A connection between fourth order surfaces and sixth order curves

1978 ◽  
pp. 123-140
Author(s):  
G. A. Utkin
Keyword(s):  
Fourth Order ◽  
Sixth Order

Relating renormalizability of D-dimensional higher-order electromagnetic and gravitational models to the classical potential at the origin

2017 ◽  
Vol 32 (08) ◽  
pp. 1750048 ◽  
Author(s):  
Antonio Accioly ◽  
Gilson Correia ◽  
Gustavo P. de Brito ◽  
José de Almeida ◽  
Wallace Herdy
Keyword(s):  
Fourth Order ◽  
Massive Gravity ◽  
Higher Order ◽  
Sixth Order ◽  
Gravity Models ◽  
Local Models ◽  
Order Model ◽  
Functional Generator ◽  
Classical Potential ◽  
Real Poles

Simple prescriptions for computing the D-dimensional classical potential related to electromagnetic and gravitational models, based on the functional generator, are built out. These recipes are employed afterward as a support for probing the premise that renormalizable higher-order systems have a finite classical potential at the origin. It is also shown that the opposite of the conjecture above is not true. In other words, if a higher-order model is renormalizable, it is necessarily endowed with a finite classical potential at the origin, but the reverse of this statement is untrue. The systems used to check the conjecture were D-dimensional fourth-order Lee–Wick electrodynamics, and the D-dimensional fourth- and sixth-order gravity models. A special attention is devoted to New Massive Gravity (NMG) since it was the analysis of this model that inspired our surmise. In particular, we made use of our premise to resolve trivially the issue of the renormalizability of NMG, which was initially considered to be renormalizable, but it was shown some years later to be non-renormalizable. We remark that our analysis is restricted to local models in which the propagator has simple and real poles.

scholarly journals Diagonally Implicit Runge–Kutta Type Method for Directly Solving Special Fourth-Order Ordinary Differential Equations with Ill-Posed Problem of a Beam on Elastic Foundation

Algorithms ◽  
2018 ◽  
Vol 12 (1) ◽  
pp. 10 ◽  
Author(s):  
Nizam Ghawadri ◽  
Norazak Senu ◽  
Firas Adel Fawzi ◽  
Fudziah Ismail ◽  
Zarina Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Elastic Foundation ◽  
Fourth Order ◽  
Sixth Order ◽  
Test Problems ◽  
Type Method ◽  
Ill Posed ◽  
Runge Kutta ◽  
Fifth Order

In this study, fifth-order and sixth-order diagonally implicit Runge–Kutta type (DIRKT) techniques for solving fourth-order ordinary differential equations (ODEs) are derived which are denoted as DIRKT5 and DIRKT6, respectively. The first method has three and the another one has four identical nonzero diagonal elements. A set of test problems are applied to validate the methods and numerical results showed that the proposed methods are more efficient in terms of accuracy and number of function evaluations compared to the existing implicit Runge–Kutta (RK) methods.

Application of a Sixth Order Generalized Stress Function for Determining Limit Loads for Plates with Triangular Penetration Patterns

2002 ◽  
Author(s):  
J. L. Gordon ◽  
D. P. Jones
Keyword(s):  
Finite Element ◽  
Fourth Order ◽  
Limit Load ◽  
Sixth Order ◽  
Numerical Representation ◽  
Order Function ◽  
Practical Applications ◽  
Perfectly Plastic ◽  
Unit Cells ◽  
Elastic Perfectly Plastic

The capability to obtain limit load solutions of plates with triangular penetration patterns using fourth order functions to represent the collapse surface has been presented in previous papers. These papers describe how equivalent solid plate elastic-perfectly plastic finite element capabilities are generated and demonstrate how such capabilities can be used to great advantage in the analysis of tubesheets in large heat exchanger applications. However, these papers have pointed out that although the fourth order functions can produce sufficient accuracy for many practical applications, there are situations where improvements in the accuracy of in-plane and transverse shear are desirable. This paper investigates the use of a sixth order function to represent the collapse surface for improved accuracy of the in-plane response. Explicit elastic-perfectly plastic finite element solutions are obtained for unit cells representing an infinite array of circular penetrations arranged in an equilateral triangular array. These cells are used to create a numerical representation of the complete collapse surfaces for a number of ligament efficiencies (h/P where h is the minimum ligament width and P is the distance between hole centers). Each collapse surface is then fit to a sixth order function that satisfies the periodicity of the hole pattern. Sixth-order collapse functions were developed for h/P values between .05 and .50. Accuracy of the sixth order and the fourth order functions are compared. It was found that the sixth order function is indeed more accurate, reducing the error from 12.2% for the fourth order function to less than 3% for the sixth order function.

scholarly journals Pinning Effects of Exchange and Magnetocrystalline Anisotropies on Skyrmion Lattice

2021 ◽  
Vol 9 ◽  
Author(s):  
Xuejin Wan ◽  
Yangfan Hu ◽  
Zhipeng Hou ◽  
Biao Wang
Keyword(s):  
Fourth Order ◽  
Magnetocrystalline Anisotropy ◽  
Sixth Order ◽  
Combined Effects ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Exchange Anisotropy ◽  
Hexagonal Symmetry ◽  
Intrinsic Anisotropy ◽  
Crystallographic Axes

Reorientation of skyrmion crystal (SkX) with respect to crystallographic axes is believed to be insensitive to anisotropies of fourth order in spin-orbit coupling, for which sixth order terms are considered for explanation. Here, we show that this is wrong due to an oversimplified assumption that SkX possesses hexagonal symmetry. When the deformation of SkX is taken into account, fourth order anisotropies such as exchange anisotropy and magnetocrystalline anisotropy have pinning (in this work, the word ‘pinning’ refers to the reorientation effects of intrinsic anisotropy terms) effects on SkX. In particular, we reproduce some experiments of MnSi and Fe1−xCoxSi by considering the effect of fourth order magnetocrystalline anisotropy alone. We reproduce the 30∘ rotation of SkX in Cu2OSeO3 by considering the combined effects of the exchange and magnetocrystalline anisotropies. And we use the exchange anisotropy to explain the reorientation of SkX in VOSe2O5.

scholarly journals A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

2021 ◽  
Vol 3 (1) ◽  
pp. 8-21 ◽  
Author(s):  
Mst. Sharmin Banu ◽  
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Exact Result ◽  
Sixth Order ◽  
Initial Value ◽  
Step Size ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fourth Order Method

In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

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