Sixth-order Derivative Research Articles

Approximation of BPS Skyrme model using modified Lagrangian Skyrmion

One of the nuclear atomic models represented by Skyrmion was the Skyrme model. This model was a modified nonlinear sigma model with a Skyrme field where the classical solution use generalized sixth order terms and potential terms. The binding energy that will be studied in the Skyrme SU(2) model is to generalize the second order nonlinear sigma model terms with sixth order derivative terms. The Lagrangian will be obtained for these two terms to find the BPS (Bogomolny Prasad Sommerfield) solution for the profile function numerically. The result of numerical calculation will be used to calculate static energy and rotational energy, where the characteristic of the nucleus can be observed from these two energies. Furthermore, the value of the coupling constant in the Lagrangian Skyrmion will be calculated from the static energy and rotational energy obtained previously. These values are expected to help in the application of Skyrme model for many research physics field.

A Tenth-Order Sixth-Derivative Block Method for Directly Solving Fifth-Order Initial Value Problems

This work proposes a hybrid block numerical method of tenth order for the direct solution of fifth-order initial value problems. The formulas that constitute the block method are derived from a continuous approximation obtained through interpolation and collocation techniques. In order to obtain better accuracy, sixth-order derivatives are incorporated to develop the formulas. The main characteristics of the method are analyzed, namely, the order, local truncation errors, zero-stability, consistency and convergence. The proposed strategy performs well, as shown by some numerical examples and the corresponding efficiency curves. Compared to existing numerical methods in the literature, the proposed method is competitive and the numerical approximations it provides are significantly close to the precise solutions.

Variable bandwidth kernel regression estimation

In this paper we propose a variable bandwidth kernel regression estimator fori.i.d. observations in ℝ2to improve the classical Nadaraya-Watson estimator. The bias is improved to the order ofO(hn4) under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.

An Approach towards Motion-Tolerant PPG-Based Algorithm for Real-Time Heart Rate Monitoring of Moving Pigs.

Animal welfare remains a very important issue in the livestock sector, but monitoring animal welfare in an objective and continuous way remains a serious challenge. Monitoring animal welfare, based upon physiological measurements instead of the audio–visual scoring of behaviour, would be a step forward. One of the obvious physiological signals related to welfare and stress is heart rate. The objective of this research was to measure heart rate (beat per minutes) in pigs with technology that soon will be affordable. Affordable heart rate monitoring is done today at large scale on humans using the Photo Plethysmography (PPG) technology. We used PPG sensors on a pig′s body to test whether it allows the retrieval of a reliable heart rate signal. A continuous wavelet transform (CWT)-based algorithm is developed to decouple the cardiac pulse waves from the pig. Three different wavelets, namely second, fourth and sixth order Derivative of Gaussian (DOG), are tested. We show the results of the developed PPG-based algorithm, against electrocardiograms (ECG) as a reference measure for heart rate, and this for an anaesthetised versus a non-anaesthetised animal. We tested three different anatomical body positions (ear, leg and tail) and give results for each body position of the sensor. In summary, it can be concluded that the agreement between the PPG-based heart rate technique and the reference sensor is between 91% and 95%. In this paper, we showed the potential of using the PPG-based technology to assess the pig′s heart rate.

Suppression of superfluid stiffness near a Lifshitz-point instability to finite-momentum superconductivity

We derive the effective Ginzburg-Landau theory for finite momentum (FFLO/PDW) superconductivity without spin population imbalance from a model with local attraction and repulsive pair-hopping. We find that the GL free energy must include up to sixth order derivatives of the order parameter, providing a unified description of the interdependency of zero and finite momentum superconductivity. For weak pair-hopping the phase diagram contains a line of Lifshitz points where vanishing superfluid stiffness induces a continuous change to a long wavelength Fulde-Ferrell (FF) state. For larger pair-hopping there is a bicritical region where the pair-momentum changes discontinuously. Here the FF type state is near degenerate with the Larkin-Ovchinnikov (LO) or Pair-Density-wave (PDW) type state. At the intersection of these two regimes there is a "Super-Lifshitz" point with extra soft fluctuations. The instability to finite momentum superconductivity occurs for arbitrarily weak pair-hopping for sufficiently large attraction suggesting that even a small repulsive pair-hopping may be significant in a microscopic model of strongly correlated superconductivity. Several generic features of the model may have bearing on the cuprate superconductors, including the suppression of superfluid stiffness in proximity to a Lifshitz point as well as the existence of subleading FFLO order (or vice versa) in the bicritical regime.

Flexure Analysis of Laminated Composite and Sandwich Beams Using Timoshenko Beam Theory

The static behaviour of laminated composite and sandwich beams subjected to various sets of boundary conditions is investigated by using the Timoshenko beam theory and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. In order to solve the problem, a SSPH code which consists of up to sixth order derivative terms in Taylor series expansion is developed. The validation and convergence studies are performed by solving symmetric and anti-symmetric cross-ply composite beam problems with various boundary conditions and aspect ratios. The results in terms of mid-span deflections, axial and shear stresses are compared with those from previous studies to validate the accuracy of the present method. The effects of fiber angle, lay-up and aspect ratio on mid-span displacements and stresses are studied. At the same time, the problems not only for the convergence analysis but also for the extensive analysis are also solved by using the Euler-Bernoulli beam theory for comparison purposes.

Generalized Skyrme model with the loosely bound potential

We study a generalization of the loosely bound Skyrme model which consists of the Skyrme model with a sixth-order derivative term - motivated by its fluid-like properties - and the second-order loosely bound potential - motivated by lowering the classical binding energies of higher-charged Skyrmions. We use the rational map approximation for the Skyrmion of topological charge B=4, calculate the binding energy of the latter and estimate the systematic error in using this approximation. In the parameter space that we can explore within the rational map approximation, we find classical binding energies as low as 1.8% and once taking into account the contribution from spin-isospin quantization we obtain binding energies as low as 5.3%. We also calculate the contribution from the sixth-order derivative term to the electric charge density and axial coupling.

Gummel Symmetry Test on charge based drain current expression using modified first-order hyperbolic velocity-field expression

Gummel Symmetry Test (GST) has been a benchmark industry standard for MOSFET models and is considered as one of important tests by the modeling community. BSIM4 MOSFET model fails to pass GST as the drain current equation is not symmetrical because drain and source potentials are not referenced to bulk. BSIM6 MOSFET model overcomes this limitation by taking all terminal biases with reference to bulk and using proper velocity saturation (v-E) model. The drain current equation in BSIM6 is charge based and continuous in all regions of operation. It, however, adopts a complicated method to compute source and drain charges. In this work we propose to use conventional charge based method formulated by Enz for obtaining simpler analytical drain current expression that passes GST. For this purpose we adopt two steps: (i) In the first step we use a modified first-order hyperbolic v-E model with adjustable coefficients which is integrable, simple and accurate, and (ii) In the second we use a multiplying factor in the modified first-order hyperbolic v-E expression to obtain correct monotonic asymptotic behavior around the origin of lateral electric field. This factor is of empirical form, which is a function of drain voltage (vd) and source voltage (vs). After considering both the above steps we obtain drain current expression whose accuracy is similar to that obtained from second-order hyperbolic v-E model. In modified first-order hyperbolic v-E expression if vd and vs is replaced by smoothing functions for the effective drain voltage (vdeff) and effective source voltage (vseff), it will as well take care of discontinuity between linear to saturation regions of operation. The condition of symmetry is shown to be satisfied by drain current and its higher order derivatives, as both of them are odd functions and their even order derivatives smoothly pass through the origin. In strong inversion region and technology node of 22nm the GST is shown to pass till sixth-order derivative and for weak inversion it is shown till fifth-order derivative. In the expression of drain current major short channel phenomena like vertical field mobility reduction, velocity saturation and velocity overshoot have been taken into consideration.

Black hole Skyrmion in a generalized Skyrme model

We study a Skyrme-like model with the Skyrme term and a sixth-order derivative term as higher-order terms, coupled to gravity and we construct Schwarzschild black hole Skyrme hair. We find, surprisingly, that the sixth-order derivative term alone cannot stabilize the black hole hair solutions; the Skyrme term with a large enough coefficient is a necessity.

Finite difference method for sixth-order derivatives of differential equations in buckling of nanoplates due to coupled surface energy and non-local elasticity theories

ABS TRACT: In this article, finite difference method (FDM) is used to solve sixth-order derivatives of differential equations in buckling analysis of nanoplates due to coupled surface energy and non-local elasticity theories. The uniform temperature change is used to study thermal effect. The small scale and surface energy effects are added into the governing equations using Eringen’s non-local elasticity and Gurtin-Murdoch’s theories, respectively. Two different boundary conditions including simply-supported and clamped boundary conditions are investigated. The numerical results are presented to demonstrate the difference between buckling obtained by considering the surface energy effects and that obtained without the consideration of surface properties. The results show that the finite difference method can be used as a powerful method to determine the mechanical behavior of nanoplates. In addition, this method can be used to solve higher-order derivatives of differential equations with different types of boundary condition with little computational effort. Moreover, it is observed that the effects of surface properties tend to increase in thinner and larger nanoplates; and vice versa.

Fractional Skyrmions and their molecules

We study a Skyrme-type model with a quadratic potential for a field with ${S}^{2}$ vacua. We consider two flavors of the model, the first is the Skyrme model and the second has a sixth-order derivative term instead of the Skyrme term; both with the added quadratic potential. The model contains molecules of half Skyrmions, each of them is a global (anti)monopole with baryon number $1/2$. We numerically construct solutions with baryon numbers one through six, and find stable solutions which look like beads on rings. We also construct a molecule with fractional Skyrmions having the baryon numbers $1/3+2/3$, by adding a linear potential term.

Baryonic torii: Toroidal baryons in a generalized Skyrme model

We study a Skyrme-type model with a potential term motivated by Bose-Einstein condensates (BECs), which we call the BEC Skyrme model. We consider two flavors of the model, the first is the Skyrme model and the second has a sixth-order derivative term instead of the Skyrme term; both with the added BEC-motivated potential. The model contains toroidally shaped Skyrmions and they are characterized by two integers P and Q, representing the winding numbers of two complex scalar fields along the toroidal and poloidal cycles of the torus, respectively. The baryon number is B=PQ. We find stable Skyrmion solutions for P=1,2,3,4,5 with Q=1, while for P=6 and Q=1 it is only metastable. We further find that configurations with higher Q>1 are all unstable and split into Q configurations with Q=1. Finally we discover a phase transition, possibly of first order, in the mass parameter of the potential under study.

Incarnations of Skyrmions

Skyrmions can be transformed into lumps or baby-Skyrmions by being trapped inside a domain wall. Here we find that they can also be transformed into sine-Gordon kinks when confined by vortices, resulting in confined Skyrmions. We show this both by an effective field theory approach and by direct numerical calculations. The existence of these trapped and confined Skyrmions does not rely on higher-derivative terms when the host solitons are flat or straight. We also construct a Skyrmion as a twisted vortex ring in a model with a sixth-order derivative term.

Baryonic sphere: A spherical domain wall carrying baryon number

We construct a spherical domain wall which has baryon charge distributed on a sphere of finite radius in a Skyrme model with a sixth-order derivative term and a modified mass term. Its distribution of energy density likewise takes the form of a sphere. In order to localize the domain wall at a finite radius we need a negative coefficient in front of the Skyrme term and a positive coefficient of the sixth order derivative term to stabilize the soliton. Increasing the pion mass pronounces the shell-like structure of the configuration.

Sixth order derivative free family of iterative methods

In this study, we develop a four-parameter family of sixth order convergent iterative methods for solving nonlinear scalar equations. Methods of the family require evaluation of four functions per iteration. These methods are totally free of derivatives. Convergence analysis shows that the family is sixth order convergent, which is also verified through the numerical work. Though the methods are independent of derivatives, computational results demonstrate that family of methods are efficient and demonstrate equal or better performance as compared with other six order methods, and the classical Newton method.

A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schr�dinger equation*1

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method . We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods–Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required. Title of program: ShdEq.nb Catalogue number: ADTT Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTT Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: none Computer for which the program is designed and others on which it has been tested: The program has been designed for the microcomputer and been tested on the microcomputer. Computers: IBM PC Operating systems under which the program has been tested: Windows XP Programming language used: Mathematica 4.2 Memory required to execute with typical data: 51 712 bytes No. of bytes in distributed program, including test data, etc.: 45 381 No. of lines in distributed program, including test data, etc.: 7311 Distribution format: tar gzip file CPC Program Library subprograms used: no Nature of physical problem: Numerical integration of one-dimensional or radial Schrödinger equation to find the eigenvalues for a bound states and phase shift for a continuum state. Method of solution: Using a two-step method twelfth-order method to integrate a Schrödinger equation numerically from both two ends and the connecting conditions at the matching point, an eigenvalue for a bound state or a resonant state with a given phase shift can be found. Restrictions on the complexity of the problem: The analytic form of the potential function and its high-order derivatives must be known. Typical running time: Less than one second. Unusual features of the program: Take advantage of the high-order derivatives of the potential function and efficient algorithm, the program can provide all the numerical solution of a given Schrödinger equation, either a bound or a resonant state, with a very high precision and within a very short CPU time. The program can apply to a very broad range of problems because the method has a very large interval of periodicity.

A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method. We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods–Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required. Program summary Title of program: ShdEq.nb Catalogue number: ADTT Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTT Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: none Computer for which the program is designed and others on which it has been tested: The program has been designed for the microcomputer and been tested on the microcomputer. Computers: IBM PC Operating systems under which the program has been tested: Windows XP Programming language used: Mathematica 4.2 Memory required to execute with typical data: 51 712 bytes No. of bytes in distributed program, including test data, etc.: 45 381 No. of lines in distributed program, including test data, etc.: 7311 Distribution format: tar gzip file CPC Program Library subprograms used: no Nature of physical problem: Numerical integration of one-dimensional or radial Schrödinger equation to find the eigenvalues for a bound states and phase shift for a continuum state. Method of solution: Using a two-step method twelfth-order method to integrate a Schrödinger equation numerically from both two ends and the connecting conditions at the matching point, an eigenvalue for a bound state or a resonant state with a given phase shift can be found. Restrictions on the complexity of the problem: The analytic form of the potential function and its high-order derivatives must be known. Typical running time: Less than one second. Unusual features of the program: Take advantage of the high-order derivatives of the potential function and efficient algorithm, the program can provide all the numerical solution of a given Schrödinger equation, either a bound or a resonant state, with a very high precision and within a very short CPU time. The program can apply to a very broad range of problems because the method has a very large interval of periodicity.

Exponential variational expansion in relative coordinates for highly accurate bound state calculations in four-body systems

Exponential variational expansions in relative coordinates are considered for four-body systems. All matrix elements needed for bound-state calculations are expressed as linear combinations of fifth- and sixth-order derivatives of a basic four-body integral. Computation of the basic four-body integral and its derivatives is performed directly, i.e., without any use of the branch tracking in the complex plane that is required in the Fromm/Hill approach, and by methods that take into account the termwise singularities of the formulas. The final computational procedure is relatively simple, physically transparent, and numerically stable. The methods are illustrated with sample data that show the importance of a singularity-canceling approach and that the increased precision thereby made possible permits more accurate wave function optimization than heretofore.

Sixth order derivative expansion of one-loop effective actions

A covariant derivative expansion is carried out up to sixth order for the functional determinant Tr log (iϖμ + Aμ(X)2 + U(X)) An efficient method proposed by Chan is used which makes no assumptions on the internal symmetry group of the external fields Aμ (χ) and U (χ). The relation between our formula and the Seeley-DeWitt coefficients is established.

Direct derivative spectrophotometric determination of nitrazepam and clonazepam in biological fluids

The use of fifth order derivative spectra allows the direct determination of nitrazepam in urine at 388 nm with a limit of detection of 1 microgram ml-1. The determination of nitrazepam in blood plasma can be carried out directly by measurement at 402 nm in the fourth order derivative spectra with a limit of detection of 1.5 micrograms ml-1. Clonazepam can be determined directly in urine samples at 384 nm by using sixth order derivative spectra with a limit of detection of 1 microgram ml-1 and in blood plasma by using fourth order derivative spectra at 384 nm with a limit of detection of 0.5 ppm.