On a nonlinear third-order equation

2017 ◽  
Vol 102 (1-2) ◽  
pp. 3-11 ◽  
Author(s):  
A. I. Aristov
Keyword(s):  
Order Equation ◽  
Third Order

A Geometrical Solution for the Inverse Kinematics of Three Revolute Manipulators

1993 ◽  
Author(s):  
Eugene F. Fichter
Keyword(s):  
Fourth Order ◽  
Inverse Kinematics ◽  
Solution Procedure ◽  
Order Equation ◽  
Algebraic Solution ◽  
Computer Implementation ◽  
Third Order ◽  
Fourth Order Equation ◽  
Inverse Kinematics Problem

Abstract Points of intersection of a circle and a torus are used to find a solution to the inverse kinematics problem for a three revolute manipulator. Both geometrical and algebraic solution procedures are discussed. The algebraic procedure begins with a third order equation instead of the usual fourth order equation. Since the procedure is basically geometrical it lends itself to a computer implementation which graphically displays each steps in the solution procedure. The potential of this approach for both design and pedagogy is discussed.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

Strongly Oscillatory and Nonoscillatory Subspaces of Linear Equations

1975 ◽  
Vol 27 (1) ◽  
pp. 106-110 ◽  
Author(s):  
J. Michael Dolan ◽  
Gene A. Klaasen
Keyword(s):  
Linear Equation ◽  
Linear Space ◽  
Nontrivial Solution ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Oscillatory Solutions ◽  
The Third ◽  
All Solutions

Consider the nth order linear equationand particularly the third order equationA nontrivial solution of (1)n is said to be oscillatory or nonoscillatory depending on whether it has infinitely many or finitely many zeros on [a, ∞). Let denote respectively the set of all solutions, oscillatory solutions, nonoscillatory solutions of (1)n. is an n-dimensional linear space. A subspace is said to be nonoscillatory or strongly oscillatory respectively if every nontrivial solution of is nonoscillatory or oscillatory. If contains both oscillatory and nonoscillatory solutions then is said to be weakly oscillatory.

The Use of Quartz as an Internal Pressure Standard in High-Pressure Crystallography

1997 ◽  
Vol 30 (4) ◽  
pp. 461-466 ◽  
Author(s):  
R. J. Angel ◽  
D. R. Allan ◽  
R. Miletich ◽  
L. W. Finger
Keyword(s):  
High Pressure ◽  
Equation Of State ◽  
Order Equation ◽  
Unit Cell ◽  
Unit Cell Parameters ◽  
Third Order ◽  
Single Crystal Diffraction ◽  
Cell Parameters ◽  
Diamond Anvil ◽  
Pressure Standard

The unit-cell parameters of quartz, SiO2, have been determined by single-crystal diffraction at 22 pressures to a maximum pressure of 8.9 GPa (at room temperature) with an average precision of 1 part in 9000. Pressure was determined by the measurement of the unit-cell volume of CaF2 fluorite included in the diamond-anvil pressure cell. The variation of quartz unit-cell parameters with pressure is described by: a −4.91300 (11) = −0.0468 (2) P + 0.00256 (7) P 2 − 0.000094 (6) P 3, c − 5.40482 (17) = − 0.03851 (2) P + 0.00305 (7) P 2 − 0.000121 (6) P 3, where P is in GPa and the cell parameters are in ångstroms. The volume–pressure data of quartz are described by a Birch–Murnaghan third-order equation of state with parameters V 0 = 112.981 (2) å3, K T0 = 37.12 (9) GPa and K′ = 5.99 (4). Refinement of K′′ in a fourth-order equation of state yielded a value not significantly different from the value implied by the third-order equation. The use of oriented quartz single crystals is proposed as an improved internal pressure standard for high-pressure single-crystal diffraction experiments in diamond-anvil cells. A measurement precision of 1 part in 10 000 in the volume of quartz leads to a precision in pressure measurement of 0.009 GPa at 9 GPa.

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