scholarly journals An efficient method based on progressive interpolation for solving non-linear equations

2016 ◽  
Vol 61 ◽  
pp. 67-72 ◽  
Author(s):  
Xiao-Diao Chen ◽  
Yubao Zhang ◽  
Jiaer Shi ◽  
Yigang Wang
Keyword(s):  
Efficient Method ◽  
Linear Equations ◽  
Non Linear ◽  
Non Linear Equations ◽  
Progressive Interpolation

An efficient method for determining the critical modification coefficient for cylindrical worm gearing in which the worm is generated by a curve

2001 ◽  
Vol 215 (12) ◽  
pp. 1745-1755 ◽  
Author(s):  
C-K Lee ◽  
C-K Chen
Keyword(s):  
Efficient Method ◽  
Linear Equations ◽  
Contact Ratio ◽  
Non Linear ◽  
Worm Gearing ◽  
System Equations ◽  
Curvature Information ◽  
Maximum Contact ◽  
Non Linear Equations

This paper proposes an efficient method for the determination of the critical modification coefficient (CMC) for the cylindrical worm gearing in which the worm is generated by a curve. The term CMC is defined as the modification coefficient that causes the first boundary points to be just on the boundary of the area of meshing. Solved from a system of five non-linear equations, the CMC not only avoids gear undercutting, but also produces the maximum contact ratio. In developing the system equations, the equation of non-undercutting is derived directly from the curvature information of the generating curve and the parameters of relative motion, without the need to derive any partial derivatives to the equation of the generating surface. By using an equivalent generation mechanism rather than the real one, an explicit form of equation of meshing is created. The equation of meshing created can simplify the form of the equation of non-undercutting, reduce the number of non-linear equations and unknowns in the system equations, increase the speed of convergence and decrease numerical instability. Based on the proposed method, a computer program is created and applied to analyse the CMCs of four commonly used worm gearings: ZA, ZN, ZE and ZC3.

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

An elliptic boundary value problem for strongly non-linear equations in unbounded domains

1977 ◽  
Vol 77 (3-4) ◽  
pp. 217-230 ◽  
Author(s):  
Vesa Mustonen
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Equations ◽  
Unbounded Domains ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Non Linear ◽  
Linear Elliptic Boundary ◽  
Non Linear Equations

SynopsisThe existence of a variational solution is shown for the strongly non-linear elliptic boundary value problem in unbounded domains. The proof is a generalisation to Orlicz-Sobolev space setting of the idea introduced in [15] for the equations involving polynomial non-linearities only.

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