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Second Derivatives (second + derivative)
Selected AbstractsADAPTIVE CHANGE IN THE RESOURCE-EXPLOITATION TRAITS OF A GENERALIST CONSUMER: THE CEOLUTION AND COEXISTENCE OF GENERALISTS AND SPECIALISTSEVOLUTION, Issue 3 2006Peter A. Abrams Abstract Mathematical models of consumer-resource systems are used to explore the evolution of traits related to resource acquisition in a generalist consumer species that is capable of exploiting two resources. The analysis focuses on whether evolution of traits determining the capture rates of two resources by a consumer species produce one generalist, two specialists, or all three types, when all types are characterized by a common fitness function. In systems with a stable equilibrium, evolution produces one generalist or two specialists, depending on the second derivative of the trade-off relationship. When there are sustained population fluctuations, the nature of the trade-off between the consumer's capture rates of the two resources still plays a key role in determining the evolutionary outcome. If the trade-off is described by a choice variable between zero and one that is raised to a power n, polymorphic states are possible when n > 1, which implies a positive second derivative of the curve. These states are either dimorphism, with two relatively specialized consumer types, or trimorphism, with a single generalist type and two specialists. Both endogenously driven consumer-resource cycles, and fluctuations driven by an environmental variable affecting resource growth are considered. Trimorphic evolutionary outcomes are relatively common in the case of endogenous cycles. In contrast to a previous study, these trimorphisms can often evolve even when new lineages are constrained to have phenotypes very similar to existing lineages. Exogenous cycles driven by environmental variation in resource growth rates appear to be much less likely to produce a mixture of generalists and specialists than are endogenous consumer-resource cycles. [source] Two Novel Triterpenes from the Leaves of Ficus microcarpaHELVETICA CHIMICA ACTA, Issue 5 2004Yueh-Hsiung Kuo Two novel triterpenes, 29(20,19)abeolupane-3,20-dione (4) and 19,20-secoursane-3,19,20-trione (5), besides (3,)-3-hydroxy-29(20,19)abeolupan-20-one (2), lupenone, and , -amyrone (6), were isolated from the leaves of Ficus microcarpa and were characterized by spectroscopic means, including 2D-NMR techniques and chemical methods. Compound 4 is the second derivative having the 29(20,19)abeolupane skeleton, and 5 is a novel skeleton. A biosynthetic pathway to 5 is proposed (Scheme). [source] An efficient co-rotational formulation for curved triangular shell elementINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2007Zhongxue Li Abstract A 6-node curved triangular shell element formulation based on a co-rotational framework is proposed to solve large-displacement and large-rotation problems, in which part of the rigid-body translations and all rigid-body rotations in the global co-ordinate system are excluded in calculating the element strain energy. Thus, an element-independent formulation is achieved. Besides three translational displacement variables, two components of the mid-surface normal vector at each node are defined as vectorial rotational variables; these two additional variables render all nodal variables additive in an incremental solution procedure. To alleviate the membrane and shear locking phenomena, the membrane strains and the out-of-plane shear strains are replaced with assumed strains in calculating the element strain energy. The strategy used in the mixed interpolation of tensorial components approach is employed in defining the assumed strains. The internal force vector and the element tangent stiffness matrix are obtained from calculating directly the first derivative and second derivative of the element strain energy with respect to the nodal variables, respectively. Different from most other existing co-rotational element formulations, all nodal variables in the present curved triangular shell formulation are commutative in calculating the second derivative of the strain energy; as a result, the element tangent stiffness matrix is symmetric and is updated by using the total values of the nodal variables in an incremental solution procedure. Such update procedure is advantageous in solving dynamic problems. Finally, several elastic plate and shell problems are solved to demonstrate the reliability, efficiency, and convergence of the present formulation. Copyright © 2007 John Wiley & Sons, Ltd. [source] Preliminary study on the use of near-infrared reflectance spectroscopy to assess nitrogen content of undried wheat plantsJOURNAL OF THE SCIENCE OF FOOD AND AGRICULTURE, Issue 1 2007Alejandro Morón Abstract Near-infrared reflectance (NIR) spectroscopy combined with chemometrics was used to assess nitrogen (N) and dry matter content (DM) and chlorophyll in whole-wheat plant (Triticum aestivum L). Whole-wheat plant samples (n = 245) were analysed by reference method and by visible and NIR spectroscopy, in fresh (n = 182) and dry (n = 63) presentations, respectively. Calibration equations were developed using partial least squares (PLS) and validated using full cross-validation (leave-one-out method). Coefficient of determination in calibration (R2CAL) and the standard error of cross-validation (SECV) for N content in fresh sample presentation, after second derivative, were 0.89 (SECV: 0.64%), 0.86 (SECV: 0.66%) and 0.82 (SECV: 0.74%) using the visible + NIR, NIR and visible wavelength regions, respectively. Dry sample presentation gave better R2CAL and SECV for N compared with fresh presentation (R2CAL > 0.90, SECV < 0.20%) using visible + NIR. The results demonstrated that NIR is a suitable method to assess N concentration in wheat plant using fresh samples (unground and undried). Copyright © 2006 Society of Chemical Industry [source] Maximum likelihood estimation of a latent variable time-series modelAPPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 1 2001Francesco Bartolucci Abstract Recently, Fridman and Harris proposed a method which allows one to approximate the likelihood of the basic stochastic volatility model. They also propose to estimate the parameters of such a model maximising the approximate likelihood by an algorithm which makes use of numerical derivatives. In this paper we propose an extension of their method which enables the computation of the first and second analytical derivatives of the approximate likelihood. As will be shown, these derivatives may be used to maximize the approximate likelihood through the Newton,Raphson algorithm, with a saving in the computational time. Moreover, these derivatives approximate the corresponding derivatives of the exact likelihood. In particular, the second derivative may be used to compute the standard error of the estimator and confidence intervals for the parameters. The paper presents also the results of a simulation study which allows one to compare our approach with other existing approaches. Copyright © 2001 John Wiley & Sons, Ltd. [source] Characterization and reconstruction of a rock fracture surface by geostatisticsINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 9 2002A. Marache Abstract It is well understood that, in studying the mechanical and hydromechanical behaviour of rock joints, their morphology must be taken into account. A geostatistical approach has been developed for characterizing the morphology of fracture surfaces at a decimetre scale. This allows the analysis of the spatial variability of elevations, and their first and second derivatives, with the intention of producing a model that gives a numerical three-dimensional (3D) representation of the lower and upper surfaces of the fracture. Two samples (I and II) located close together were cored across a natural fracture. The experimental data are the elevations recorded along profiles (using recording steps of 0.5 and 0.02 mm, respectively, for the samples I and II). The goal of this study is to model the surface topography of sample I, so getting estimates for elevations at each node of a square grid whose mesh size will be, for mechanical purposes, no larger than the recording step. Since the fracture surface within the sample core is not strictly horizontal, geostatistical methods are applied to residuals of elevations of sample I. Further, since structural information is necessary at very low scale, theoretical models of variograms of elevations, first and second derivatives are fitted using data of both that sample I and sample II. The geostatistical reconstructions are computed using kriging and conditional simulation methods. In order to validate these reconstructions, variograms and distributions of experimental data are compared with variograms and distributions of the fitted data. Copyright © 2002 John Wiley & Sons, Ltd. [source] Adaptive mesh technique for thermal,metallurgical numerical simulation of arc welding processesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2008M. Hamide Abstract A major problem arising in finite element analysis of welding is the long computer times required for a complete three-dimensional analysis. In this study, an adaptative strategy for coupled thermometallurgical analysis of welding is proposed and applied in order to provide accurate results in a minimum computer time. The anisotropic adaptation procedure is controlled by a directional error estimator based on local interpolation error and recovery of the second derivatives of different fields involved in the finite element calculation. The methodology is applied to the simulation of a gas,tungsten-arc fusion line processed on a steel plate. The temperature field and the phase distributions during the welding process are analyzed by the FEM method showing the benefits of dynamic mesh adaptation. A significant increase in accuracy is obtained with a reduced computational effort. Copyright © 2007 John Wiley & Sons, Ltd. [source] Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2005Jean-François Remacle Abstract An anisotropic adaptive analysis procedure based on a discontinuous Galerkin finite element discretization and local mesh modification of simplex elements is presented. The procedure is applied to transient two- and three-dimensional problems governed by Euler's equation. A smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified. The mesh metric field in smooth portions of the domain is controlled by a Hessian matrix constructed using a variational procedure to calculate the second derivatives. The transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain. Copyright © 2004 John Wiley & Sons, Ltd. [source] A return map algorithm for general isotropic elasto/visco-plastic materials in principal spaceINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2004Luciano Rosati Abstract We describe a methodology for solving the constitutive problem and evaluating the consistent tangent operator for isotropic elasto/visco-plastic models whose yield function incorporates the third stress invariant . The developments presented are based upon original results, proved in the paper, concerning the derivatives of eigenvalues and eigenprojectors of symmetric second-order tensors with respect to the tensor itself and upon an original algebra of fourth-order tensors obtained as second derivatives of isotropic scalar functions of a symmetric tensor argument . The analysis, initially referred to the small-strain case, is then extended to a formulation for the large deformation regime; for both cases we provide a derivation of the consistent tangent tensor which shows the analogy between the two formulations and the close relationship with the tangent tensors of the Lagrangian description of large-strain elastoplasticity. Copyright © 2004 John Wiley & Sons, Ltd. [source] Time accurate consistently stabilized mesh-free methods for convection dominated problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2003Antonio Huerta Abstract The behaviour of high-order time stepping methods combined with mesh-free methods is studied for the transient convection,diffusion equation. Particle methods, such as the element-free Galerkin (EFG) method, allow to easily increase the order of consistency and, thus, to formulate high-order schemes in space and time. Moreover, second derivatives of the EFG shape functions can be constructed with a low extra cost and are well defined, even for linear interpolation. Thus, consistent stabilization schemes can be considered without loss in the convergence rates. Copyright © 2003 John Wiley & Sons, Ltd. [source] Application of a new differential quadrature methodology for free vibration analysis of platesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2003G. Karami Abstract A new methodology is introduced in the differential quadrature (DQ) analysis of plate problems. The proposed approach is distinct from other DQ methods by employing the multiple boundary conditions in a different manner. For structural and plate problems, the methodology employs the displacement within the domain as the only degree of freedom, whereas along the boundaries the displacements as well as the second derivatives of the displacements with respect to the co-ordinate variable normal to the boundary in the computational domain are considered as the degrees of freedom for the problem. Employing such a procedure would facilitate the boundary conditions to be implemented exactly and conveniently. In order to demonstrate the capability of the new methodology, all cases of free vibration analysis of rectangular isotropic plates, in which the conventional DQ methods have had some sort of difficulty to arrive at a converged or accurate solution, are carried out. Excellent convergence behaviour and accuracy in comparison with exact results and/or results obtained by other approximate methods were obtained. The analogous DQ formulation for a general rectangular plate is derived and for each individual boundary condition the general format for imposing the given conditions is devised. It must be emphasized that the computational efforts of this new methodology are not more than for the conventional differential quadrature methods. Copyright © 2002 John Wiley & Sons, Ltd. [source] A hybrid Padé ADI scheme of higher-order for convection,diffusion problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2010Samir KaraaArticle first published online: 8 SEP 200 Abstract A high-order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection,diffusion problems. The scheme employs standard high-order Padé approximations for spatial first and second derivatives in the convection-diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22: 87,110) are applied for the time integration. The approximate factorization imposes a second-order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher-order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC-based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high-order ADI schemes for solving unsteady convection-diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given. Copyright © 2009 John Wiley & Sons, Ltd. [source] Fifth-order Hermitian schemes for computational linear aeroacousticsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2007Article first published online: 17 APR 200, G. Capdeville Abstract We develop a class of fifth-order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth-order accuracy. A one-dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy,Kovaleskaya (the ,,-P5 scheme'); the second one relies on a semi-discrete form and evolves in time the discrete unknowns by using a five-stage Runge,Kutta method (the ,RGK-P5 scheme'). Although the RGK-P5 scheme shares the same local spatial interpolator with the ,-P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi-dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210:133,170; J. Comput. Phys. 2006; 217:530,562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the ,reconstruction stage', we define a fifth-order local spatial interpolator based on an upwind stencil. In the ,decomposition stage', we decompose the time derivatives into simple wave contributions. In the ,evolution stage', we use these fluctuations to update either by a Cauchy,Kovaleskaya procedure or by a five-stage Runge,Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the ,evolution stage', two fifth-order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK-P5 appears as the best compromise between simplicity and accuracy, while the ,-P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright © 2007 John Wiley & Sons, Ltd. [source] On accurate boundary conditions for a shape sensitivity equation methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2006R. Duvigneau Abstract This paper studies the application of the continuous sensitivity equation method (CSEM) for the Navier,Stokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve flow derivatives at the boundary. Thus, accurate flow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate flow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least-squares procedure to evaluate accurate first and second derivatives of the flow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The flow and sensitivity fields are solved using an adaptive finite-element method. The proposed methodology is first verified on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity fields for realistic problems is then demonstrated. The flow and sensitivity fields for a NACA 0012 airfoil are used for fast evaluation of the nearby flow over an airfoil of different thickness (NACA 0015). Copyright © 2005 John Wiley & Sons, Ltd. [source] On convexity of MQAM's and MPAM's bit error probability functionsINTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMS, Issue 11 2009M. Naeem Abstract For MQAM and MPAM with practical values of M and Gray mapping, we provide a rigorous proof that the associated bit error probability (BEP) functions are convex of the signal-to-noise ratio per symbol. The proof employs Taylor series expansions of the BEP functions' second derivatives and term-by-term comparisons between positive and negative terms. Convexity results are useful for optimizing communication systems as in optimizing adaptive transmission policies. Copyright © 2009 John Wiley & Sons, Ltd. [source] Note on the calculation of analytical Hessians in the zeroth-order regular approximation (ZORA)INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 12 2006J. H. van Lenthe Abstract The previously proposed atomic zeroth-order regular approximation (ZORA) approach, which was shown to eliminate the gauge dependent effect on gradients and to be remarkably accurate for geometry optimizations, is tested for the calculation of analytical second derivatives. It is shown that the resulting analytic second derivatives are indeed exact within this approximation. The method proves to yield frequencies that are remarkably close to the experimental frequency for uranium hexafluoride but less satisfactory for the gold dimer. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] A comparative study of some red- and blue-shifted linear H-bonded complexes of N2JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 2 2008Sean A. C. McDowell Abstract Bond length changes, harmonic vibrational frequency shifts, and changes in the proton magnetic shielding of HX and HKrX (X = F, Cl) on complexation with N2 to form the linear red-shifted N2 , HX and linear blue-shifted N2 , HKrX complexes were determined by ab initio computations, with and without counterpoise correction, at the SCF and MP2(full) levels of theory using a 6-311++G(2d,2p) basis set. The MP2 computations agree with predictions from a perturbation theory model involving the first and second derivatives of the interaction energy with respect to displacement of the HX and HKr bond lengths from their equilibrium values in the isolated monomers. The theoretical results agree qualitatively with the experimentally observed frequency shifts, with near quantitative agreement for N2 , HKrCl. The characteristic downfield shift of the isotropic proton magnetic resonance in the red-shifted complexes was obtained, but for the blue-shifted complexes, the proton NMR shifts to higher fields. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2008 [source] Orbital-orthogonality constraints and basis-set optimizationJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 6 2006Fabio E. Penotti Abstract A new procedure is presented for introducing arbitrary orbital-orthogonality constraints in the variational optimization of otherwise nonorthogonal multiconfiguration electronic wave functions. It is based on suitable analytical changes to the expressions for the first and second derivatives of the electronic energy with respect to the independent variational parameters, and can be applied in the presence of symmetry constraints. It is tested using a second-derivative optimization procedure, the Optimized Basis Set,Generalized Multiconfiguration Spin-Coupled (OBS-GMCSC) approach, that can treat basis-function exponential parameters as variational parameters, to be optimized simultaneously with configuration, spin-coupling, and orbital coefficients. This enables rigorous optimization of basis-set exponential parameters even for fully orthogonal multiconfiguration wave functions. Test calculations are carried out, with optimized even-tempered basis sets, on Li2 and on the CH radical. For the latter, special attention is paid to the electronic spin density at the nuclei. © 2006 Wiley Periodicals, Inc. J Comput Chem 27: 762,772, 2006 [source] Dynamic Charge Equilibration-Morse stretch force field: Application to energetics of pure silica zeolitesJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 16 2002Jan Sefcik Abstract We present the Dynamic Charge Equilibration (DQEq) method for a self-consistent treatment of charge transfer in force field modeling, where atomic charges are designed to reproduce electrostatic potentials calculated quantum mechanically. Force fields coupled with DQEq allow charges to readjust as geometry changes in classical simulations, using appropriate algorithms for periodic boundary conditions. The full electrostatic energy functional is used to derive the corresponding forces and the second derivatives (hessian) for vibrational calculations. Using DQEq electrostatics, we develop a simple nonbond force field for simulation of silica molecular sieves, where nonelectrostatic interactions are described by two-body Morse stretch terms. Energy minimization calculations with the new force field yield accurate unit cell geometries for siliceous zeolites. Relative enthalpies with respect to quartz and third-law entropies calculated from harmonic vibrational analysis agree very well with available calorimetric data: calculated SiO2 enthalpies relative to ,-quartz are within 2 kJ/mol and entropies at 298 K are within 3 J/mol K of the respective experimental values. Contributions from the zero point energy and vibrational degrees of freedom were found to be only about 1 kJ/mol for the free energy of mutual transformations between microporous silica polymorphs. The approach presented here can be applied to interfaces and other oxides as well and it is suitable for development of force fields for accurate modeling of geometry and energetics of microporous and mesoporous materials, while providing a realistic description of electrostatic fields near surfaces and inside pores of adsorbents and catalysts. © 2002 Wiley Periodicals, Inc. J Comput Chem 23: 1507,1514, 2002 [source] Conformational analysis of thiopeptides: derivation of sp2 sulfur parameters for the CFF91 force fieldJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 10 2001Tran Trung Tran Abstract When a sulfur atom is used to substitute for the oxygen in peptide bonds, its bulkiness should restrict the conformational space available to an amino acid. This conformational restriction as well as the ability to confer resistance to enzymatic degradation in the body means that thio-substituted amino acids are potentially useful building blocks for drug design. To simulate the effects of thio substitution, force field parameters for sp2 sulfur are required. In this article, parameters for the thioamide group have been derived for the molecular mechanics CFF91 force field (available at http://www.ludwig.edu.au/archive/tran). The bond increment charges were obtained by fitting to ab initio charges and dipoles. The van der Waals parameters were obtained by fitting to high-resolution crystallographic data, and the nonbonded parameters were verified by comparing with experimentally derived lattice energy. The bonded parameters were derived by least-square fits to the ab initio calculated energy surfaces, i.e., conformational energy as well as their first and second derivatives of seven model thioamide molecules. When the sp2 sulfur parameters were tested on a set of seven X-ray crystallographic structures from the Cambridge Structural Database, they satisfactorily reproduced the bond lengths, bond angles, torsional angles, and nonbonded distances of all the crystal structures. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1010,1025, 2001 [source] The economic lot size of the integrated vendor-buyer inventory system derived without derivativesOPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 3 2002P.C. Yang Abstract In previous modellings of the integrated vendor,buyer system, the buyer's economic order quantity and the vendor's optimal number of deliveries are derived by setting the first derivatives to zero and solving the simultaneous equations. The Hessian matrix of second derivatives is used to prove the convexity of the objective function. This procedure can be difficult for students who lack the background of differential calculus. This study develops algebraically the optimal policy of the integrated vendor,buyer inventory system without using differential calculus. A significant cost reduction is also achieved when Goyal's model is modified. Copyright © 2002 John Wiley & Sons, Ltd. [source] Sharp regularity results on second derivatives of solutions to the Monge-Ampère equation with VMO type dataCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2009Qingbo Huang We establish interior estimates for Lp -norms, Orlicz norms, and mean oscillation of second derivatives of solutions to the Monge-Ampère equation det D2u = f(x) with zero boundary value, where f(x) is strictly positive, bounded, and satisfies a VMO-type condition. These estimates develop the regularity theory of the Monge-Ampère equation in VMO-type spaces. Our Orlicz estimates also sharpen Caffarelli's celebrated W2, p -estimates. © 2008 Wiley Periodicals, Inc. [source] | ||||||||||||||||