FACIES PATTERNS THAT DEFINE ORBITALLY FORCED THIRD-, FOURTH-, AND FIFTH-ORDER SEQUENCES OF SIXTH-ORDER CYCLES AND THEIR RELATIONSHIP TO OSTRACOD FAUNICYCLES: THE PURBECKIAN (BERRIASIAN) OF DORSET, ENGLAND

2004 ◽  
pp. 245-260 ◽  
Author(s):  
EDWIN J. ANDERSON
Keyword(s):  
Sixth Order ◽  
Fifth Order

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.

scholarly journals XXIV.— On the development of the disturbing function, upon which depend the inequalities of the motions of the planets, caused by their mutual attrac­tion

1833 ◽  
Vol 123 ◽  
pp. 559-592
Keyword(s):  
Differential Equation ◽  
Present State ◽  
Sixth Order ◽  
Disturbing Function ◽  
Algebraic Expression ◽  
First Order ◽  
The Subject ◽  
The Mean ◽  
Fifth Order ◽  
Arithmetical Calculation

The perturbations of the planets caused by their mutual attraction depend chiefly upon one algebraic expression, from the development of which all the inequalities of their motions are derived. This function is very complicated, and requires much labour and many tedious operations to expand it in a series of parts which can be separately computed according to the occasions of the astronomer. The progress of physical astronomy has undoubtedly been re­tarded by the excessive length and irksomeness attending the arithmetical calculation of the inequalities. On this subject astronomers generally and continually complain; and that their complaints are well founded, is very aptly illustrated by a paper contained in the last year’s Transactions of this Society. The disturbing function is usually expanded in parts arranged according to the powers and products of the excentricities and the inclinations of the orbits to the ecliptic; and, as these elements are always small, the resulting series decreases in every case with great rapidity. No difficulty would therefore be found in this research, if an inequality depended solely on the quantity of the coefficient of its argument in the expanded function; because the terms of the series decrease so fast, that all of them, except those of the first order, or, at most, those of the first and second orders, might be safely neglected, as pro­ducing no sensible variation in the planet’s motion. But the magnitude of an inequality depends upon the length of its period, as well as upon the coefficient of its argument. When the former embraces a course of many years, the latter, although almost evanescent in the differential equation, acquires a great mul­tiplier in the process of integration, and thus comes to have a sensible effect on the place of the planet. Such is the origin of some of the most remarkable of the planetary irregularities, and in particular, of the great equations in the mean motions of Jupiter and Saturn, the discovery of which does so much honour to the sagacity of Laplace. It is not, therefore, enough to calculate the terms of the first order, or of the first and second orders, in the expansion of the disturbing function. This is already done in most of the books that treat of physical astronomy with all the care and fulness which the importance of the subject demands, leaving little room for further improvement. In the present state of the theory of the planetary motions, it is requisite that the astronomer have it in his power to compute any term in the expansion of the disturbing function below the sixth order; since it has been found that there are inequalities depending upon terms of the fifth order, which have a sensible effect on the motions of some of the planets.

scholarly journals Multi-level stratigraphic heterogeneities in a Triassic shoal grainstone, Oman Mountains, Sultanate of Oman: Layer-cake or shingles?

GeoArabia ◽  
2015 ◽  
Vol 20 (2) ◽  
pp. 115-142
Author(s):  
Michael Obermaier ◽  
Nicklas Ritzmann ◽  
Thomas Aigner
Keyword(s):  
Fluid Flow ◽  
Fourth Order ◽  
Carbonate Ramp ◽  
Sixth Order ◽  
Small Scale ◽  
Lateral Migration ◽  
Architectural Elements ◽  
Layer Cake ◽  
Multi Level ◽  
Fifth Order

ABSTRACT A fundamental question in the correlation of 1-D sedimentologic data is whether to use a layer-cake or shingled correlation approach. The resulting reservoir geometry has important implications for the characterization of reservoir heterogeneities and fluid flow. On the Saiq Plateau in Oman, epeiric carbonate ramp deposits of the Triassic Sudair Formation are well exposed and can be investigated in detail over several kilometers. There, reservoir heterogeneities on different scales have been documented by creating various outcrop wall panels and 2-D correlations. Multi-level architectural elements with different depositional geometries were discovered, which were linked to a sequence-stratigraphic hierarchy consisting of three levels. Level 1: A “layer-cake”-type stratigraphic architecture with minor thickness variations over several kilometers becomes apparent when correlating fourth-order cycle set boundaries. Level 2: The correlation of fifth-order cycle boundaries reflects horizontally continuous geometries, within which, however, internal grainstone layers were discovered to be arranged in a shingled fashion. Muddy layers in between these shingles illustrate sixth-order mini-cycle boundaries. Level 3: Within sixth-order mini-cycles another scale of a shingle-like architecture can be observed. Amalgamated cm-thick grainstone units form thin wedges with subtle but clearly inclined dipping geometry. Fourth-order cycle sets and fifth-order cycles can be traced over several kilometers, and therefore assumed to be related to allocyclic stratigraphic processes. The internal shingle geometries within fifth-order cycles are traceable over 100s of meters and presumably reflect an autocyclic lateral migration of a shoal complex. Cm-thick shingling grainstone wedges within sixth-order mini-cycles are interpreted as storm-related spill deposits. Their event-driven character is reflected by frequent amalgamation and reworking of the preceding deposits. The results of this study of epeiric carbonate ramp deposits suggest that a “layer-cake” correlation approach is appropriate when correlating 10s of m-thick grainstone units over a distance of several kilometers. However in the documented example, these thick grainstone units consist internally of small-scale architectural elements, which show inclined geometries and require a shingled correlation approach. These small-scale heterogeneities within an overall “layer-cake” architecture might have an impact on fluid flow in similar subsurface reservoirs and should be taken into account for detailed reservoir correlations and static reservoir models.

II. On a new auxiliary equation in the theory of equations of the fifth order

1862 ◽  
Vol 11 ◽  
pp. 134-137
Keyword(s):  
General Theory ◽  
Single Equation ◽  
The Other ◽  
Sixth Order ◽  
Auxiliary Equation ◽  
Resolvent Equation ◽  
Root Of Unity ◽  
Quintic Equation ◽  
Solution Of Equations ◽  
Fifth Order

Considering the equation of the fifth order, or quintic equation, (*) ( v , 1) 5 = ( v — x 1 )( v — x 2 ) ( v — x 3 ) ( v — x 4 ) ( v — x 5 ) = 0, and putting as usual fω = x 1 + ωx 2 + ω 2 x 3 + ω 3 x 4 + ω 4 x 5 , where ω is an imaginary fifth root of unity, then, according to Lagrange’s general theory for the solution of equations, fω is the root of an equation of the order 24, called the Resolvent Equation, but the solution whereof depends ultimately on an equation of the sixth order, viz. ( fω ) 5 , ( fω 2 ) 5 , ( fω 3 ) 5 , ( fω 4 ) 5 are the roots of an equation of the fourth order, each coefficient whereof is determined by an equation of the sixth order; and moreover the other coefficients can be all of them rationally expressed in terms of any one coefficient assumed to be known; the solution thus depends on a single equation of the sixth order.

On the developement of the disturbing function upon which depend the inequalities of the motions of the planets, caused by their mutual attraction

1837 ◽  
Vol 3 ◽  
pp. 209-210
Keyword(s):  
Algebraic Function ◽  
Sixth Order ◽  
Disturbing Function ◽  
Mutual Attraction ◽  
The Mean ◽  
Fifth Order ◽  
Moderate Length

The progress of physical astronomy has been retarded by the excessive labour requisite for the arithmetical computation of the inequalities in the motions of the planets, arising from the perturbations produced by their mutual attractions. If an inequality depended solely on the quantity of the coefficient of its argument in the expanded algebraic function, the difficulty of computation would not be great, since, from the smallness of the elements on which it depends, namely, the eccentricities and the inclinations of the orbits to the ecliptic, the resulting series decreases, in every case, with great rapidity: but as its magnitude depends also upon the length of its period, the coefficient of its argument will, when this period embraces many years, acquire, in the process of integration, a high multiplier, and comes thus to have a sensible effect on the place of the planet. Such is the origin of some of the most remarkable of the planetary inequalities, and, in particular, of the great equations in the mean motions of Jupiter and Saturn. It is necessary, therefore, that the astronomer be furnished with the means of computing any term in the expansion of the disturbing function below the sixth order; since it has been found that there are inequalities depending upon terms of the fifth order, which have a sensible effect on the motions of some of the planets. The object of the author in the present paper is to give the function such a form that the astronomer may have it in his power to select any inequality he may wish to examine, and to compute the coefficient of its argument by an arithmetical process of moderate length. The investigation comprehends every argument not passing the fifth order; but as the formulae are regular, the method may be extended indefinitely to any order.

XIII. On a new auxiliary equation in the theory of equations of the fifth order

1861 ◽  
Vol 151 ◽  
pp. 263-276 ◽  
Keyword(s):  
Single Equation ◽  
The Other ◽  
Sixth Order ◽  
Auxiliary Equation ◽  
Resolvent Equation ◽  
Root Of Unity ◽  
Quintic Equation ◽  
Solution Of Equations ◽  
Fifth Order ◽  
The Given

Considering the equation of the fifth order, or quintic equation, (*)( v , 1) 5 = ( v — x 1 ) ( v — x 2 ) ( v — x 3 ) ( v — x 4 ) ( v — x 5 ) = 0, and putting as usual fω = x 1 + ωx 2 + ω 2 x 3 + ω 3 x 4 + ω 4 x 5 , where ω is an imaginary fifth root of unity, then, according to Lagrange’s general theory for the solution of equations, fω is the root of an equation of the order 24, called the Resolvent Equation, but the solution whereof depends ultimately on an equation of the sixth order, viz. ( fω ) 5 , ( fω 2 ) 5 , ( fω 3 ) 5 , ( fω 4 ) 5 are the roots of an equation of the fourth order, each coefficient whereof is determined by an equation of the sixth order; and moreover the other coefficients can be all of them rationally expressed in terms of any one coefficient assumed to be known; the solution thus depends on a single equation of the sixth order. In particular the last coefficient, or ( fω . fω 2 . fω 3 . fω 4 ) 5 , is determined by an equation of the sixth order; and not only so, but its fifth root, or fω . fω 2 . fω 3 . fω 4 , (which is a rational function of the roots, and is the function called by Mr. Cockle the Resolvent Product), is also determined by an equation of the sixth order: this equation may be called the Resolvent-Product Equation. But the recent researches of Mr. Cockle and Mr. Harley show that the solution of an equation of the fifth order may be made to depend on an equation of the sixth order, originating indeed in, and closely connected with, the resolvent-product equation, but of a far more simple form; this is the auxiliary equation referred to in the title of the present memoir. The connexion of the two equations, and the considerations which led to the new one, will be pointed out in the sequel; but I will here state synthetically the construction of the auxiliary equation. Representing for shortness the roots ( x 1 , x 2 , x 3 , x 4 , x 5 ) of the given quintic equation by 1, 2, 3, 4, 5, and putting moreover 12345 = 12+23+34+45+51, &c.

scholarly journals Trajectory Planning and Optimization for a Par4 Parallel Robot Based on Energy Consumption

2019 ◽  
Vol 9 (13) ◽  
pp. 2770
Author(s):  
Xiaoqing Zhang ◽  
Zhengfeng Ming
Keyword(s):  
Energy Consumption ◽  
Trajectory Planning ◽  
Design Method ◽  
Mechanical Energy ◽  
Parallel Robot ◽  
Sixth Order ◽  
Function Parameter ◽  
Pick And Place ◽  
Order Polynomial ◽  
Fifth Order

A study on trajectory planning and optimization for a Par4 parallel robot was carried out, based on energy consumption in high-speed picking and placing. In the end-effector operating space of the Par4 parallel robot, the rectangular transition of the pick-and-place trajectory was rounded by a Lamé curve. A piecewise design method was adopted to accomplish trajectory shape planning for displacement, velocity and acceleration. To make the Par4 robot’s end run more smoothly and to reduce residual vibration, asymmetric fifth-order and sixth-order polynomial motion laws were employed. With the aim of reaching the minimum mechanical energy consumption for the Par4 parallel robot, the recently proposed Grey Wolf Optimizer (GWO) algorithm was adopted to optimize the planning trajectory. The validity of the design method was verified by experiments, and it was found that the minimum mechanical energy consumption of the optimal trajectory planned under the law of fifth-order polynomial motion is lower than that of sixth-order polynomial motion. In addition, the experiments also revealed the optimal values of Parameters e and f, which were the parameters of the Lamé curve function. Parameter e can be calculated as half the pick-up span for the minimum mechanical energy consumption, unlike parameter f, whose optimal value depends on specific circumstances such as the pick-and-place coordinates and the pick-up height.

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