Taylor-type Expansion Research Articles

Derivative Formulas and Gradient of Functions with Non-Independent Variables

Stochastic characterizations of functions subject to constraints result in treating them as functions with non-independent variables. By using the distribution function or copula of the input variables that comply with such constraints, we derive two types of partial derivatives of functions with non-independent variables (i.e., actual and dependent derivatives) and argue in favor of the latter. Dependent partial derivatives of functions with non-independent variables rely on the dependent Jacobian matrix of non-independent variables, which is also used to define a tensor metric. The differential geometric framework allows us to derive the gradient, Hessian, and Taylor-type expansions of functions with non-independent variables.

Pointwise estimates for degenerate Kolmogorov equations with $$L^p$$-source term

The aim of this paper is to establish new pointwise regularity results for solutions to degenerate second-order partial differential equations with a Kolmogorov-type operator of the form $$\begin{aligned} {\mathscr {L}}:=\sum _{i,j=1}^m \partial ^2_{x_i x_j } +\sum _{i,j=1}^N b_{ij}x_j\partial _{x_i}-\partial _t, \end{aligned}$$where \((x,t) \in {{\mathbb {R}}}^{N+1}\), \(1 \le m \le N\) and the matrix \(B:=(b_{ij})_{i,j=1,\ldots ,N}\) has real constant entries. In particular, we show that if the modulus of \(L^p\)-mean oscillation of \({\mathscr {L}}u\) at the origin is Dini, then the origin is a Lebesgue point of continuity in \(L^p\) average for the second-order derivatives \(\partial ^2_{x_i x_j} u\), \(i,j=1,\ldots ,m\), and the Lie derivative \(\left( \sum _{i,j=1}^N b_{ij}x_j\partial _{x_i}-\partial _t\right) u\). Moreover, we are able to provide a Taylor-type expansion up to second order with an estimate of the rest in \(L^p\) norm. The proof is based on decay estimates, which we achieve by contradiction, blow-up and compactness results.

Taylor-type expansions in terms of exponential polynomials

Taylor-type expansions in terms of exponential polynomials

The rotation-torsion spectrum of CD2HOH

New transitions are reported in the submillimeter wave, terahertz, and far-infrared rotation-torsion spectrum of doubly-deuterated methanol CD2HOH. The newly assigned transitions allow us to spectroscopically characterize torsional states with 0⩽K⩽12 and 0⩽vt⩽2. Three line position analyses are carried out. In the first one, restricted to rotation-torsion lines involving torsional states with 3⩽K⩽12 and vt⩽2, rotational energies are evaluated with a J(J+1) Taylor-type expansion for each torsional state. 4853transitions were accounted for with a unitless standard deviation of 11.4. In the second analysis, 126torsional subband centers are fitted to obtain refined torsional parameters including the hindering potential. In the third analysis, 5911rotation-torsion transitions involving torsional states with vt⩽2, and J⩽26 are reproduced with a unitless standard deviation of 3.2 using a four-dimensional rotation-torsion fitting Hamiltonian.

Expanding solutions of quasilinear parabolic equations

<p style='text-indent:20px;'>By using the theory of maximal <inline-formula><tex-math id="M1">\begin{document}$ L^{q} $\end{document}</tex-math></inline-formula>-regularity and methods of singular analysis, we show a Taylor's type expansion–with respect to the geodesic distance around an arbitrary point–for solutions of quasilinear parabolic equations on closed manifolds. The powers of the expansion are determined explicitly by the local geometry, whose reflection to the solutions is established through the local space asymptotics.

A second order analysis of McKean–Vlasov semigroups

We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean–Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. Bismut–Elworthy–Li formulae for the gradient and the Hessian of the integro-differential operators associated with these expansions are also presented. The article also provides explicit Dyson–Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon. We illustrate the impact of these results with a second order extension of the Alekseev–Gröbner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos properties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities.

Derivative Free Fourth Order Solvers of Equations with Applications in Applied Disciplines

This paper develops efficient equation solvers for real- and complex-valued functions. An earlier study by Lee and Kim, used the Taylor-type expansions and hypotheses on higher than first order derivatives, but no derivatives appeared in the suggested method. However, we have many cases where the calculations of the fourth derivative are expensive, or the result is unbounded, or even does not exist. We only use the first order derivative of function Ω in the proposed convergence analysis. Hence, we expand the utilization of the earlier scheme, and we study the computable radii of convergence and error bounds based on the Lipschitz constants. Furthermore, the range of starting points is also explored to know how close the initial guess should be considered for assuring convergence. Several numerical examples where earlier studies cannot be applied illustrate the new technique.

Discretization Error Cancellation in the Plane-Wave Approximation of Periodic Hamiltonians with Coulomb Singularities

In solid-state physics, energies of molecular systems are usually computed with a plane-wave discretization of Kohn-Sham equations. A priori estimates of plane-wave convergence for periodic Kohn-Sham calculations with pseudopotentials have been proved , however in most computations in practice, plane-wave cut-offs are not tight enough to target the desired accuracy. It is often advocated that the real quantity of interest is not the value of the energy but of energy differences for different configurations. The computed energy difference is believed to be much more accurate because of `discretization error cancellation', since the sources of numerical errors are essentially the same for different configurations. For periodic linear Hamiltonians with Coulomb potentials, error cancellation can be explained by the universality of the Kato cusp condition. Using weighted Sobolev spaces, Taylor-type expansions of the eigenfunctions are available yielding a precise characterization of this singularity. This then gives an explicit formula of the first order term of the decay of the Fourier coefficients of the eigenfunctions. It enables one to prove that errors on eigenvalue differences are reduced but converge at the same rate as the error on the eigenvalue.

Refined Beam Elements for the Free Vibration Analysis of Stepped Shafts With and Without Transition Area

Investigation of the free vibration of stepped shafts with and without transition area is presented in this paper. The refined one-dimensional beam theories are chosen to be used based on the Carrera Unified Formulation (CUF) in the present work. According to CUF, Taylor-type expansion is employed to describe the displacement field over cross section of shafts. With the implementation of Hamilton’s principle, the governing differential equations and the related mass matrix, the stiffness matrix and the load vector are obtained. To obtain the numerical solutions, the finite method is adopted in this paper. The accuracy and reliability of the present model are demonstrated by comparing with classical beam model (Timoshenko beam model) and a solid model generated in the commercial software ANSYS. Meanwhile, the influences of step ratio, step location, boundary conditions and the slope of generatrix attached to the transition area on vibration characteristics are studied.

A perturbation analysis of stochastic matrix Riccati diffusions

Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces.

The torsional and rotation–torsion spectra of CD2HOH

The torsional and rotation–torsion spectra of the doubly deuterated species of methanol CD2HOH have been analyzed using submillimeter wave, terahertz, and far infrared spectra. 101 torsional subbands, with subband centers ranging from 2.3 to 626cm−1, were assigned. Analysis of these yielded kinetic energy and hindering potential parameters of the torsional Hamiltonian describing the large amplitude internal rotation of the CD2H methyl group with respect to the hydroxyl group. 3271 rotation and rotation–torsion transitions, involving the 24 torsional levels up to e1 with 3⩽K⩽10, were assigned and fitted approximating the rotational energy of each torsional level with a Taylor-type expansion in J(J+1). The rotational structure of 48 torsional subbands involving torsional levels higher than e1 has also been analyzed. In most cases, only the Q branch could be observed and assigned.

Refined dynamic stiffness elements applied to free vibration analysis of generally laminated composite beams with arbitrary boundary conditions

The Carrera Unified Formulation (CUF) is used in this paper to develop higher-order beam theories for composite laminates. In the framework of the CUF, the three-dimensional displacement field is approximated as a truncated Taylor-type expansion series of the generalized displacements, which lie on the beam axis. The truncation of the series determines the theory order N, which is a free parameter of the formulation. The strong form of the principle of virtual displacements is used to obtain the equations of motion and the natural boundary conditions for the laminated beam in free vibration. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The Wittrick–Williams algorithm is applied to solve the transcendental eigenvalue problem resulting from the present approach. Composite beams with arbitrary boundary conditions, geometries and lamination schemes can be analysed with the proposed method. Numerical investigations are carried out and the results are compared with reference solutions from the literature and with solutions from commercial finite elements codes.

Free vibration analysis of laminated beam by polynomial, trigonometric, exponential and zig-zag theories

A number of refined beam theories are discussed in this paper to trace the free vibration response of laminated beams, including thin-walled boxes. By expanding the unknown displacement variables over the beam section axes using Taylor type expansions, trigonometric series, exponential, hyperbolic and zig-zag functions, many new displacement fields were obtained and, for the first time, evaluated for the dynamic analyses of composite structures. The finite element method is used to derive governing equations in weak form. These equations are written using the unified formulation introduced by the first author, in terms of fundamental nuclei, whose forms do not depend on the expansions used. The natural frequencies are compared with results available in the literature or with those obtained by the finite element models related to commercial software. A number of analyses were conducted to compare various theories, including Euler–Bernoulli and Timoshenko models. The advantages/disadvantages of using the different theories are discussed for significant problems related to laminated beams as well as thin-walled boxes. It is shown that refined kinematic theories are able to yield a very accurate evaluation of fundamental as well as higher mode frequencies in a way comparable to three-dimensional analysis, but it is obtained with a strong reduction of computational costs.

Buckling of thin-walled beams by a refined theory

The buckling of thin-walled structures is presented using the 1D finite element based refined beam theory formulation that permits us to obtain N-order expansions for the three displacement fields over the section domain. These higher-order models are obtained in the framework of the Carrera unified formulation (CUF). CUF is a hierarchical formulation in which the refined models are obtained with no need for ad hoc formulations. Beam theories are obtained on the basis of Taylor-type and Lagrange polynomial expansions. Assessments of these theories have been carried out by their applications to studies related to the buckling of various beam structures, like the beams with square cross section, I-section, thin rectangular cross section, and annular beams. The results obtained match very well with those from commercial finite element softwares with a significantly less computational cost. Further, various types of modes like the bending modes, axial modes, torsional modes, and circumferential shell-type modes are observed.

Buckling of composite thin walled beams by refined theory

This work presents the buckling analysis of laminated composite thin walled structures by the 1D finite element based unified higher-order models obtained within the framework of the Carrera Unified Formulation (CUF). In the present study, the refined beam theories are obtained on the basis of Taylor-type expansions. The finite element analysis has been chosen to easily handle arbitrary geometries as well as boundary conditions. Buckling behavior of laminated composite beam and flat panels are analyzed to illustrate the efficacy of the present formulation and various types of buckling modes are observed depending on the geometrical and material parameters. It is observed that the lower order models are unable to deal with torsion.

Refined beam elements with only displacement variables and plate/shell capabilities

This paper proposes a refined beam formulation with displacement variables only. Lagrange-type polynomials, in fact, are used to interpolate the displacement field over the beam cross-section. Three- (L3), four- (L4), and nine-point (L9) polynomials are considered which lead to linear, quasi-linear (bilinear), and quadratic displacement field approximations over the beam cross-section. Finite elements are obtained by employing the principle of virtual displacements in conjunction with the Unified Formulation (UF). With UF application the finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumptions made (L3, L4, or L9). Additional refined beam models are implemented by introducing further discretizations over the beam cross-section in terms of the implemented L3, L4, and L9 elements. A number of numerical problems have been solved and compared with results given by classical beam theories (Euler-Bernoulli and Timoshenko), refined beam theories based on the use of Taylor-type expansions in the neighborhood of the beam axis, and solid element models from commercial codes. Poisson locking correction is analyzed. Applications to compact, thin-walled open/closed sections are discussed. The investigation conducted shows that: (1) the proposed formulation is very suitable to increase accuracy when localized effects have to be detected; (2) it leads to shell-like results in case of thin-walled closed cross-section analysis as well as in open cross-section analysis; (3) it allows us to modify the boundary conditions over the cross-section easily by introducing localized constraints; (4) it allows us to introduce geometrical boundary conditions along the beam axis which lead to plate/shell-like cases.

Approximating [formula omitted]-time differentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor type expansion

On utilizing the spectral representation of selfadjoint operators in Hilbert spaces, some approximations for the n -time differentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor type expansions are given.

Scanning anisotropy parameters in complex media

Parameter estimation in an inhomogeneous anisotropic medium offers many challenges; chief among them is the trade-off between inhomogeneity and anisotropy. It is especially hard to estimate the anisotropy anellipticity parameter η in complex media. Using perturbation theory and Taylor’s series, I have expanded the solutions of the anisotropic eikonal equation for transversely isotropic (TI) media with a vertical symmetry axis (VTI) in terms of the independent parameter η from a generally inhomogeneous elliptically anisotropic medium background. This new VTI traveltime solution is based on a set of precomputed perturbations extracted from solving linear partial differential equations. The traveltimes obtained from these equations serve as the coefficients of a Taylor-type expansion of the total traveltime in terms of η. Shanks transform is used to predict the transient behavior of the expansion and improve its accuracy using fewer terms. A homogeneous medium simplification of the expansion provides classical nonhyperbolic moveout descriptions of the traveltime that are more accurate than other recently derived approximations. In addition, this formulation provides a tool to scan for anisotropic parameters in a generally inhomogeneous medium background. A Marmousi test demonstrates the accuracy of this approximation. For a tilted axis of symmetry, the equations are still applicable with a slightly more complicated framework because the vertical velocity and δ are not readily available from the data.

Unified Formulation Applied to Free Vibrations Finite Element Analysis of Beams with Arbitrary Section

This paper presents hierarchical finite elements on the basis of the Carrera Unified Formulation for free vibrations analysis of beam with arbitrary section geometries. The displacement components are expanded in terms of the section coordinates, (x, y), using a set of 1-D generalized displacement variables. N-order Taylor type expansions are employed. N is a free parameter of the formulation, it is supposed to be as high as 4. Linear (2 nodes), quadratic (3 nodes) and cubic (4 nodes) approximations along the beam axis, (z), are introduced to develop finite element matrices. These are obtained in terms of a few fundamental nuclei whose form is independent of both N and the number of element nodes. Natural frequencies and vibration modes are computed. Convergence and assessment with available results is first made considering different type of beam elements and expansion orders. Additional analyses consider different beam sections (square, annular and airfoil shaped) as well as boundary conditions (simply supported and cantilever beams). It has mainly been concluded that the proposed model is capable of detecting 3-D effects on the vibration modes as well as predicting shell-type vibration modes in case of thin walled beam sections.

Unified Formulation Applied to Free Vibrations Finite Element Analysis of Beams with Arbitrary Section

This paper presents hierarchical finite elements on the basis of the Carrera Unified Formulation for free vibrations analysis of beam with arbitrary section geometries. The displacement components are expanded in terms of the section coordinates, (x, y), using a set of 1-D generalized displacement variables. N-order Taylor type expansions are employed. N is a free parameter of the formulation, it is supposed to be as high as 4. Linear (2 nodes), quadratic (3 nodes) and cubic (4 nodes) approximations along the beam axis, (z), are introduced to develop finite element matrices. These are obtained in terms of a few fundamental nuclei whose form is independent of both N and the number of element nodes. Natural frequencies and vibration modes are computed. Convergence and assessment with available results is first made considering different type of beam elements and expansion orders. Additional analyses consider different beam sections (square, annular and airfoil shaped) as well as boundary conditions (simply supported and cantilever beams). It has mainly been concluded that the proposed model is capable of detecting 3-D effects on the vibration modes as well as predicting shell-type vibration modes in case of thin walled beam sections.