Order Partial Derivatives Research Articles

On Opial-type inequalities with higher order partial derivatives

In the present paper we establish some new Opial-type inequalities involving higher order partial derivatives. Our results in special cases yield some of the recent results on Opial’s inequality and provide new estimates on inequalities of this type. As an application, we prove the uniqueness of the solution of the initial value problem involving higher order partial differential equations.

Caloric curve for nuclear liquid–gas phase transition in relativistic mean-field hadronic model

The main thermodynamical properties of the first order phase transition of the relativistic mean-field (RMF) hadronic model were explored in the isobaric, the canonical and the grand canonical ensembles on the basis of the method of the thermodynamical potentials and their first derivatives. It was proved that the first order phase transition of the RMF model is the liquid-gas type one associated with the Gibbs free energy $G$. The thermodynamical potential $G$ is the piecewise smooth function and its first order partial derivatives with respect to variables of state are the piecewise continuous functions. We have found that the energy in the caloric curve is discontinuous in the isobaric and the grand canonical ensembles at fixed values of the pressure and the chemical potential, respectively, and it is continuous, i.e. it has no plateau, in the canonical and microcanonical ensembles at fixed values of baryon density, while the baryon density in the isotherms is discontinuous in the isobaric and the canonical ensembles at fixed values of the temperature. The general criterion for the nuclear liquid-gas phase transition in the canonical ensemble was identified.

Detection of Microcalcification Clusters Using Hessian Matrix and Foveal Segmentation Method on Multiscale Analysis in Digital Mammograms

Mammography is the most efficient technique for detecting and diagnosing breast cancer. Clusters of microcalcifications have been mainly targeted as a reliable early sign of breast cancer and their earliest detection is essential to reduce the probability of mortality rate. Since the size of microcalcifications is very tiny and may be overlooked by the observing radiologist, we have developed a Computer Aided Diagnosis system for automatic and accurate cluster detection. A three-phased novel approach is presented in this paper. Firstly, regions of interest that corresponds to microcalcifications are identified. This can be achieved by analyzing the bandpass coefficients of the mammogram image. The suspicious regions are passed to the second phase, in which the nodular structured microcalcifications are detected based on eigenvalues of second order partial derivatives of the image and microcalcification pixels are segmented out by exploiting the foveal segmentation in multiscale analysis. Finally, by combining the responses coming out from the second order partial derivatives and the foveal method, potential microcalcifications are detected. The detection performance of the proposed method has been evaluated by using 370mammograms. The detection method has a TP ratio of 97.76% with 0.68 false positives per image. We have examined the performance of our computerized scheme using free-response operating characteristics curve.

Saturation properties of isospin asymmetric nuclear matter

Abstract The saturation properties of the isospin asymmetric nuclear matter, ANM, are studied microscopically using the density dependent M3Y-Paris and M3Y-Reid effective interactions in their CDM3Y- K versions. To do so, two-dimensional expansion of the energy per nucleon of ANM with respect to its density, ρ , and isospin asymmetry, I ≡ ( ρ n − ρ p ) / ρ , has been used within a suitable density range. Within this framework, the ANM saturation density, energy per nucleon and incompressibility as functions of the isospin asymmetry, I , up to its eighth order are derived in simple analytical formulas. These formulas link explicitly the ANM saturation properties to the CDM3Y- K density dependence forms through 24 characteristic quantities and coefficients which represent the different order partial derivatives of the energy with respect to ρ and I . The results show that the different terms up to I 8 , but with only 18 characteristic coefficients, are needed to describe reasonably the different ANM saturation properties. Up to four times the saturation density of nuclear matter, some properties such as the energy per nucleon of ANM and pure neutron matter are well expressed by their expansion up to only the quadratic term in I . Based on both Paris and Reid effective interactions and within the symmetric nuclear matter saturation incompressibility range of K 0 = 220 – 250 MeV , the different coefficients of the ANM incompressibility are obtained with the values of K 02 = − 348 ± 57 MeV , K 04 = 35 ± 31 MeV and K 06 = 4 ± 13 MeV .

Local electron tomography using angular variations of surface tangents: Stomo version 2

Local electron tomography using angular variations of surface tangents: Stomo version 2

On Regularity and Extension of Green’s Operator on Bounded Smooth Domains

We prove regularity and extension results for Green’s operators that are associated to strictly elliptic second order divergence-type linear PDO’s with coefficients in \(C^{1,\alpha}(\overline{\Omega})\). Here α ∈ (0, 1) and Ω ⊂ R n , n ≥ 3, is a bounded C 2,α domain. The regularity result gives boundary estimates for the derivatives up to order (2 + α) of the associated Green’s function. With the aid of this regularity result, we then extend the Green’s operator to a globally defined integral operator whose second order partial derivatives are Calderon–Zygmund singular integrals. We also show that, under reasonable a priori assumptions, the C 2,α regularity of the domain is necessary for the aforementioned extension of the Green’s operator to a weakly singular integral operator, belonging to the class \({\rm{SK}}^{-2}_{{\bf{R}}^n}(\alpha)\).

Sensitivity, probabilistic and stochastic analysis of the thermo-piezoelectric phenomena in solids by the stochastic perturbation technique

The main aim here is to present the application of the generalized stochastic perturbation technique to thermo-piezoelectric analysis of solid continua. The general nth order Taylor series representation for all random input parameters and the state functions is employed to formulate the coupled thermo-electro-elasticity equilibrium equations of the additional order; a determination of any probabilistic moments and characteristics is described; the discretization of the problem in terms of the Stochastic perturbation-based Finite Element Method is also provided. Since this expansion includes the lowest order partial derivatives, the structural sensitivity analysis using direct differentiation is performed at the same time with probabilistic modeling contrasted with the Monte-Carlo simulation results. The probabilistic approach is extended here towards an accounting for the stochastic ageing processes, which appear frequently in aggressive external environments and under dynamic excitation. The two parametric stochastic process with Gaussian initial value and ageing velocity is tested for this purpose. The entire procedure is tested on the example of the thermo-electro-elastic pulsation of the beam modeled analytically using the symbolic software MAPLE, where polynomial approximations, design sensitivities, probabilistic moments and their histories are computed and visualized.

On the gradient of curvature of the plumblines of the Earth’s normal gravity field and its isocurvature lines

This paper presents an approach to determine the gradient of curvature of the normal plumblines at a point P above the ellipsoid and introduces a new geometrical object which is the isocurvature line. The assumed facts are the coordinates of the point P and the formula for the normal gravity potential U. For the determination of the gradient of the normal plumbline curvature k at the point P we define a small circle on the meridian plane of P whose center is at the point P. The circle has the radius of one meter and interior D. In this circle we construct a curvature replacement function to approximate the curvature function k. This replacement function is a quotient of polynomials hence it is easy to find its partial derivatives at the point P. For the construction of replacement function we make the assumption that in the interior of the circle D the first order partial derivatives of U behave linearly and the second order partial derivatives have constant values which equal their value at the point P. Then we set the gradient of the curvature function to be equal with the gradient of the aforementioned replacement function at P. An isocurvature line of the normal gravity field passing through a point P is a curve such that the value of the function of the plumblines’ curvature k is constant and equals k(P). We give a formula to find the direction of the isocurvature line on the meridian plane and we prove that there are infinitely many isocurvature lines passing through the point P and they all lie on a special surface, the isocurvature surface.

Schwarz-pick estimates for bounded holomorphic functions on classical domains

In this paper, Schwarz-Pick estimates for high order Fréchet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.

Local Collocation Approach for Solving Turbulent Combined Forced and Natural Convection Problems

AbstractAn application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number к – ε model with Launder and Sharma and Abe et al. closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF’s. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin’s fractional step method. The adaptive upwinding technique, proposed by Lin and Atluri , is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura for Reynolds number 4494, based on the channel width, and Grash of number 9.6×105. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.

On the local structure of optimal measures in the multi-marginal optimal transportation problem

We consider an optimal transportation problem with more than two marginals. We use a family of semi-Riemannian metrics derived from the mixed, second order partial derivatives of the cost function to provide upper bounds for the dimension of the support of the solution.

Multivariate sigmoidal neural network approximation

Here we study the multivariate quantitative constructive approximation of real and complex valued continuous multivariate functions on a box or RN, N∈N, by the multivariate quasi-interpolation sigmoidal neural network operators. The "right" operators for our goal are fully and precisely described. This approximation is derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the logarithmic sigmoidal function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Multivariate hyperbolic tangent neural network approximation

Here we study the multivariate quantitative approximation of real and complex valued continuous multivariate functions on a box or RN, N∈N, by the multivariate quasi-interpolation hyperbolic tangent neural network operators. This approximation is derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the hyperbolic tangent function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

On a Fractional Master Equation

A fractional order time‐independent form of the wave equation or diffusion equation in two dimensions is obtained from the standard time‐independent form of the wave equation or diffusion equation in two‐dimensions by replacing the integer order partial derivatives by fractional Riesz‐Feller derivative and Caputo derivative of order α, β, 1 < ℜ(α) ≤ 2 and 1 < ℜ(β) ≤ 2 respectively. In this paper, we derive an analytic solution for the fractional time‐independent form of the wave equation or diffusion equation in two dimensions in terms of the Mittag‐Leffler function. The solutions to the fractional Poisson and the Laplace equations of the same kind are obtained, again represented by means of the Mittag‐Leffler function. In all three cases, the solutions are represented also in terms of Fox′s H‐function.

Schwarz–Pick estimates for holomorphic mappings from the polydisk to the unit ball

In this paper, Schwarz–Pick estimates of arbitrary order partial derivatives for holomorphic mappings from the polydisk to the unit ball are presented. We generalize the early work on Schwarz–Pick estimates of higher order derivatives for bounded holomorphic functions on the unit disk and the polydisk.

Generalized stochastic finite element method in elastic stability problems

The main issue of this paper is the stability analysis of elastic systems with random parameters using the Generalized Stochastic Finite Element Method. The Taylor expansion with random coefficients of nth order is used to express all random functions and to determine up to fourth order probabilistic moments of the critical force or critical pressure. The response function method assists to determine higher order partial derivatives of the structural response instead of the Direct Differentiation Method employed widely before. This approach is examined on the classical Euler problem, 2D and 3D steel frames as well as in addition to the cylindrical shell with some geometrical parameters defined as the Gaussian variables. The comparison of the GSFEM versus the Monte-Carlo simulation on the Euler problem proves the probabilistic convergence of this new technique.

Orientation-Preserving Rod Elements for Real-Time Thin-Shell Simulation

We propose a new computation model for simulating elastic thin shells at interactive rates. Existing graphical simulation methods are mostly based on dihedral angle energy functions, which need to compute the first order and second order partial derivatives with respect to current vertex positions as bending forces and stiffness matrices. The symbolic derivatives are complicated in nonisometric element deformations. To simplify computing the derivatives, instead of directly constructing the dihedral angle energy, we use the orientation change energy of mesh edges. A continuum-mechanics-based orientation-preserving rod element model is developed to provide the bending forces. The advantage of our method is simple bending force and stiffness matrix computation, since in the rod model, we apply a novel incremental construction of the deformation gradient tensor to linearize both tensile and orientation deformations. Consequently, our model is efficient, easy to implement, and supports both quadrilateral and triangle meshes. It also treats shells and plates uniformly.

A Method for Rapid Matching Based on Second Order Partial Derivative

Our goal is to enhance matching speed which is important for image engineering. Second Partial Derivative operator in Harris corner detector is directly used to compute the similarity between corners. Then initial matches are obtained. The algorithm is contrasted with normalized cross-correlation and method based on horizontal and vertical gradient. Its computational complexity is reduced and matching speed is improved effectively because this method only adopts the addition and subtraction operations. Experiments on several real images test the matching speed, the matching precision and matching rate of the algorithm. The results demonstrate that the algorithm not only have higher speed but also get higher matching precision and correct matching rate. Even though the stereo image pairs have brightness differences, it still performs rather well.

膜系光谱系数对膜层参数的一阶和二阶偏导数的计算模型

膜系光谱系数对膜层参数的一阶和二阶偏导数的计算模型

Formula omitted] piecewise cubic quasi-interpolants on a 6-direction mesh

We study two kinds of quasi-interpolants (abbr. QI) in the space of C 2 piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinity norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples.