Sixth Derivative Research Articles

High order quark number susceptibilities in hot QCD from lattice electrostatic QCD

Building on the experience of [1], we develop a formalism to construct operators for higher derivatives of the pressure in hot QCD with respect to the quark chemical potential $\mu$. We provide formulae for the operators up to the sixth derivative, and obtain continuum-extrapolated results from lattice EQCD at zero and finite $\mu$ and at six different pairs of temperature T and number of massless quark flavors $n_f$. Our data is benchmarked against full-QCD lattice and perturbative results, allowing to judge the quality of the perturbative series expansion in EQCD and the dimensional reduction procedure as a whole.

New hybrid method for direct numerical solution of nonlinear second, third and fourth orders ordinary differential equations

A novel and efficient algorithm for the concurrent numerical solution of second, third and fourth orders of ordinary differential equations is studied in this article. Collocation and interpolation technique was employed in the derivation of the method and power series approximate solution was used as the interpolating polynomial. The fourth derivative of the power series was collocated at the entire grid and off-grid points, while the fifth and sixth derivatives of the polynomial were collocated at the endpoint only. Proper investigation of the basic properties of the method was done. The results showed that the new block method applied on a nonlinear second, third and fourth orders of ordinary differential equations were better in terms of accuracy than the existing methods. The proposed method takes away the burden of developing a separate method for the solution of second, third and fourth order initial value problem of ordinary differential equations.

Convergence Criteria of Three Step Schemes for Solving Equations

We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence.

New Efficient Numerical Model for Solving Second, Third and Fourth Order Ordinary Differential Equations Directly

This article presents a two-step hybrid linear multistep block method for solving second, third and fourth order initial value problems of ordinary differential equations directly. The derivation of the method was done using collocation and interpolation techniques while approximated power series was used as an interpolating polynomial. The fourth derivative of the power series is collocated at the entire grid and off grid points while the fifth and sixth derivatives of the polynomial are collocated at the end point only. The basic properties of the developed method, that is, order, error constant, zero stability, region of absolute stability, convergence and consistence of the method are properly investigated. The numerical results demonstrated that the scheme developed handles second, third and fourth order ordinary differential equations efficiently and also better in accuracy when compared with existing methods. The proposed method takes away the burden of developing separate method for the solution of second, third and fourth order initial value problem of ordinary differential equations.

Local convergence comparison between two novel sixth order methods for solving equations

Abstract The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

Local Convergence of a Family of Iterative Methods with Sixth and Seventh Order Convergence Under Weak Conditions

In this paper, we study a local convergence analysis of a family of iterative methods with sixth and seventh order convergence for nonlinear equations, which was established by [Cordero et al. [2010] in “A family of iterative methods with sixth and seventh order convergence for nonlinear equations,” Math. Comput. Model. 52, 1190–1496]. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative. In our work, we make an attempt to study and establish a local convergence theorem by using only hypotheses the first derivative of the function and Lipschitz constants. We can also obtain error bounds and radii of convergence based on our results. Hence, the applicability of the methods is expanded. Moreover, we consider some different numerical examples and obtain the radii of convergence centered at the solution for different parameter values [Formula: see text] of the family. Furthermore, the basins of attraction of the family with different parameter values are also studied, which allow us to distinguish between the good and bad members of the family in terms of convergence and stable properties, and help us find the members with better or the best stable behavior.

New three-stages symmetric six-step finite difference method with vanished phase-lag and its derivatives up to sixth derivative for second order initial and/or boundary value problems with periodical and/or oscillating solutions

In this paper, for the first time in the literature, we develop a symmetric three-stages six-step method with the following characteristics; the method A detailed theoretical, numerical and computational analysis is also presented. The above analyses consist of: The theoretical, numerical and computational achievements lead to the conclusion that the new produced three-stages symmetric six-step pair is more efficient than other known or recently developed finite difference pairs of the literature.

Local convergence of a Newton–Traub composition in Banach spaces

We study the local convergence of a sixth-order Newton–Traub method to approximate a locally-unique solution of a system of nonlinear equations. Earlier studies show convergence under hypotheses on the sixth derivative or even higher, although only the first derivatives are used in the method. The convergence in this study is shown under hypotheses only on the first derivative. Our analysis avoids the usual Taylor expansions requiring higher order derivatives but uses generalized Lipschitz–Holder-type conditions only on the first derivative. Moreover, our new approach provides computable radius of convergence as well as error bounds on the distances involved and estimates on the uniqueness of the solution based on some functions appearing in these generalized conditions. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives which may not exist or may be very expensive or impossible to compute. Therefore, we do not know how close to the solution the initial point should be for convergence of the method. That is the initial point is a shot in the dark with the approaches using Taylor expansions. The convergence order is computed using computational order of convergence or approximate computational order of convergence which do not require usage of higher derivatives. This technique can be applied to any iterative method using Taylor expansions involving high order derivatives. In this sense the applicability of the method is expanded. Finally, numerical examples are provided to show that the present results apply to solve equations in cases where earlier results cannot apply.

Local convergence for multipoint methods using only the first derivative

We present a local convergence analysis of two methods in order to approximate a locally unique solution of an equation in a Banach space setting. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative (or even higher order derivatives) although only the first derivative appears in these methods. We only show convergence using hypotheses on the first derivative. We also provide computable: error bounds, radii of convergence as well as uniqueness results for the solution based on Lipschitz constants not given in earlier studies. The computational order of convergence is also used to determine the order of convergence. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

High algebraic order Runge–Kutta type two-step method with vanished phase-lag and its first, second, third, fourth, fifth and sixth derivatives

A Runge–Kutta type twelfth algebraic order two-step method with vanished phase-lag and its first, second, third, fourth, fifth and sixth derivatives are developed in this paper. The construction of the method, the local truncation error (lte) of the newly obtained method, the comparative error analysis of the new method with the corresponding method with constant coefficients and the stability (interval of periodicity) of the new method using frequency for the scalar test equation different than the frequency used in the scalar test equation for phase-lag analysis are studied in this paper. Finally, an application of the newly obtained method to the coupled differential equations of the Schrödinger type is also presented in this paper. From the presented numerical results, the efficiency of the newly proposed method is shown. It is noted that for the first time in the literature a multistep method with vanished phase-lag and its derivatives up to sixth order is developed.

A fourth order accurate approximation of the first and pure second derivatives of the Laplace equation on a rectangle

In this paper, we discuss an approximation of the first and pure second order derivatives for the solution of the Dirichlet problem on a rectangular domain. The boundary values on the sides of the rectangle are supposed to have the sixth derivatives satisfying the Holder condition. On the vertices, besides the continuity condition, the compatibility conditions, which result from the Laplace equation for the second and fourth derivatives of the boundary values, given on the adjacent sides, are also satisfied. Under these conditions a uniform approximation of order $O(h^{4})$ (h is the grid size) is obtained for the solution of the Dirichlet problem on a square grid, its first and pure second derivatives, by a simple difference scheme. Numerical experiments are illustrated to support the analysis made.

On some quadrature rules with Laplace end corrections

Abstract We investigate quadrature rules with Laplace end corrections that depend on a parameter β. Specific values of β yield sixth order rules. We apply our results to approximating the sum of slowly converging series s = Σi=1∞ f(i + 1/2) where f ∈ C 6 with its sixth derivative of constant sign on [m, ∞) and ∫ m∞ f(x)dx is known for m ∈ ℕ. Several examples show the efficiency of this method. This paper continues the results from [Solak W., Szydełko Z., Quadrature rules with Gregory-Laplace end corrections, J. Comput. Appl. Math., 1991, 36(2), 251–253].

Beyond vibrational self‐consistent‐field methods: Benchmark calculations for the fundamental vibrations of ethylene

AbstractA number of different methods for calculation of vibrational energies are reviewed and tested for calculation of the fundamental vibrational frequencies for ethylene. The methods tested for the description of the vibrational states are vibrational self‐consistent field (VSCF), vibrational configuration interaction (VCI), vibrational Møller–Plesset theory (VMP), and vibrational coupled cluster (VCC). The convergence of the different methods toward the full vibrational configuration interaction (FVCI) result is discussed for a modest‐sized one‐mode basis. For larger one‐mode basis sets results are presented for selected methods. Different representations of the potential energy surface are compared including full and partial quartic force field as well as expansions using up to sixth derivatives of the potential energy surface. Using MP2 electronic structure theory, the electronic structure basis set convergence is tested. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Soft-edged magnet models for higher-order beam-optics map codes

Continuously varying surface and volume source-density distributions are used to model magnetic fields inside of cylindrical volumes. From these distributions, a package of subroutines computes on-axis generalized gradients and their derivatives at arbitrary points on the magnet axis for input to the numerical map-generating subroutines of the Lie-algebraic map code Marylie. In the present version of the package, the magnet menu includes: (1) cylindrical current-sheet or radially thick current distributions with either open boundaries or with a surrounding cylindrical boundary with normal field lines (which models high-permeability iron), (2) Halbach-type permanent multipole magnets, either as sheet magnets or as radially thick magnets, (3) modeling of arbitrary fields inside a cylinder by use of a fictitious current sheet. The subroutines provide on-axis gradients and their z derivatives to essentially arbitrary order, although in the present third- and fifth-order Marylie only the zeroth through sixth derivatives are needed. The formalism is especially useful in beam-optics applications, such as magnetic lenses, where realistic treatment of fringe-field effects is needed.

Spatial frequency processing in inferred PC- and MC-pathways

The goal of this study was to investigate the role of inferred parvocellular (PC) and magnocellular (MC) pathways in spatial contrast sensitivity. Localized, spatially narrow-band patterns (sixth derivatives of Gaussians, D6s) were presented at various peak spatial frequencies. When the D6 appeared on a pulsed luminance pedestal (Pulsed-Pedestal Paradigm), the spatial contrast sensitivity showed a band-pass shape with good contrast sensitivity at medium spatial frequencies. When the D6 appeared on a steady luminance pedestal (Steady-Pedestal Paradigm), the spatial contrast sensitivity showed a low-pass shape with decreased sensitivity at high spatial frequencies. The band-pass CSF was interpreted as reflecting PC-pathway mediation; the lower spatial frequency region of the low-pass CSF as reflecting MC-pathway mediation.

Spatial-frequency and orientation tuning in psychophysical end-stopping.

A psychophysical analog to cortical receptive-field end-stopping has been demonstrated previously in spatial filters tuned to a wide range of spatial frequencies (Yu & Levi, 1997a). The current study investigated tuning characteristics in psychophysical spatial filter end-stopping. When a D6 (the sixth derivative of a Gaussian) target is masked by a center mask (placed in the putative spatial filter center), two end-zone masks (placed in the filter end-zones) reduce thresholds. This "end-stopping" effect (the reduction of masking induced by end-zone masks) was measured at various spatial frequencies and orientations of end-zone masks. End-stopping reached its maximal strength when the spatial frequency and/or orientation of the end-zone masks matched the spatial frequency and/or orientation of the target and center mask, showing spatial-frequency tuning and orientation tuning. The bandwidths of spatial-frequency and orientation tuning functions decreased with increasing target spatial frequency. At larger orientation differences, however, end-zone masks induced a secondary facilitation effect, which was maximal when the spatial frequency of end-zone masks equated the target spatial frequency. This facilitation effect might be related to certain types of contour and texture perception, such as perceptual pop-out.

Continuous monitoring of intermediates and final products of oxidation of low density lipoprotein by means of UV-spectroscopy.

The most widely used routine technique for determination of LDL 'oxidizability' is the continuous monitoring of the absorption at 234 nm in the UV spectrum of LDL, following the addition of an oxidation promotor such as copper ions. This absorption is commonly attributed to the conjugated dienes formed upon oxidation as the major intermediate, namely the hydroperoxides of polyunsaturated fatty acids, mostly linoleate hydroperoxides. These, however, are not the only products of oxidation that absorb light at 234 nm. Other products, particularly 7-ketocholesterol, also absorb light at the same wavelength. Furthermore, enals and dienals also absorb in the wavelength range of 210-300 nm. The aim of the present work was to develop a simple spectroscopic method for more detailed investigation of the kinetics of lipoprotein oxidation. The method is based on continuous measurement of the UV spectrum in the wavelength range of 210-300 nm and subsequent decomposition of the spectra into four absorption bands due to hydroperoxides, 7-ketocholesterol, dienals and enals. The sixth derivatives of the spectra, recorded during the first seven hours of copper-induced oxidation of LDL were used to monitor the growth and subsequent decay of the hydroperoxides. The resultant time course, in conjunction with difference spectra obtained after the concentration of these intermediates decay to zero, enabled us to determine the spectra of the other oxidation products and, by that, to evaluate their time dependencies. Based on these results, we present a series of four simple equations that can be used to evaluate the concentrations of the individual products of LDL oxidation from UV absorption measurements of the mixtures ar merely four different wavelengths. The resultant time dependencies of the accumulation of four major products of lipid oxidation are consistent with published data obtained through separation and chemical analysis. This simple method can be used for more meaningful routine kinetic measurements of lipids oxidation.

Effect of spatial sampling on grating resolution and letter identification.

The purpose of the study was to determine whether spatial sampling has equivalent effects on visual performance for grating and letter stimuli, two optotypes that are frequently used in the clinical assessment of visual function. Test targets consisted of five-bar square-wave gratings, spatial D6 (sixth derivative of a Gaussian) patterns, or individual Sloan letters, presented for 255 ms in the center of an adapting field. Spatial sampling was introduced by replacing random regions of the targets with occluding elements that had the same luminance as the adapting field, so that only random samples of the test target were visible on each trial. The occluding elements consisted of 4 x 4, 8 x 8, or 16 x 16 pixel arrays (subtending 0.67, 1.33, and 2.67 arcmin, respectively), and the percentage of the test target that remained unoccluded ranged from 3% to 100%. Visual acuity and contrast sensitivity for the spatially sampled targets were measured in two visually normal subjects. Results were compared with the predicted effect of the reduction in effective contrast that was introduced by the sampling paradigm. For grating stimuli the effect of spatial sampling was equivalent to the reduction in effective stimulus contrast produced by the sampling, regardless of occluding-element size. For letter stimuli the reduction in effective stimulus contrast predicted most, but not all, of the loss of visual acuity and contrast sensitivity for sampled targets, and the deficit in performance increased when larger occluding elements were used.(ABSTRACT TRUNCATED AT 250 WORDS)

Flaw sizing from pulse-echo ultrasonics: A model based approach.

Inversion of pulse-echo ultrasonic signals for flaw characteristics requires accounting for the excitation signal of the transducer, transducer characteristics, and diffraction of sound. The approach most often taken is to remove the influence of these factors by deconvolution using a signal obtained by reflecting the ultrasound from a flat surface. This flat surface needs to be located at the same distance from the transducer as the flaw which is a great disadvantage for an arbitrarily located flaw. This paper suggests an alternate technique in which rather simple time models account for the above influences. The goal is to describe the flaw from its scattering response. Instead of choosing the impulse response, there is a definite advantage of seeking the ‘‘ramp’’ response. The ramp response is defined such that two time derivatives of the ramp response yields the impulse response. The ramp response approximates the ‘‘area profile’’ function of the scatterer. An integral of the area profile function yields the volume of the flaw scatter. The findings show the output (echo) signal from the transducer can be expressed as the sixth time derivative of the convolution of the desired ramp response with the transducer function twice, the radiation function twice, plus the smoothing function due to attenuation, and the excitation pulse. The convolution of the transducer, radiation, and attenuation functions is ‘‘Gaussian’’ in shape. The experimenter has some control over the time width of these discontinuous functions: by choice of thickness of the piezoelectric in the transducer; and, by choosing a transducer so that the scatterer is in the farfield of the transducer. The functions describing radiation and the transducer enter twice as they are involved in transmission of the sound pulse and its reception after being scattered by a flaw. Assuming that the ‘‘Gaussian’’ shaped function is designed to be insignificant, the output contains the sixth derivative of the convolution of the area profile function and the exciting function to the transducer. By exciting the transducer with a well-behaved, narrow Gaussian function of time, the output signal can be deconvolved using the sixth derivative of this Gaussian yielding the desired output, an approximation to the area profile function. In cases where the ‘‘Gaussian’’ shaped function is not insignificant, it can be deconvolved from the above answer, within limits. In this case, the length of the area profile function should not be significantly shorter than the ‘‘Gaussian’’ shaped function.