Second Derivative Of Function Research Articles

Integrable theories and generalized graded Maillet algebras

We present a general formalism to investigate the integrable properties of a large class of non-ultralocal models which in principle allows the construction of the corresponding lattice versions. Our main motivation comes from the subsector of the string theory on AdS5 × S5 in the uniform gauge, where such type of non-ultralocality appears in the resulting Alday–Arutyunov–Frolov (AAF) model. We first show how to account for the second derivative of the delta function in the Lax algebra of the AAF model by modifying Maillet’s r- and s-matrices formalism, and derive a well-defined algebra of transition matrices, which allows for the lattice formulation of the theory. We illustrate our formalism on the examples of the bosonic Wadati–Konno–Ichikawa–Shimizu (WKIS) model and the two-dimensional free massive Dirac fermion model, which can be obtained by a consistent reduction of the full AAF model, and give the explicit forms of their corresponding r-matrices.

On best‐fit corotated frames for 3D continuum finite elements

SUMMARYWe discuss a strategy to construct corotated frames for three‐dimensional continuum finite elements by minimizing deformations within the frame. We find that irrespective of the type of element and the number of nodes, using a quaternion parametrization of rotations, this minimization is naturally stated as computing the smallest eigenvalue of a 4 × 4 matrix. The simplicity of this smallest eigenvalue plays a crucial role when linearizing the kinematics. Ensuant quaternion algebra, although lengthy, results in remarkably simple formulas for projections that arise in the linearized kinematics. The exact stiffness matrix does not require computation of the second derivative of the rotation function and is also given by a simple formula. As a result, the implementation of this corotational formulation becomes particularly straightforward. Furthermore, in contrast to other results in the literature, the stiffness matrix for elements with translational DOFs is symmetric. For illustration, this corotational formulation is applied to a solid‐shell finite element, and numerical results are presented. Copyright © 2014 John Wiley & Sons, Ltd.

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability

Based on Huang’s accurate tri-sectional nonlinear kinematic equation (1997), a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external moving boundary, which are different from the classical Stefan problem in heat conduction: The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transformation, the nonlinear partial differential equation (PDE) system is transformed into a linear PDE system. Then an analytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact analytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensitive effects of the dimensionless variable on the dimensionless pressure distribution and dimensionless pressure gradient distribution becomemore serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensitive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient (TPG) has a great effect on the external moving boundary but has little effect on the internal moving boundary. Open image in new window

Second derivative of Voigt function

In high-temperature high-pressure environment, the measurement precisions of tunable diode laser absorption spectroscopy and other laser spectrum technologies are influenced by spectral overlap because of Doppler and Lorentz broadenings. One of the potential methods to improve precision is to use the second derivative spectral signal, which has less overlap. This paper deals with the second derivative of Voigt function. The integration of its second derivative from negative to positive infinity is proved to be zero. And the analytical results of its second derivative minimum and the maxima or minima of its even-order derivatives are obtained. It is also shown that there is the relationship between the ratio of second derivative maximum point location to zero point location and the ratio of Lorentz half-width to Doppler half-width. These results provide the basis for inversing precision information from second derivative spectral signal.

Continuous-time method and its discretization to inverse problem of intensity-modulated radiation therapy treatment planning

We propose a novel approach for solving box-constrained inverse problems in intensity-modulated radiation therapy (IMRT) treatment planning based on the idea of continuous dynamical methods and split-feasibility algorithms. Our method can compute a feasible solution without the second derivative of an objective function, which is required for gradient-based optimization algorithms. We prove theoretically that a double Kullback–Leibler divergence can be used as the Lyapunov function for the IMRT planning system.Moreover, we propose a non-negatively constrained iterative method formulated by discretizing a differential equation in the continuous method. We give proof for the convergence of a desired solution in the discretized system, theoretically. The proposed method not only reduces computational costs but also does not produce a solution with an unphysical negative radiation beam weight in solving IMRT planning inverse problems.The convergence properties of solutions for an ill-posed case are confirmed by numerical experiments using phantom data simulating a clinical setup.

The Interlaminar Stresses Analysis of Composite Laminated Plates Based on the Generalized Higher-Order Global-Local Plate Theory

A generalized higher-order global-local theory was presented. The transverse shear stresses can be got directly through the constitutive equation without using the equilibrium equation. The second derivative of interpolation function was deduced. The hammer integration of triangular area coordinate method was applied to solve the multiple integration problem of the element stiffness matrix. The order choice of numerical integration was discussed and results obtained through two different integration orders were compared. The flow of how to compile a FORTRAN program was given. A moderately thick composite laminated plate was analyzed via finite element method (FEM) based on the theory and results were compared with that of Paganos three-dimensional elasticity. It shows that the interlaminar stresses are accurate for moderately thick laminated plates.

Propagation and scattering of waves in inhomogeneous optical media

One of the most remarkable features of wave dynamics in inhomogeneous media is the development of late-time ‘tails’. These tails reflect the backscattering of the waves by some effective scattering potential. The inhomogeneity of an optical medium can be characterized by a nonuniform refractive index \U0001d4a9(x). In this work we present a systematic analysis of the optical tail phenomenon for waves propagating in generic inhomogeneous media (with a generic form of the inhomogeneous refractive function \U0001d4a9(x)). We show that if \U0001d4a9‴(x) approaches zero faster than \U0001d4a9″(x) far away from the source, then the optical late-time tail is determined solely by the second spatial derivative of the inhomogeneous function \U0001d4a9(x). On the other hand, if \U0001d4a9‴(x) approaches zero at the same rate as \U0001d4a9″(x), then the late-time tail characterizing the optical wave is dominated by the entire set of derivatives of the inhomogeneous refractive function.

On the existence of supply function equilibria

Supply function equilibria are used in the analysis of divisible good auctions with a large number of identical objects to be sold or bought. An important example occurs in wholesale electricity markets. Despite the substantial literature on supply function equilibria the existence of a pure strategy Nash equilibria for a uniform price auction in asymmetric cases has not been established in a general setting. In this paper we prove the existence of a supply function equilibrium for a duopoly with asymmetric firms having convex non-decreasing marginal costs, with decreasing concave demand subject to an additive demand shock, provided that the second derivative of the demand function is small enough and not increasing. The proof is constructive and also gives insight into the structure of the equilibrium solutions.

SU‐E‐T‐661: Nonlinear Continuous Method for Non‐Negatively Constrained Inverse Problem of IMRT Planning

Purpose: The intensity‐modulated radiation therapy (IMRT) is an effective therapeutic technique for the treatment of tumors in the human body without inflicting serious damage to normal tissues. IMRT plans include an optimization strategy for minimizing an objective function of radiation beam weights. We present a novel approach for solving non‐negatively constrained inverse problems in IMRT treatment planning, based on the idea of continuous‐time dynamical methods using nonlinear differential equations. Our method can compute a feasible solution without the second derivative of an objective function, which is required for gradient‐based optimization algorithms such as conjugate gradient and steepest descent methods. Methods: We first consider the stability of an equilibrium, which corresponds to the desired planning in the IMRT system, by the use of Lyapunov's stability theorem. Then we perform some phantom experiments using phantom data simulating a clinical setup. Results: We prove theoretically that a Kullback‐Leibler (KL) divergence can be a Lyapunov function for the IMRT planning system, in consistent case. It means that the KL‐divergence measure decreases along the solution to the differential equation. Additionally, we show that the intensity of any radiation beam is not negative. Because the planning system can be created as an analog electronic circuit, its implementation in actual hardware yields dramatically faster planning as a physical phenomenon with real parallel computing, than software‐based iterative methods. Conclusion: The proposed method provides not only the reduction of a computational cost but also no production of a solution with an unphysical negative radiation beam coefficient in solving IMRT planning inverse problems. Moreover the theoretical results on convergence properties of solutions are confirmed by numerical experiments. Based on the results, we found that we can obtain a feasible solution violating the physical dose constraints minimally, even if the system is inconsistent.

Identities from infinite-dimensional symmetries of Herglotz variational functional

This paper formulates and proves a theorem which provides the identities corresponding to infinite-dimensional symmetry groups of the functional defined by the generalized variational principle of Herglotz in the case of several independent variables. It contains the classical second Noether theorem as a special case. The equations satisfied by the extrema of this functional, when the Lagrangian density depends on first and second order partial derivatives of the argument functions, are found. We apply the theorem to find two new identities satisfied by the four-potential of the electromagnetic field propagating in a conductive medium. We also obtain an identity corresponding to a gauge symmetry of the Klein-Gordon equation with dissipation/generation. This identity becomes a continuity law when the wave function is a solution of this equation.

Derivatives of maximum‐entropy basis functions on the boundary: Theory and computations

SUMMARYIn this paper, we obtain explicit expressions to evaluate the derivatives of maximum‐entropy (max‐ent) basis function on the boundary of a convex domain. In the max‐ent formulation, the basis functions are obtained by maximizing a concave functional subjected to linear constraints (reproducing conditions). In doing so, it is found that the Lagrange multipliers blow up when x ∈ ∂Ω, and the expressions for the derivatives of the max‐ent basis functions in Ω are of an indeterminate form for points on ∂Ω. We appeal to l'Hôpital's rule to derive expressions to determine the derivatives of the basis functions. We consider the Shannon entropy functional and the relative entropy functional with different choices of the prior weight function. The first‐order derivatives of all basis functions are bounded. In contrast, on an irregular grid with a certain nodal spacing, some of the second derivatives of the basis functions are unbounded on the boundary. Necessary and sufficient conditions on the priors to obtain bounded Lagrange multipliers are established. Optimal convergence rates for fourth‐order problems are demonstrated for a Galerkin approach with a quadratically complete partition‐of‐unity enriched max‐ent approximation.Copyright © 2013 John Wiley & Sons, Ltd.

A Numerical Method for Estimating the Variance of Age at Maximum Growth Rate in Growth Models

Studies on maturation and body composition mention age at peak height velocity (PHV) as an important measure that could predict adulthood outcome. The age at PHV is often derived from growth models such as the triple logistic fitted to the stature (height) data. Theoretically, for a well-behaved growth function, age at PHV could be obtained by setting the second derivative of the growth function to zero and solving for age. Such a solution obviously depends on the parameters of the growth function. Therefore, the uncertainty in the estimation of age at PHV resulting from the uncertainty in the estimation of the growth model, need to be accounted for in the models in which it is used as a predictor. Explicit expressions for the age at PHV and, consequently the variance of the estimate of the age at PHV, do not exist for some of the commonly used nonlinear growth functions, such as the triple logistic function. Once an estimate of this variance is obtained, it could be incorporated in subsequent modeling either through measurement error models or by using the inverse variances as weights. A numerical method for estimating the variance is implemented. The accuracy of this method is demonstrated through comparisons in models where explicit solution for the variance exists. The method of estimating the variance is illustrated by applying to growth data from the Fels study and subsequently used as weights in modeling two adulthood outcomes from the same study.

Analytical sensitivity and Hessian matrix analysis of PSD functions for random seismic responses

This paper describes three analytical methods to calculate sensitivity and Hessian matrix of structural Power Spectral Density (PSD) functions for random seismic responses. The methods are derived based on Complete Quadratic Combination method (CQC), Square Root of the Sum of Squares method (SRSS) and Pseudo-Excitation Method (PEM) by direct differentiation. A planar frame and a building are demonstrated. The results indicate the first and second derivatives of PSD functions, achieved by CQC-based method, PEM-based method, are exactly same. The first and second derivatives achieved by SRSS-based method are almost the same as that obtained by other two methods. PEM-based method is the most efficient method. SRSS-based method is much more efficient than CQC-based method.

Design of Shaped PVDF Radiation Mode Sensor and its Application in Active Structural Acoustic Control System

This paper presents a new strategy for the design of radiation mode sensors by using shaped PVDF films for one-dimension structures. Based on a modal approach, the shape of a PVDF sensor is analytically determined in such a way that the output signal of the sensor is directly proportional to amplitude of a particular radiation mode. Other modes are filtered out. A general expression of the PVDF sensor shape is obtained. It is found that the modal sensor shape can be expressed as a function of the second spatial derivative of the structural mode shape function. Finally, with an example of a vibrating beam, the proposed PVDF radiation mode sensor is used in an active structural acoustic control system for reduction of the structural-borne noise.

Optical study of bulk and thin-film tin dioxide

We grew nano-crystalline SnO2 thin films on Si substrates by using RF magnetron sputtering deposition method. We examined the microstructure of the SnO2 thin films via X-ray diffraction (XRD) and the optical properties of the SnO2 thin films via photoluminescence (PL) spectroscopy and spectroscopic ellipsometry. We obtained the dielectric functions (e = e1 + ie2) of the SnO2 thin films in the visible to deep ultraviolet spectral range, as well as the anisotropic dielectric functions of the bulk SnO2 single crystal. The band edge blue shift and the larger peak maximum of e2z compared to those of e2x, in the directions parallel and perpendicular to the optical c-axis, respectively, are consistent with calculations found in the literature. We estimated the critical point energies by using the second derivatives of the dielectric functions with respect to energy, and the results matched well with the calculated band structures in the literature. We also found critical point energies in the experimental data which were forbidden by selection rules in the calculations. The PL study of the SnO2 thin films and bulk single crystals showed similar PL peaks associated with the band gap energy, suggesting that even the RT-grown thin film contained a very small number of crystalline nanograins even though the XRD spectra did not show any crystalline peaks.

Impact of nonlinearity and discontinuity on the spatial scaling effects of the leaf area index retrieved from remotely sensed data

This article addresses the major scaling problems in leaf area index (LAI) retrieval for a heterogeneous surface associated with (1) the nonlinearity in the relationships between remotely sensed reflectances and LAI products, (2) the discontinuity caused by the mixture of contrasting cover types that is categorized as the dominating type within a large-scale pixel, and (3) the algorithm for the dominant cover type being used for the retrieval of the LAI in that large-scale mixed pixel. Through mathematical analysis, two scaling models (a component-based model and a pixel-based model) are proposed on the basis of the Taylor series expansion with the corresponding textural and contextural parameters (i.e. variance–covariance matrices and component fractions) to correct for the scaling effects among LAI products at different scales. These models express the magnitude of the scaling effects for the nonlinear and discontinuous situations as a function of (1) the degree of nonlinearity quantified by the second derivative of the retrieval function, (2) the spatial heterogeneity quantified by variance–covariance matrices, and (3) the component fractions in the large-scale mixed pixel. To evaluate the proposed scaling models, a scaling correction test is performed and analysed on a SPOT (Système Pour l'Observation de la Terre) image for two vegetation types. The component fractions have proven to be the main reason for the scaling effects in a mixed pixel. Compared to the results before scaling, using either of the two proposed models greatly reduces the retrieval errors that the scaling effects cause. The relative scaling effects of the LAI may be up to 55% in an uncorrected, large-scale mixed pixel. However, the relative scaling errors can be as low as 2% with the intra-component textural parameters and about 13% with the intra-pixel textural parameters. Because the scaling effects can be corrected for the spatial heterogeneity caused either by density changes within the same cover or by cover type changes, our work indicates that the proposed scaling models are promising and feasible.

Exact analytical solutions of moving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient

The dimensionless mathematical models of one-dimensional flow in the semi-infinite long porous media with threshold pressure gradient are built for the two cases of constant flow rate and constant production pressure on the inner boundaries. Through formula deduction, it is found that the velocity of the moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on the moving boundary, which is very different from the classical heat-conduction Stefan problems. However, by introducing some similarity transformation from Stefan problems, the exact analytical solutions of the dimensionless mathematical models are obtained, which can be used for strict validation of approximate analytical solutions, numerical solutions and pore-scale network modeling for the flow in porous media with threshold pressure gradient. Comparison curves of the dimensionless pressure distributions and the transient dimensionless production pressure under different values of dimensionless threshold pressure gradient are plotted from the exact analytical solutions of problems of the flow in semi-infinite long porous media with and without threshold pressure gradient. It is shown that for the case of constant flow rate the effect of the dimensionless threshold pressure gradient on the dimensionless pressure distributions and the transient dimensionless production pressure is not very obvious; in contrast, for the case of constant production pressure the effect on the dimensionless pressure distributions is more obvious especially at the larger dimensionless distance near the moving boundary; and for the case of constant production pressure, the smaller the dimensionless threshold pressure gradient is, the larger the dimensionless pressure is, and the further the pressure disturbance area reaches.

Corrected discrete least-squares meshless method for simulating free surface flows

Abstract In this paper, a modified version of discrete least-squares meshless (DLSM) method is used to simulate free surface flows with moving boundaries. DLSM is a newly developed meshless approach in which a least-squares functional of the residuals of the governing differential equations and its boundary conditions at the nodal points is minimized with respect to the unknown nodal parameters. The meshless shape functions are also derived using the Moving Least Squares (MLS) method of function approximation. The method is, therefore, a truly meshless method in which no integration is required in the computations. Since the second order derivative of the MLS shape function are known to contain higher errors compared to the first derivative, a modified version of DLSM method referred to as corrected discrete least-squares meshless (corrected DLSM) is proposed in which the second order derivatives are evaluated more accurately and efficiently by combining the first order derivatives of MLS shape functions with a finite difference approximation of the second derivatives. The governing equations of fluid flow (Navier–Stokes) are solved by the proposed method using a two-step pressure projection method in a Lagrangian form. Three benchmark problems namely; dam break, underwater rigid landslide and Scott Russell wave generator problems are used to test the accuracy of the proposed approach. The results show that proposed corrected DLSM can be employed to simulate complex free surface flows more accurately.

A quadratically convergent Newton method for vector optimization

We propose a Newton method for solving smooth unconstrained vector optimization problems under partial orders induced by general closed convex pointed cones. The method extends the one proposed by Fliege, Graña Drummond and Svaiter for multicriteria, which in turn is an extension of the classical Newton method for scalar optimization. The steplength is chosen by means of an Armijo-like rule, guaranteeing an objective value decrease at each iteration. Under standard assumptions, we establish superlinear convergence to an efficient point. Additionally, as in the scalar case, assuming Lipschitz continuity of the second derivative of the objective vector-valued function, we prove q-quadratic convergence.

Design of piezoelectric modal sensor for non-uniform Euler–Bernoulli beams with rectangular cross-section by using differential transformation method

In this paper, a solution to the problem of finding the shape of piezoelectric modal sensors for non-uniform Euler–Bernoulli beams with rectangular cross-sections is proposed by using the differential transformation method (DTM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. The modal sensor shape is expressed as a linear function of the second spatial derivative of the structural mode shape function as well as the beam width and thickness function. Based on the DTM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes for beams consisting of an arbitrary number of steps can be determined through a recursive way. The solution is obtained by solving a set of algebraic equations with only five unknown parameters. Furthermore, the method can be extended to obtain an approximate solution of the second spatial derivative of the mode shapes for any type of non-uniform beams. Then the shapes of the piezoelectric modal sensors for arbitrary non-uniform Euler–Bernoulli beam with rectangular cross-section are obtained. Finally, several numerical examples are given to demonstrate the feasibility of the proposed modal sensors with various boundary conditions.