Taylor Series Representation Research Articles

Fekete-Szegö and Hankel inequalities related to the sine function

ABSTRACT The Fekete-Szegö inequality is one of the inequalities for the coefficients which associated with the famous Bieberbach conjecture. Other issues associated with this inequality are determining the Hankel determinant denoted as Hd inequalities which are involved in the study of the singularities and integral coefficients in the Taylor series representations. In this investigation, we discuss such inequalities for certain mappings g for which ( g ′ ( z ) ) α [ 2 z g ′ ( z ) g ( z ) − g ( − z ) ] ( 1 − α ) ≺ 1 + sin ⁡ z , ( 0 ≤ α ≤ 1 ) lies in the image domain starlike about 1 and symmetric about real axis. Particularly, we obtain certain inequality for functions involving Poisson distribution. Furthermore, we also find the second Hankel determinant and other inequalities related to this newly defined class.

Computing the Mittag-Leffler function of a matrix argument

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions).

A Precision Benchmark Suite for Nuclear Reactor Point Kinetics Equations via Converged Accelerated Taylor Series (CATS)

Extreme benchmarks of 10 or more places for the point kinetics equations for time-dependent nuclear reactor power transients are rare. Therefore, to establish an extreme benchmark, we employ a Taylor series (TS) with continuous analytical continuation to solve the ordinary differential equations of point kinetics including feedback. Nonlinear Wynn-epsilon convergence acceleration confirms the highly precise solutions for neutron and precursor densities. Through adaptive partitioning of time intervals, the proposed Converged Accelerated Taylor Series, or CATS algorithm in double precision, automatically performs successive mesh refinement to obtain high-precision initial conditions for each subinterval, with the intent to reduce propagation error. Confirmation of 10 to 12 places comes from comparison to the BEFD (Backward Euler Finite Difference) algorithm in quadruple precision also developed by the author. We report benchmark results for common cases found in the literature including step, ramp, zigzag, and sinusoidal prescribed reactivity insertions and insertions with nonlinear adiabatic Doppler feedback. We also establish a suite of new prescribed reactivity insertions and insertions with feedback, based on reactivities with Taylor series representations as suggested by the CATS algorithm.

Determining internal coordinate sets for optimal representation of molecular vibration.

Arising from the harmonic approximation in solving the vibrational Schrödinger equation, normal modes dissect molecular vibrations into distinct degrees of freedom. Normal modes are widely used as they give rise to descriptive vibrational notations and are convenient for expanding anharmonic potential energy surfaces as an alternative to higher-order Taylor series representations. Usually, normal modes are expressed in Cartesian coordinates, which bears drawbacks that can be overcome by switching to internal coordinates. Considering vibrational notations, normal modes with delocalized characters are difficult to denote, but internal coordinates offer a route to clearer notations. Based on the Hessian, normal mode decomposition schemes for a given set of internal coordinates can describe a normal mode by its contributions from internal coordinates. However, choosing a set of internal coordinates is not straightforward. While the Hessian provides unique sets of normal modes, various internal coordinate sets are possible for a given system. In the present work, we employ a normal mode decomposition scheme to choose an optimal set. Therefore, we screen reasonable sets based on topology and symmetry considerations and rely on a metric that minimizes coupling between internal coordinates. Ultimately, the Nomodeco toolkit presented here generates internal coordinate sets to find an optimal set for representing molecular vibrations. The resulting contribution tables can be used to clarify vibrational notations. We test our scheme on small to mid-sized molecules, showing how the space of definable internal coordinate sets can significantly be reduced.

Octonionic monogenic and slice monogenic Hardy and Bergman spaces

Abstract In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

A spatial local method for solving 2D and 3D advection-diffusion equations

PurposeThis study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.Design/methodology/approachThe parabolic AD equations are reduced to the nonhomogeneous elliptic system of partial differential equations by utilizing the Chebyshev spectral collocation method (ChSCM) in the temporal variable. The implicit-explicit local differential transform method (IELDTM) is constructed over two- and three-dimensional meshes using continuity equations of the neighbor representations with either explicit or implicit forms in related directions. The IELDTM yields an overdetermined or underdetermined system of algebraic equations solved in the least square sense.FindingsThe IELDTM has proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over the existing numerical techniques is optimizing the local spatial degrees of freedom. It has been proven that the IELDTM provides more accurate results with far fewer degrees of freedom than the finite difference, finite element and spectral methods.Originality/valueThis study shows the derivation, applicability and performance of the IELDTM for solving 2D and 3D advection-diffusion equations. It has been demonstrated that the IELDTM can be a competitive numerical method for addressing high-space dimensional-parabolic partial differential equations (PDEs) arising in various fields of science and engineering. The novel ChSCM-IELDTM hybridization has been proven to have distinct advantages, such as continuous utilization of time integration and optimized formulation of spatial approximations. Furthermore, the novel ChSCM-IELDTM hybridization can be adapted to address various other types of PDEs by modifying the theoretical derivation accordingly.

Runge-Kutta Methods for Rough Differential Equations

We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (B-series) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying Runge-Kutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a Runge-Kutta scheme that is based on the increments of the driver of the rough differential equation. This simplified method can be easily implemented and is computational cheap since it is derivative-free. We provide a full characterization of this implementable Runge-Kutta method meaning that we provide necessary and sufficient algebraic conditions for an optimal order of convergence in case that the driver, e.g., is a fractional Brownian motion with Hurst index $\frac{1}{4} < H \leq \frac{1}{2}$. We conclude this paper by conducting numerical experiments verifying the theoretical rate of convergence.

A Note on Taylor Series Representation of the Nield-Kuznetsov Function of the Second Kind

Taylor and Maclaurin series, and polynomial approximations of the Standard Nield-Kuznetsov function of the second kind are obtained in this work. Convergence and error criteria are developed. The obtained series represent alternatives to existing asymptotic and ascending series approximations of this integral function, and are expected to provide an efficient method of computation.

Taylor and Maclaurin Series Representations of the Nield-Kuznetsov Function of the First Kind

Taylor and Maclaurin series and polynomial approximations of the Standard Nield-Kuznetsov function of the first kind are obtained in this work. Convergence and error criteria are developed. The obtained series represent alternatives to the existing asymptotic and ascending series approximations of this integral function, and are expected to provide an efficient method of computation that is valid for all values of the argument.

Scattering of light by a parity-time-symmetric dipole beyond the first Born approximation

The scattering of light by localized three-dimensional dielectrics having gain and loss defies the usual Born approximation, since the material can increase the amplitude of the incident field within the scatterer, making the weak-scattering assumption invalid. The convergence of the Born series is rarely discussed in analytical treatments, as the state of exhaustion is reached after calculating the very first few terms of the series. Even if all the terms are obtained, the series will certainly be of a divergent type in general, thus invalidating the equality of the scattered field to its Taylor-series representation. We present here a simple localized material model of a dielectric having parity-time ($\mathcal{PT}$) symmetry, consisting of a $\mathcal{PT}$-symmetric dipole, such that all the terms in the Born series are analytically evaluated and a closed-form expression is obtained in the far-zone approximation. The scattered field is then analyzed by using Pad\'e approximants in order to obtain convergent representations of the scattered radiation and to compare with the exact solution. This allows us to study the role of the gain and loss parameter in strong-scattering regimes and to demonstrate the remarkable properties of Pad\'e approximants when applied to scattering.

An implicit-explicit local method for parabolic partial differential equations

PurposeThe purpose of this article is to derive an implicit-explicit local differential transform method (IELDTM) in dealing with the spatial approximation of the stiff advection-diffusion-reaction (ADR) equations.Design/methodology/approachA direction-free numerical approach based on local Taylor series representations is designed for the ADR equations. The differential equations are directly used for determining the local Taylor coefficients and the required degrees of freedom is minimized. The complete system of algebraic equations is constructed with explicit/implicit continuity relations with respect to direction parameter. Time integration of the ADR equations is continuously utilized with the Chebyshev spectral collocation method.FindingsThe IELDTM is proven to be a robust, high order, stability preserved and versatile numerical technique for spatial discretization of the stiff partial differential equations (PDEs). It is here theoretically and numerically shown that the order refinement (p-refinement) procedure of the IELDTM does not affect the degrees of freedom, and thus the IELDTM is an optimum numerical method. A priori error analysis of the proposed algorithm is done, and the order conditions are determined with respect to the direction parameter.Originality/valueThe IELDTM overcomes the known disadvantages of the differential transform-based methods by providing reliable convergence properties. The IELDTM is not only improving the existing Taylor series-based formulations but also provides several advantages over the finite element method (FEM) and finite difference method (FDM). The IELDTM offers better accuracy, even when using far less degrees of freedom, than the FEM and FDM. It is proven that the IELDTM produces solutions for the advection-dominated cases with the optimum degrees of freedom without producing an undesirable oscillation.

Fast robust optimization using bias correction applied to the mean model

Ensemble methods are remarkably powerful for quantifying geological uncertainty. However, the use of the ensemble of reservoir models for robust optimization (RO) can be computationally demanding. The straightforward computation of the expected net present value (NPV) requires many expensive simulations. To reduce the computational burden without sacrificing accuracy, we present a fast and effective approach that requires only simulation of the mean reservoir model with a bias correction factor. Information from distinct controls and model realizations can be used to estimate bias for different controls. The effectiveness of various bias-correction methods and a linear or quadratic approximation is illustrated by two applications: flow optimization in a one-dimensional model and the drilling-order problem in a synthetic field model. The results show that the approximation of the expected NPV from the mean model is significantly improved by estimating the bias correction factor, and that RO with mean model bias correction is superior to both RO performed using a Taylor series representation of uncertainty and deterministic optimization from a single realization. Use of the bias-corrected mean model to account for model uncertainty allows the application of fairly general optimization methods. In this paper, we apply a nonparametric online learning methodology (learned heuristic search) for efficiently computing an optimal or near-optimal robust drilling sequence on the REEK Field example. This methodology can be used either to optimize a complete drilling sequence or to optimize only the first few wells at a reduced cost by limiting the search depths.

Quasi-spherical collapse of matter in ΛCDM

We report the findings of new exact analytical solutions to the cosmological fluid equations, namely for the case where the initial conditions are perturbatively close to a spherical top-hat profile. To do so we enable a fluid description in a Lagrangian-coordinates approach, and prove the convergence of the Taylor-series representation of the Lagrangian displacement field until the time of collapse ("shell-crossing"). This allows the determination of the time for quasi-spherical collapse, which is shown to happen generically earlier than in the spherical case. For pedagogical reasons, calculations are first given for a spatially flat universe that is only filled with a non-relativistic component of cold dark matter (CDM). Then, the methodology is updated to a $\Lambda$CDM Universe, with the inclusion of a cosmological constant $\Lambda>0$.

Coefficient Inequalities of Functions Associated with Petal Type Domains

In the theory of analytic and univalent functions, coefficients of functions’ Taylor series representation and their related functional inequalities are of major interest and how they estimate functions’ growth in their specified domains. One of the important and useful functional inequalities is the Fekete-Szegö inequality. In this work, we aim to analyze the Fekete-Szegö functional and to find its upper bound for certain analytic functions which give parabolic and petal type regions as image domains. Coefficient inequalities and the Fekete-Szegö inequality of inverse functions to these certain analytic functions are also established in this work.

Particle trajectories and mass transport under mechanically generated nonlinear water waves

A theoretical model is developed to study the kinematics of water particles and mass transport of wave-induced flow generated in a closed flume. A semi-analytical pseudo-spectral solution to the problem of mechanically generated nonlinear water waves propagating at a constant depth is derived under the potential flow assumptions. Velocity potential and free-surface displacement functions are represented in a form based on Fourier series. The eigenfunction expansions are resolved by means of a Fast Fourier Transform technique and a higher order time-stepping procedure. The derived nonlinear solution of the wavemaker problem is based on Taylor series representation of the free-surface and wavemaker boundary conditions expanded up to an arbitrary truncation order. This technique is employed to study the evolution of a velocity field and particle trajectories under regular gravity waves generated in a wave flume. The instantaneous velocity fields are integrated in time to calculate the particle trajectories and time-averaged spatial distribution of Lagrangian and Eulerian mean velocities. The theoretical results are compared with the available experimental data. A good agreement is observed, which confirms the applicability of the presented theory to the problems of transport processes associated with the propagation of water waves and their generation in the flume.

An exact and explicit implied volatility inversion formula

In this paper, we develop an exact and explicit (model-independent) Taylor series representation of the implied volatility based on the novel applications of an extended Faà di Bruno formula under the operator calculus setting, and the Lagrange inversion theorem. We rigorously establish that our formula converges to the true implied volatility as the truncation order increases. Numerical examples illustrate the remarkable accuracy and efficiency of the formula. The formula distinguishes from previous literature as it converges to the true exact implied volatility, is a closed-form formula whose coefficients are explicitly determined and do not involve numerical iterations.

An analytic approach for the study of pulsar spindown

In this work we develop an analytic approach to study pulsar spindown. We use the monopolar spindown model by Alvarez and Carramiñana (2004 Astron. Astrophys. 414 651–8), which assumes an inverse linear law of magnetic field decay of the pulsar, to extract an all-order formula for the spindown parameters using the Taylor series representation of Jaranowski et al (1998 Phys. Rev. D 58 6300). We further extend the analytic model to incorporate the quadrupole term that accounts for the emission of gravitational radiation, and obtain expressions for the period P and frequency f in terms of transcendental equations. We derive the analytic solution for pulsar frequency spindown in the absence of glitches. We examine the different cases that arise in the analysis of the roots in the solution of the non-linear differential equation for pulsar period evolution. We provide expressions for the spin-down parameters and find that the spindown values are in reasonable agreement with observations. A detection of gravitational waves from pulsars will be the next landmark in the field of multi-messenger gravitational wave astronomy.

An Exact and Explicit Implied Volatility Inversion Formula

In this paper, we develop an exact and explicit (model-independent) Taylor series representation of the Black-Scholes implied volatility in terms of market-observed option prices. Based on an extended Faa di Bruno formula under the operator calculus setting, we first derive the Taylor series of the Black-Scholes formula with respect to the volatility around a pre-chosen positive initial value. Then we apply the Lagrange inversion theorem to explicitly invert the Taylor series to obtain the implied volatility formula. We rigorously establish that our formula converge to the true implied volatility as the truncation order increases, and choose the initial value as the model-independent upper bound of the true implied volatility. Numerical examples illustrate the remarkable accuracy of the formula. Our formula distinguishes from previous literature in that it converges to the true exact implied volatility, is a closed-form formula whose coefficients are explicitly determined and do not involve numerical iterations, and is extremely efficient, which makes it suitable for industrial implementation and adoption.

Parametric Model Order Reduction via Balanced Truncation with Taylor Series Representation

This paper presents a new method for parametric model order reduction based on balanced truncation. Parametric model order reduction seeks to generate low-order models from larger models without losing the dependence on a parameter. Using a Taylor expansion of the original system, a Taylor expansion of the balanced system can be obtained. In contrast to interpolation-based approaches for the solution of the parametric model order reduction problem, the proposed approach permits calculation of the reduced system as well as the corresponding projection matrix for different parameter values with reduced computation power. This bypasses the problem of incompatible subspaces from different snapshot points potentially occurring in interpolation based approaches that can lead to unexpected behavior up to instability. The presented method can handle multidimensional parameter spaces. Sufficient conditions for the convergence of the Taylor series of the balanced system based on holomorphic functions are derived. The truncation step as well as error bounds are discussed. A Bernoulli beam model is used as an example to demonstrate the performance of the technique.

Fractional Hida-Malliavan derivatives and series representations of fractional conditional expectations

We represent fractional conditional expectations (an extension of the classical conditional expectations) of a functional of fractional Brownian motion as a convergent series in L 2 (P H ) space. When the target random variable is some function of a discrete trajectory of fractional Brownian motion, we obtain a backward Taylor series representation; when the target functional is generated by a continuous fractional filtration, the series representation is obtained by applying a ”frozen path” operator and an exponential operator to the functional. We provide three examples where our representation gives useful series expansions of expectations in several models.