Second-order Partial Derivatives Research Articles

A new formula for bivariate Hermite interpolation on variable step grids and its application to image interpolation.

In this paper, we present a novel formula of the bivariate Hermite interpolating (BHI) polynomial in the case of support points arranged on a grid with variable step. This expression is applicable when interpolation of a bivariate function is required, given its value and the values of its partial derivatives of arbitrarily high order, at the support points. The proposed formula is a generalization of an existing formula for the bivariate Hermite polynomial. It is also algebraically much simpler, thus can be computed more efficiently. In order to apply Hermite interpolation to image interpolation, we simplify the proposed (BHI) to handle support points on a regular unit-step grid. The values of image partial derivatives are arithmetically approximated using compact finite differences. The proposed method is being assessed in a number of image interpolation experiments that include a synthetic image, for which the values of the partial derivatives are computed analytically, as well as a collection of images from different medical modalities. The proposed BHI with up to second-order image partial derivatives, outperforms the convolution-based interpolation methods, as well as generalized interpolation methods with the same number of support points that was compared with, in the majority of image interpolation experiments. The computational load of the proposed BHI is calculated and its behaviour with respect to its controlling parameters is investigated.

Comparative statics for oligopoly: A generalized result

We perform comparative statics for a general model of asymmetric oligopoly and derive a concise formula for the response of one firm to a marginal change in its rival’s strategic variable, taking into account the responses of all other firms. We obtain the conditions under which the sign of this response coincides with that of the mixed second-order partial derivative of the firm’s payoff function. We then propose a distinction between gross and net strategic relationships (i.e., strategic substitute and complement).

An improvement on the upper bounds of the magnitudes of derivatives of rational triangular Bézier surfaces

New bounds on the magnitudes of the first- and second-order partial derivatives of rational triangular Bézier surfaces are presented. Theoretical analysis shows that the proposed bounds are tighter than the existing ones. The superiority of the proposed new bounds is also illustrated by numerical tests.

An improving fringe analysis method based on the accuracy of S-transform profilometry

The S transform, as a simple and popular technique for space\ ime–frequency analysis, has been introduced in optical three-dimensional surface shape measurement in recent years. Based on the S transform, S transform “ridge” method (STR) and S transform filtering method (STF) have been proposed to extract the phase information from the single deformed fringe pattern. This paper focuses on studying the STR in fringe pattern analysis. In previous researches about the STR, a linear constraint, which assumed that the phase was locally expressed as first-order Taylor expansion with respect to x and y directions, was implicitly added. Actually at least the second-order partial derivatives of the phase in each local area should be taken into account because they are related to the local curvature of the height distribution of the tested object. Therefore, the traditional STR has larger phase measuring errors in those areas with the rapid height variation on the tested object. This paper proposes an improved STR method, in which the phase is approximately expressed as a quadric in each local area. The phase extraction formula based on the quadric is derived, and the phase correction is carried out as well. Both the simulations and the experiments verify that a more accurate phase map can be obtained by the improved method compared with that by the traditional STR, especially in the areas where height variation is steep.

Social Welfare Maximization of Multimodal Transportation

Multimodal urban transportation systems exhibit complex inter actions between components, including users, multimodal transportation facilities, supply side agencies, and operators. Although these interactions are obvious, rigorous quantitative methods for optimizing control variables across modes of transportation on real-world networks are deficient. A social welfare maximization model was established for joint optimization of bus fare, rail transit fare, and congestion tolls for private cars. The authors determined the optimality conditions and the second-order partial derivatives for this optimization problem. Because of the complexity of the multimodal urban transportation system, the objective function of social welfare has no closed form and is extremely expensive to evaluate. The authors therefore proposed a simulation-based framework for evaluation of the objective function and optimization of three decision variables across multiple travel modes. Several metamodels (i.e., mathematical functions that approximate the true shape of an unknown, nonlinear, and complex objective function) were adopted to approximate the highly nonlinear input–output mappings in the urban system. This is the first study to develop a simulation-based method for joint optimization of transit and road network operations. The case study applied the simulation-based optimization framework to the Sino–Singapore ecocity in Tianjin, China, by using VISSUM as the urban systems simulator. An “ecocity” is defined by the authors as “a thriving city that is socially harmonious, environmentally friendly, resource-efficient, and a model for sustainable development.” Results show that metamodels can accurately approximate the real objective function and produce good suboptimal and near-optimal solutions. The optimal combination of transit fares and congestion tolls significantly outperform those under two baseline scenarios. The optimal solutions also suggest that extreme transit fares (too high or too low) or congestion tolls are contrary to welfare-maximizing objectives. The proposed method can be applied for joint optimization of other multimodal planning and operational strategies, such as investment and operational decisions across various modes of transportation.

Regularized collocation for spherical harmonics gravitational field modeling

Motivated by the problem of satellite gravity gradiometry, which is the reconstruction of the Earth gravity potential from the satellite data provided in the form of the second-order partial derivatives of the gravity potential at a satellite altitude, we discuss a special regularization technique for solving this severely ill-posed problem in a spherical framework. We are especially interested in the regularized collocation method. As a core ingredient we present an a posteriori parameter choice rule, namely the weighted discrepancy principle, and prove its order optimality. Finally, we illustrate our theoretical findings by numerical results for the computation of the Fourier coefficients of the gravitational potential directly from the noisy synthetic data.

Application of the Störmer—Verlet-Like Symplectic Method to the Wave Equation

A fourth-order three-stage symplectic integrator similar to the second-order Störmer—Verlet method has been proposed and used before [Chin. Phys. Lett. 28 (2011) 070201; Eur. Phys. J. Plus 126 (2011) 73]. Continuing the work initiated in the publications, we investigate the numerical performance of the integrator applied to a one-dimensional wave equation, which is expressed as a discrete Hamiltonian system with a fourth-order central difference approximation to a second-order partial derivative with respect to the space variable. It is shown that the Störmer—Verlet-like scheme has a larger numerical stable zone than either the Störmer—Verlet method or the fourth-order Forest—Ruth symplectic algorithm, and its numerical errors in the discrete Hamiltonian and numerical solution are also smaller.

Uniform in Bandwidth Law of the Iterated Logarithm for a Transformation Kernel Estimator of Copulas

In this paper, we establish a uniform in bandwidth law of the iterated logarithm for the Transformation kernel estimator of bivariate copulas introduced in Omelka et al. (2009). To this end, we make use of a general empirical process approach inspired by the works in Mason and Swanepoel (2011). We obtain the asymptotic order of the maximal deviation of this estimator from its expectation. Then, we show that the bias converges asymptotically to zero at the same order provided that the second-order partial derivatives of the copula exist and are bounded.We also propose a bandwidth selection method by using a cross-validation approach. Finally, we compare in a simulation study the performances of the Transformation kernel estimator by considering two different methods of selecting the bandwidth. Keywords: Copula function; Kernel estimation; Transformation estimator; Law of the iterated logarithm; Uniform in bandwidth consistency; Cross-validation method

Mathematical modelling of multilayer thin body deformation

We construct a theory of multilayer thin bodies within the framework of the threedimensional moment theory by using an efficient parametrization of a multilayer thin domain; in contrast to classic approaches, several base surfaces and an analytic method with Legendre and Chebyshev polynomial systems are used. Geometric characteristics typical for the proposed parametrizations are introduced into consideration. A fundamental theorem for a multilayer thin domain is formulated. Various representations of the equations of motion, the heat influx, and the constitutive relations of physical and heat content are presented for the new body domain parametrization. The definition of the kth order moment of a certain quantity with respect to an orthonormal system of second-kind Chebyshev polynomials is given. The expressions of moments of first- and second-order partial derivatives of a certain tensor field are obtained, and this is also done for some important expressions required for constructing different variants of the thin body theory. Various variants of the equations of motion in moments with respect to Legendre and Chebyshev polynomial systems are also obtained. The interlayer conditions are written down under various connections of adjacent layers of a multilayer body.

Asymptotics of empirical copula processes under non-restrictive smoothness assumptions

Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in Stute's representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.

Advanced Chebyshev expansion for identification of smart structures

There are strong needs for health-monitoring of continuous, multi-dimensional structures as bridges or aircraft wings. Identification based on vibration analysis offer a promising option to fulfil such needs. The present contribution is dedicated to the use of Chebyshev orthogonal functions to expand the response of a system. Based on this expansion, a novel computation of the partial derivatives is presented. Therefore the partial differential equations of motion can be written as algebraic equations. An exact expansion method is firstly presented. An enhanced differentiation method is then developed. For this aim, a case-study is provided: a cantilever beam which involves a fourth-order partial derivative with respect to space and a second-order partial derivative with respect to time of the displacement. In this example, the identification error is reduced in comparison to classical methods. The proposed method has demonstrated it efficiency for identification of distributed parameter (material properties, structure dimensions). Therefore this identification method is well adapted for multi-dimensional structure monitoring.

On Agarwal–Pang-type integral inequalities

We establish some new Agarwal–Pang-type inequalities including second-order partial derivatives. In special cases, our results yield some correlated results and provide new estimates for the inequalities of this type.

Spectral Combination of Spherical Gradiometric Boundary-Value Problems: A Theoretical Study

The Earth’s gravity potential can be determined from its second-order partial derivatives using the spherical gradiometric boundary-value problems which have three integral solutions. The problem of merging these solutions by spectral combination is the main subject of this paper. Integral estimators of biased- and unbiased-types are presented for recovering the disturbing gravity potential from gravity gradients. It is shown that only kernels of the biased-type integral estimators are suitable for simultaneous downward continuation and combination of gravity gradients. Numerical results show insignificant practical difference between the biased and unbiased estimators at sea level and the contribution of far-zone gravity gradients remains significant for integration. These contributions depend on the noise level of the gravity gradients at higher levels than sea. In the cases of combining the gravity gradients, contaminated with Gaussian noise, at sea and 250 km levels the errors of the estimated geoid heights are about 10 and 3 times smaller than those obtained by each integral.

Geometric phase in the Kitaev honeycomb model and scaling behaviour at critical points

In this paper a geometric phase of the Kitaev honeycomb model is derived and proposed to characterize the topological quantum phase transition. The simultaneous rotation of two spins is crucial to generate the geometric phase for the multi-spin in a unit-cell unlike the one-spin case. It is found that the ground-state geometric phase, which is non-analytic at the critical points, possesses zigzagging behavior in the gapless $B$ phase of non-Abelian anyon excitations, but is a smooth function in the gapped $A$ phase. Furthermore, the finite-size scaling behavior of the non-analytic geometric phase along with its first- and second-order partial derivatives in the vicinity of critical points is shown to exhibit the universality. The divergent second-order derivative of geometric phase in the thermodynamic limit indicates the typical second-order phase transition and thus the topological quantum phase transition can be well described in terms of the geometric-phase.

Composed complex-cue histograms: An investigation of the information content in receptive field based image descriptors for object recognition

Recent work has shown that effective methods for recognizing objects and spatio-temporal events can be constructed based on histograms of receptive field like image operations.This paper presents the results of an extensive study of the performance of different types of receptive field like image descriptors for histogram-based object recognition, based on different combinations of image cues in terms of Gaussian derivatives or differential invariants applied to either intensity information, color-opponent channels or both. A rich set of composed complex-cue image descriptors is introduced and evaluated with respect to the problems of (i) recognizing previously seen object instances from previously unseen views, and (ii) classifying previously unseen objects into visual categories.It is shown that there exist novel histogram descriptors with significantly better recognition performance compared to previously used histogram features within the same class. Specifically, the experiments show that it is possible to obtain more discriminative features by combining lower-dimensional scale-space features into composed complex-cue histograms. Furthermore, different types of image descriptors have different relative advantages with respect to the problems of object instance recognition vs. object category classification. These conclusions are obtained from extensive evaluations on two mutually independent data sets.For the task of recognizing specific object instances, combined histograms of spatial and spatio-chromatic derivatives are highly discriminative, and several image descriptors in terms rotationally invariant (intensity and spatio-chromatic) differential invariants up to order two lead to very high recognition rates. For category classification, primary information is contained in both first-and second-order derivatives, where second-order partial derivatives constitute the most discriminative cue.Dimensionality reduction by principal component analysis and variance normalization prior to training and recognition can in many cases lead to a significant increase in recognition or classification performance. Surprisingly high recognition rates can even be obtained with binary histograms that reveal the polarity of local scale-space features, and which can be expected to be particularly robust to illumination variations.An overall conclusion from this study is that compared to previously used lower-dimensional histograms, the use of composed complex-cue histograms of higher dimensionality reveals the co-variation of multiple cues and enables much better recognition performance, both with regard to the problems of recognizing previously seen objects from novel views and for classifying previously unseen objects into visual categories.

Erratum to “Semiparametric estimation in copula models”

Note that ψ is decreasing and continuous. We have The inequality above is meant to hold for all sufficiently large n which may depend on ω in the underlying probability space. This completes the proof of the claim. Remark. With the above result, we can give a detailed proof of the strong Bahadur–Kiefer type result stated in Stute 1984. If we assume that C is twice continuously differentiable on and that all the second-order partial derivatives are bounded, then, as stated in Tsukahara 2000, one can show that , a.s. Together with the result we proved above, we have , a.s., establishing (4.2) in Stute 1984. The above assumption on the second-order partial derivatives is rather restrictive, and Segers (2010) recently established a similar almost sure rate under weaker conditions on them. The author is very grateful to Jaeyong Lee and Ivan Kojadinovic for pointing out the error. He would also like to thank Johan Segers for pointing out imprecise statements in the earlier version and for suggesting publishing the correction.

Regularization using different surfaces and the second-order derivatives of 1/r

We study the dependence on the surface Σ : g(x) = 1, where g(x) is continuous in ℝ n ∖{0} and homogeneous of degree 1, of the distributional regularization ℛΣ(K(x))∈𝒟′(ℝ n ), of a homogeneous function K defined in ℝ n ∖{0}. We also consider the dependence on Σ of the regularization PfΣ(K(x)) given by the Hadamard finite part of the limit when this limit does not exist. We study, in particular, the results on surface dependence for the kernels the ordinary second-order derivatives in ℝ3, and recover the formulae for the second-order generalized derivatives obtained by Hnizdo [V. Hnizdo, Generalized second-order partial derivatives of 1/r, Eur. J. Phys. 32 (2011), pp. 287–297].

Valuation of Two-Factor Interest Rate Contingent Claims Using Green's Theorem

Over the years a number of two-factor interest rate models have been proposed that have formed the basis for the valuation of interest rate contingent claims. This valuation equation often takes the form of a partial differential equation that is solved using the finite difference approach. In the case of two-factor models this has resulted in solving two second-order partial derivatives leading to boundary errors, as well as numerous first-order derivatives. In this article we demonstrate that using Green's theorem, second-order derivatives can be reduced to first-order derivatives that can be easily discretized; consequently, two-factor partial differential equations are easier to discretize than one-factor partial differential equations. We illustrate our approach by applying it to value contingent claims based on the two-factor CIR model. We provide numerical examples that illustrate that our approach shows excellent agreement with analytical prices and the popular Crank–Nicolson method.

Potential fields and their partial derivatives produced by a 2D homogeneous polygonal source: A summary with some revisions

We summarized and revised the present forward modeling methods for calculating the gravity- and magnetic-field components and their partial derivatives of a 2D homogeneous source with a polygonal cross section. The responses of interest include the gravity-field components and their first- and second-order partial derivatives and the magnetic-field components and their first-order partial derivatives. The revised formulas consist of several basic quantities that are common in all the formulations. A singularity appears when the observation point coincides with a polygon vertex. This singularity is removable for the gravity formulas but not for the others. The compact forms of the revised formulas make them easy to implement. We compare the gravity- and magnetic-field components and their partial derivatives produced by a 2D prism whose polygonal cross section approximates a cylinder with the corresponding analytical fields and partial derivatives of the cylinder. The perfect fittings presented by both data sets confirm the reliability of the updated formulas.

On order reduction of non-linear equations of mechanics and mathematical physics, new integrable equations and exact solutions

Some classes of non-linear equations of mechanics and mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where a first-order partial derivative is taken as a new independent variable and a second-order partial derivative is taken as the new dependent variable. The results obtained are used for order reduction of hydrodynamic equations (Navier–Stokes, Euler, and boundary layer) and deriving exact solutions to these equations. Associated Bäcklund transformations are constructed for evolution equations of general form (special cases include Burgers, Korteweg-de Vries, and many other non-linear equations of mathematical physics). A number of new integrable non-linear equations, inclusive of the generalized Calogero equation, are considered.