Order Partial Derivatives Research Articles

Acoustic analysis of anisotropic poroelastic multilayered systems

The proposed method allows for an extended analysis of the wave analysis, internal powers, and acoustic performance of anisotropic poroelastic media within semi-infinite multilayered systems under arbitrary excitation. Based on a plane wave expansion, the solution is derived from a first order partial derivative as proposed by Stroh. This allows for an in-depth analysis of the mechanisms controlling the acoustic behaviour in terms of internal powers and wave properties in the media. In particular, the proposed approach is used to highlight the influence of the phenomena intrinsic to anisotropic poroelastic media, such as compression-shear coupling related to the material alignment, the frequency shift of the fundamental resonance, or the appearance of particular geometrical coincidences in multilayered systems with such materials.

Schwarps: Locally Projective Image Warps Based on 2D Schwarzian Derivatives

Image warps -or just warps- capture the geometric deformation existing between two images of a deforming surface. The current approach to enforce a warp's smoothness is to penalize its second order partial derivatives (Bookstein in IEEE Trans Pattern Anal Mach Intell 11:567---585, 1989; Rueckert et al. in IEEE Trans Med Imaging 18:712---721, 1999). Because this favors locally affine warps, this fails to capture the local projective component of the image deformation. This may have a negative impact on applications such as image registration and deformable 3D reconstruction. We propose a novel penalty designed to smooth the warp while capturing the deformation's local projective structure. Our penalty is based on equivalents to the Schwarzian derivatives, which are projective differential invariants exactly preserved by homographies. We propose a methodology to derive a set of partial differential equations with only homographies as solutions. We call this system the Schwarzian equations and we explicitly derive them for 2D functions using differential properties of homographies. We name as Schwarp a warp which is estimated by penalizing the residual of Schwarzian equations. Experimental evaluation shows that Schwarps outperform existing warps in modeling and extrapolation power, and lead to better results in three deformable reconstruction methods, namely, shape reconstruction in shape-from-template, camera calibration in Shape-from-Template and Non-Rigid Structure-from-Motion.

Using Differentiation Term by Term Theorem to Study the Partial Differential Problems

This article takes the mathematical software Maple as the auxiliary tool to study the partial differential problem of two types of multivariable functions. We can obtain the infinite series forms of any order partial derivatives of these two types of functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we propose two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying our answers by using Maple.

A density result for Sobolev functions and functions of higher order bounded variation with additional integrability constraints

We prove density of smooth functions in subspaces of Sobolev- and higher order $BV$-spaces of kind $W^{m,p}(\Omega)\cap L^q(\Omega-D)$ and $BV^m(\Omega)\cap L^q(\Omega-D)$, respectively, where $\Omega\subset\mathbb{R}^n$ ($n\in\mathbb{N}$) is an open and bounded set with suitably smooth boundary, $m<n$ is a positive integer, $1\leq p<\infty$ s.t. $mp<n$, $D\Subset\Omega$ is a sufficiently regular open subset and $q> np/(n-mp)$. Here we say that a $W^{m-1,1}(\Omega)$-function is of $m$-th order bounded variation ($BV^m$) if its $m$-th order partial derivatives in the sense of distributions are finite Radon measures. This takes up earlier results by C. Tietz and the author concerning functions with merely one order of differentiability which emerged in the context of a variational problem related to image analysis. In the connection of our methods we also investigate a question concerning the boundary traces of $W^{1,p}(\Omega)\cap L^q(\Omega)$-functions.

A Systematic Analysis on Analyticity of Semisimple Eigenvalues of Matrix-Valued Functions

In this paper we study the existence of analytic eigenvalue functions of an analytic matrix-valued function $L(\lambda,\rho)$. Instead of proposing sufficient conditions for each individual case as in the literature, we propose a systematic scheme to discuss the existence of analytic eigenvalue functions of $L(\lambda,\rho)$ when $\lambda_0$ is a semisimple eigenvalue of $L(\lambda,0)$. We show that $\lambda(\rho)=\lambda_0+\rho\mu(\rho)$ is an eigenvalue of $L(\lambda,\rho)$ if and only if $\mu(\rho)$ is an eigenvalue of another analytic matrix-valued function $P(\mu,\rho)$ which is constructed based on the first order (partial) derivatives of $L(\lambda,\rho)$ at $(\lambda_0,0)$. Based on this result, a systematic scheme is proposed to check whether there exist analytic eigenvalue functions of $L(\lambda,\rho)$. This systematic scheme covers existing sufficient conditions in the literature, and can lead to much more general conditions.

Small analytic solutions to the Hartree equation

We consider the Cauchy problem for the Hartree equation in space dimension d ≥ 3 . We assume that the interaction potential V belongs to the weak L d / 2 space. We prove that if the initial data ϕ is sufficiently small in the L 2 -sense and either ϕ or its Fourier transform F ϕ satisfies a real-analytic condition, then the solution u ( t ) is also real-analytic for any t ≠ 0 . We also prove that if ϕ and V satisfy some strong condition, then u ( t ) can be extended to an entire function on C d for any t ≠ 0 . A part of our method is applicable to the final value problem. We remark that no L 2 smallness condition is imposed on first and higher order partial derivatives of ϕ and F ϕ .

Calculation of the Classic-Curvature and the Intensity-Curvature Term Before Interpolation of a Bivariate Polynomial

This paper presents the calculation of the classic-curvature and the intensity-curvature term before interpolation of a bivariate polynomial model function. The classic-curvature is termed as yc (x, y) and the intensity-curvature term before interpolation is termed as E0. The classic-curvature is defined as the sum of the four second order partial derivatives of the bivariate polynomial. The intensity-curvature term before interpolation is defined as the integral of the product between the pixel intensity value termed as f(0, 0) and the classic-curvature calculated at the origin of the coordinate system of the pixel. This paper presents an application of the calculation of classic-curvature and the intensity- curvature term before interpolation using two- dimensional Magnetic Resonance Imaging (MRI) data and reports for the first time in the literature on the behavior of the intensity-curvature term before interpolation. Index Terms—Magnetic Resonance Imaging (MRI), classic-curvature, intensity-curvature term before interpolation, intensity-curvature measurement approaches, first order partial derivative, second order partial derivative, bivariate polynomial, model function, image.

Gradient‐based back‐propagation dynamical iterative learning scheme for the neuro‐fuzzy inference system

AbstractIn this paper, a gradient‐based back propagation dynamical iterative learning algorithm is proposed for structure optimization and parameter tuning of the neuro‐fuzzy system. Premise and consequent parameters of the neuro‐fuzzy model are initialized randomly and then tuned by the proposed iterative algorithm. The learning algorithm is based on the first order partial derivative of the output with respect to the structure parameters. The first order derivative of the model output with respect to the structure parameters determines the sensitivity of the model to structure parameters. The sensitivity values are then used to set the tuning factors and parameters updating step sizes. Therefore, an adaptive dynamical iterative scheme is achieved which adapts the learning procedure to the current state of the performance during the optimization process. Larger tuning step sizes make the convergence speed higher and vice versa. In this regard, this parameter is treated according to the calculated sensitivity of the model to the parameter. The proposed learning algorithm is compared with the least square back propagation method, genetic algorithm and chaotic genetic algorithm in the neuro‐fuzzy model structure optimization. Smaller mean square error and shorter learning time are sought in this paper, and the performance of the proposed learning algorithm is versified regarding these criteria.

On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in [formula omitted] or [formula omitted

We consider weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in L1 or L∞ norms. We provide conditions on the weights under which the corresponding spaces have equivalent norms with constants independent of, or only polynomially dependent on the number of variables.

Construction of wave-free potentials and multipoles in a two-layer fluid having free-surface boundary condition with higher-order derivatives

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation (or Helmholtz equation) arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.

Control point based exact description of trigonometric/hyperbolic curves, surfaces and volumes

Using the normalized B-bases of vector spaces of trigonometric and hyperbolic polynomials of finite order, we specify control point configurations for the exact description of the zeroth and higher order (mixed partial) derivatives of integral curves and (hybrid) multivariate surfaces determined by coordinate functions that are exclusively given either by traditional trigonometric or hyperbolic polynomials in each of their variables. Based on homogeneous coordinates and central projection, we also propose algorithms for the control point and weight based exact description of the zeroth order (partial) derivative of the rational counterpart of these integral curves and surfaces. The core of the proposed modeling methods relies on basis transformation matrices with entries that can be efficiently obtained by order elevation.

Differentiability properties of the minimum time function for normal linear systems

A result by Sussmann [32], implying that the minimum time function T for normal linear systems is analytic out of a locally finite union of analytic submanifolds, is revisited. The original proof relies on classical properties of subanalytic sets. A constructive and simpler proof is given of a part of it: we show that the singular, i.e., the nondifferentiability, set of T is contained in a lower dimensionally rectifiable set which can be identified through suitable properties of the switching function. Our approach extends to large times the strategy initiated by Hájek in [23]. Furthermore, a result on the propagation of singularities of non-Lipschitz type is proved. Finally, we give explicit formulas for the propagation of first and second order partial derivatives of T along optimal trajectories and we prove that normal cones to the epigraph of T at each (x,T(x)) and to the sublevel of T at x have the same dimension.

Performance Verification of a Fuzzy Wavelet Neural Network in the First Order Partial Derivative Approximation of Nonlinear Functions

Approximation of the first order partial derivative of a function modeled from a set of discrete data is the requirement of several applications. However, using a direct method for calculating the partial derivative from a set of discrete points is preferred rather differentiating the function which is obtained by modeling the discrete dataset. In this paper, the first order partial derivative of a fuzzy wavelet neural network structure is calculated to act as a direct differentiator. The structure of the network is described and its parameters are tuned by an adaptive gradient-based back propagation learning algorithm. It is shown that the proposed model outperforms the adaptive neuro-fuzzy inference-based and feed forward neural network-based differentiators in approximating the first order partial derivatives of multi-variable nonlinear functions.

Predictive Control with Parameter Adaptation to Achieve α-Protection in the RECONFIGURE Benchmark in the Presence of Icing

Undetected ice accretions on aerodynamic surfaces can deeply change the dynamic behaviour of aircraft, leading to poor performance of the automatic control systems. This is characterised by oscillatory behaviour, and overshoot of setpoints and ight envelope protection values. To mitigate this undesirable behaviour, this paper applies predictive control in combination with online estimation of the first and second order partial derivatives of the lift and pitching moment coeffcients with respect to the angle of attack using an Extended Kalman Filter, to achieve a constrained indirect adaptive ight control law. The design is evaluated on the RECONFIGURE benchmark, which is a nonlinear, high fidelity, industrially validated simulator of a large Airbus aircraft. In icing scenarios at high incidence, the resulting trajectories are shown to be better damped and more compliant with constraints when compared to a predictive control law employing only linear disturbance estimation.

Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays

In the paper we investigate the continuity properties of the mapping \({\Phi}\) which sends any non-empty compact connected hv-convex planar set K to the associated generalized conic function fK. The function fK measures the average taxicab distance of the points in the plane from the focal set K by integration. The main area of applications is geometric tomography because fK involves the coordinate X-rays’ information as second order partial derivatives (Nagy and Vincze, J Approx Theory 164: 371–390, 2012). We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the supremum-norm and the L1-norm provided that we restrict the domain to the collection of non-empty compact connected hv-convex planar sets contained in a fixed box (reference set) with parallel sides to the coordinate axes. We also have that \({\Phi^{-1}}\) is upper semi-continuous as a set-valued mapping. The upper semi-continuity establishes an approximating process in the sense that if fL is close to fK then L must be close to an element \({K^\prime}\) such that \({f_{K}=f_{K^\prime}}\). Therefore K and \({K^\prime}\) have the same coordinate X-rays almost everywhere. Lower semi-continuity is usually related to the existence of continuous selections. If a set-valued mapping is both upper and lower semi-continuous at a point of its domain it is called continuous. The last section of the paper is devoted to the case of non-empty compact convex planar sets. We show that the class of convex bodies that are determined by their coordinate X-rays coincides with the family of convex bodies K for which fK is a point of lower semi-continuity for \({\Phi^{-1}}\).

Solving the Partial Differential Problems Using Maple

This paper considers the partial differential problem of two types of multivariable functions and uses mathematicalsoftware Maple for verification. The infinite series forms of any order partial derivatives of these two types ofmultivariable functions can be obtained using binomial series and differentiation term by term theorem, which greatlyreduce the difficulty of calculating their higher order partial derivative values. On the other hand, four examples areused to demonstrate the calculations.

Numerical solution of a diffusion problem by exponentially fitted finite difference methods.

This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver.

Existence of invariant measures of stochastic systems with delay in the highest order partial derivatives

In this note, we shall consider the existence of invariant measures for a class of infinite dimensional stochastic functional differential equations with delay whose driving semigroup is eventually norm continuous. The results obtained are applied to stochastic heat equations with distributed delays which appear in such terms having the highest order partial derivatives. In these systems, the associated driving semigroups are generally non eventually compact.

The distance to the Galactic center determined by OB stars

The OB stars are concentrated near the Galactic plane and should permit a determination of the distance to the Galactic center. van Leeuwen’s new reduction of the Hipparcos catalog provides, after 824 Gould belt stars have been excluded, 6288 OB stars out to 1 kpc and Westin’s compilation an additional 112 stars between 1 kpc and 3 kpc. The reduction model involves 14 unknowns: the Oort A and B constants, the distance to the Galactic center R0, 2 second-order partial derivatives, the 3 components of solar motion, a K term, a first order partial derivative for motion perpendicular to the Galactic plane, a second-order partial for acceleration perpendicular to the plane, two terms for a possible expansion of the OB stars, and a C constant. The model is nonlinear, and the unknowns are calculated by use the simplex algorithm for nonlinear adjustment applied to 14313 equations of condition, 12694 in proper motion and 1619 in radial velocity. Various solutions were tried: an L1 solution, a least squares solution with modest (2.7 %) trim of the data, and two robust least squares solutions (biweight and Welsch weighting) with more extreme trimming. The Welsch solution seems to give the best results and calculates a distance to the Galactic center 6.72±0.39 kpc. Statistical tests show that the data are homogeneous, that the reduction model seems adequate and conforms with the assumptions used in its derivation, and that the post-fit residuals are random. Inclusion of more terms, such as streaming motion induced by Galactic density waves, degrades the solution.

On the forecast of periodic events from incomplete records

In practice several natural phenomena like wind direction, death rates, egg production ,etc. follow circular normal distribution. In this paper the problem of estimating the parameters of this distribution from incomplete records has been worked out by the method of maximum likelihood. Second order partial derivatives are given to determine the standard error of the estimates. A numerical example is added to illustrate the practical application of the method.