy. Plots and results for the Pearcey and swallowtail cuspoid integrals given in previous articles are reproduced and verified. Our subroutines significantly extend the range of application of this earlier work.

Differential quadrature element analysis using extended differential quadrature

The extended differential quadrature (EDQ) has been proposed. A certain order derivative or partial derivative of the variable function with respect to the coordinate variables at an arbitrary discrete point is expressed as a weighted linear sum of the values of function and/or its possible derivatives at all grid nodes. The grid pattern can be fixed while the selection of discrete points for defining discrete fundamental relations is flexible. This method can be used to the differential quadrature element and generalized differential quadrature element analyses.

ON SOBOLEV-VISIK-DUBINSKII TYPE INEQUALITIES


 
 
 In the present paper we establish some new integral inequalities of the Sobolev-Visik­ Dubinskii type involving functions of several independent variables and their first order partial derivatives. The analysis used in the proofs is elementary and the obtained results provide new estimates on these types of inequalities. 
 
 

Approximation of functions and their derivatives: A neural network implementation with applications

This paper reports a neural network (NN) implementation for the numerical approximation of functions of several variables and their first and second order partial derivatives. This approach can result in improved numerical methods for solving partial differential equations by eliminating the need to discretise the volume of the analysis domain. Instead only an unstructured distribution of collocation points throughout the volume is needed. An NN approximation of relevant variables for the whole domain based on these data points is then achieved. Excellent test results are obtained. It is shown how the method of approximation can then be used as part of a boundary element method (BEM) for the analysis of viscoelastic flows. Planar Couette and Poiseuille flows are used as illustrative examples.

Shape design of cylindrical probe coils for the induction of specified eddy current distributions

A sensitivity analysis scheme for designing the shape of cylindrical probe coils is presented. These coils are used for the induction of specified eddy current distributions inside cylindrical conducting specimens. This requirement is satisfied by designing the outer surface of the coil, while the excitation current density is assumed to be known and constant all over the cross-section of the coil. An appropriate error function, which is related to the desired and the estimated values of the magnetic vector potential, is minimized by applying Newton's gradient based optimization method. The estimated values of the magnetic potential are computed by an integral formulation, which is derived from the diffusion equation. During each iteration of the minimization process, the gradient and the second order partial derivatives of the error function, with respect to the design variables, are evaluated analytically. In numerical results, the method is applied to the induction of a uniform eddy current distribution parallel to the axis of a conducting cylinder.

On eigenvalues of differentiable positive definite kernels

If a positive definite kernelK(x, y) has thepth order partial derivative (∂ p /∂y p )K(x,y) continuous on the square [0,1]2, we show that the eigenvalues of the integral operator generated byK(x, y) are asymptoticallyo(1/n p+1 ). We also obtain the anticipated asymptotic estimate when (∂ p /∂y p )K(x,y) satisfies further a Lipschitz condition iny of order 0<α≤1. These results, which extend some classical estimates of I. Fredholm and H. Weyl under the additional positive definiteness assumption, are based on two interesting inequalities of K. Fan.