## Inverse and implicit functions in domain theory (2005)

Venue: | Proc. 20th IEEE Symposium on Logic in Computer Science (LICS 2005 |

Citations: | 5 - 5 self |

### BibTeX

@INPROCEEDINGS{Edalat05inverseand,

author = {Abbas Edalat and Dirk Pattinsondepartment Of Computing},

title = {Inverse and implicit functions in domain theory},

booktitle = {Proc. 20th IEEE Symposium on Logic in Computer Science (LICS 2005},

year = {2005},

pages = {417--426}

}

### OpenURL

### Abstract

C1 norm to the inverse func-tion. A similar result holds for implicit functions. Combined with the domain-theoretic model for computationalgeometry, this provides a robust technique for construction of curves and surfaces in geometric modelling and CAD. 1.

### Citations

1178 |
Optimization and Nonsmooth Analysis
- Clarke
- 1983
(Show Context)
Citation Context ...ded to multi-variable differential calculus, resulting in the no-tion of a domain-theoretic derivative, which for Lipschitz functions gives the smallest hyper-rectangle containing the Clark gradient =-=[3]-=-, as well as a domain for multi-variable differentiable functions. In this paper, we will illustrate the first main application of the domain of multi-variable differentiable functions, by using it to... |

400 | Introduction to Implicit Surfaces - Bloomenthal - 1997 |

42 |
Generative Modeling for Computer Graphics and CAD
- Snyder
- 1992
(Show Context)
Citation Context ..., where curves and surfaces are usually defined implicitly [2]. Currently, there are no robust methods to approximate an implicit surface and the most reliable technique provided by interval analysis =-=[12]-=- is only able to approximate the implicit surface without approximating its derivative. The paper thus presents a framework for a robust CAD system, where implicitly given surfaces can be effectively ... |

31 | Foundation of a computable solid modeling
- Edalat, Lieutier
- 1999
(Show Context)
Citation Context ...losed connected orientable manifold given by implicit equations such as f (x, y, z) = 0 when 0is a regular value of f . Furthermore, the domain-theoreticframework for geometric modelling developed in =-=[6]-=- combined with the results in this work lead to a domain of ori-entable closed Lipschitz manifolds. This will synthesize the domain-theoretic framework for geometry and that for dif-ferential calculus... |

20 | A domain theoretic account of Picard’s theorem
- Edalat, Pattinson
- 2004
(Show Context)
Citation Context ...developed which in particular pro-vides an effectively given domain for Lipschitz or differentiable functions. Later on, domain-theoretic techniquesfor solving initial value problems were obtained in =-=[5, 9]-=-, which enable us to approximate the unique solution of aninitial value problem given by a Lipschitz vector field up 1sto the precision required by the user. In [8], the domain-theoretic model was ext... |

20 |
Domain theory and differential calculus (functions of one variable
- Edalat, Lieutier
(Show Context)
Citation Context ... a recursion-theoretic account of the inverse function and the implicit function theorems, which are the main fundamental tools in multi-variable differential calculus and the theory of manifolds. In =-=[7]-=-, a domain-theoretic framework for differential calculus of one variable was developed which in particular provides an effectively given domain for Lipschitz or differentiable functions. Later on, dom... |

14 | An Empirical Analysis - F - 1996 |

13 | Domain-theoretic solution of differential equations (scalar fields
- Edalat, Krznarić, et al.
- 2003
(Show Context)
Citation Context ...developed which in particular pro-vides an effectively given domain for Lipschitz or differentiable functions. Later on, domain-theoretic techniquesfor solving initial value problems were obtained in =-=[5, 9]-=-, which enable us to approximate the unique solution of aninitial value problem given by a Lipschitz vector field up 1sto the precision required by the user. In [8], the domain-theoretic model was ext... |

10 | A computational model for multi-variable differential calculus
- Edalat, Lieutier, et al.
- 2005
(Show Context)
Citation Context ... value problems were obtained in [5, 9], which enable us to approximate the unique solution of aninitial value problem given by a Lipschitz vector field up 1sto the precision required by the user. In =-=[8]-=-, the domain-theoretic model was extended to multi-variable differential calculus, resulting in the notion of a domain-theoreticderivative, which for Lipschitz functions gives the smallest hyper-recta... |

4 |
The inverse function theorem for Lipschitz maps, www. math.sc.edu/∼howard
- Howard
(Show Context)
Citation Context ...orm k * k2, we write Br = {y 2R n | kyk2 < r} for the open ball around the origin with radius r and denote the closed ball by Br = {y 2 Rn |k yk2 <= r}. Theorem 1.1 Inverse Theorem for Lipschitz maps =-=[10]-=-Let Br be a closed ball containing the origin in Rn and let f : Br ! Rn with f (0) = 0, so that for some invertiblelinear map L : Rn ! Rn and some ae < 1 kL-1f (x2) - L-1f (x1) - (x2 - x1)k2 <= aekx2 ... |

4 | The constructive implicit function theorem and applications
- Bridges, Calude, et al.
- 1999
(Show Context)
Citation Context ...onstructive proof of the existence of the inverse function (or the implicit function) for a C1 function is obtained by approximation but no approximations to the derivative of the inverse in provided =-=[4]-=-. In none of these approaches, the inverse or the implicit function is obtained as a fixed point of a functional. 2. Preliminaries We briefly recall the essential notions of the domaintheoretic framew... |

1 |
Generative Modeling for ComputerGraphics and CAD
- Snyder
- 1992
(Show Context)
Citation Context ... where curves and surfaces are usually defined implicitly [2]. Currently, there are no robust meth-ods to approximate an implicit surface and the most reliable technique provided by interval analysis =-=[11]-=- is only able toapproximate the implicit surface without approximating its derivative. The paper thus presents a framework for a ro-bust CAD system, where implicitly given surfaces can be effectively ... |